The Coleman correspondence at the free fermion point

We prove that the truncated correlation functions of the charge and gradient fields associated with the massless sine-Gordon model on $\mathbb{R}^2$ with $\beta=4\pi$ exist for all coupling constants and are equal to those of the chiral densities and vector current of free massive Dirac fermions. This is an instance of Coleman's prediction that the massless sine-Gordon model and the massive Thirring model are equivalent (in the above sense of correlation functions). Our main novelty is that we prove this correspondence starting from the Euclidean path integral in the non-perturative regime of the infinite volume models. We use this correspondence to show that the correlation functions of the massless sine-Gordon model with $\beta=4\pi$ decay exponentially and that the corresponding probabilistic field is localized.


Introduction and main results
Statistical and quantum field theory in two (Euclidean) dimensions is rich and special in various ways. This manifests itself, for example, through the existence of the powerful theory of conformal field theory (CFT), the possibility of quasiparticles which are neither bosons nor fermions but instead have anyonic particle statistics, or the perhaps surprising possibility of equivalence of fermionic and bosonic field theories-known as bosonization. The two-dimensional setting also provides one of the main test cases for the understanding of strongly interacting field theories. The massless sine-Gordon model is a principal example of a non-conformal perturbation of a CFT in two dimensions. Despite the absence of conformal symmetry, there is a detailed but almost entirely conjectural description of many of its physical features, not accessible by perturbation theory, including the prediction of a mass gap for all coupling constants. These features pertain to the infinite volume theory. In this paper, we study the arguably most fundamental (and simplest) instance of this-the Coleman correspondence at the free fermion point, which we prove in the non-perturbative regime of the infinite volume models and for all coupling constants.
1.1. Coleman correspondence. The Coleman correspondence is a prototype for bosonization [12]. It states that the massless sine-Gordon model with parameters (β, z) and the massive Thirring model with parameters (g, µ) are equivalent in the sense of correlation functions when (β, z) and (g, µ) are appropriately related. This is an instance of bosonization because the sine-Gordon model is a bosonic quantum field theory while the massive Thirring model is a fermionic quantum field theory. The equivalence is especially striking when β = 4π, which corresponds to parameters of the massive Thirring model for which it becomes non-interacting (free massive Dirac fermions), while the sine-Gordon model is interacting (non-Gaussian).
Previous mathematical results have established variants of the Coleman correspondence in the perturbative regime, i.e., for small coupling constants and with finite volume interaction term (or with external mass term), see [5,15,26,46] and Section 1.3. In this article, we prove that, for β = 4π, the Coleman correspondence holds in the non-perturbative regime of the infinite volume models (without external mass term). Unlike the previous perturbative results, our proof has thus strong implications for the massless sine-Gordon model with β = 4π: we show exponential decay of correlations for all z = 0 and that the field is probabilistically localized-results that are non-perturbative in the coupling constant (and false for z = 0).
Before stating our results, we first introduce the sine-Gordon model and free Dirac fermions (both in their Euclidean versions). The massless sine-Gordon model with coupling constants β ∈ (0, 8π) and z ∈ R is defined in terms of the limit ε → 0, m → 0, L → ∞ of the probability measures ν SG(β,z|ε,m,L) (dϕ) ∝ exp 2z Λ L ε −β/4π cos( βϕ(x)) dx ν GFF(ε,m) (dϕ), (1.1) where Λ L = {x ∈ R 2 : |x| L} is the Euclidean disk of radius L, and ν GFF(ε,m) is the Gaussian free field (GFF) on R 2 with mass m > 0 regularised at scale ε > 0. Here the precise choice of the regularisation of the GFF is not important, but to be concrete, we choose ν GFF(ε,m) as the Gaussian measure supported on C ∞ (R 2 ) with covariance kernel , (1.5) has a non-trivial limit for test functions f 1 and f 2 with disjoint supports. In general, the truncated correlation function of observables O 1 , . . . , O n can be defined inductively by where the sum is over proper partitions P = (P j ) ∈ P n of [n] = {1, . . . , n}. Note that O 1 · · · O n T does not only depend on the product of the O i , and a more precise notation would be O 1 ; . . . ; O n T . However, the formal product notation without semicolons is standard and more convenient for our purposes. Equivalent to (1.6), the truncated correlations are Taylor coefficients of the logarithm of the joint moment generating function of O 1 , . . . , O n if it exists, see Appendix A.
Free fermions are defined in terms of their correlation kernel. The correlation kernel of free Dirac fermions of mass µ ∈ R is the fundamental solution of the massive Dirac operator on R 2 for which we use the representation where ∂ = 1 2 (−i∂ 0 + ∂ 1 ) and∂ = 1 2 (i∂ 0 + ∂ 1 ) and we identify x = (x 0 , x 1 ) ∈ R 2 with ix 0 + x 1 ∈ C. In terms of the modified Bessel function K 0 , this fundamental solution is explicitly given by where ∼ holds as µ → 0; see Section 6. For distinct points x 1 , . . . , x n , y 1 , . . . , y n ∈ R 2 , and any α 1 , . . . , α n , β 1 , . . . , β n ∈ {1, 2}, we then denote the correlation functions of free Dirac fermions by (1.9) The right-hand side is regarded as the definition of the left-hand side. In Appendix A, some standard operational tools for free fermions that we will use later are collected. Because S is singular, the correlations ofψ α i (x i )ψ β i (x i ) are not defined, in general, but for distinct points x 1 , . . . , x n with n > 1, the truncated correlations ofψ α i ψ β i (x i ) formally make sense and are given by where the sum is over cyclic permutations π on [n] = {1, . . . , n}. For our purposes, we again regard the right-hand side of (1.10) as the definition of the left-hand side of (1.10). (As explained in Appendix A, we note that if S was not singular, then (1.10) would be an identity that follows from (1.6) and (1.9) without restriction to distinct points. Alternatively one could thus define the truncated correlations as limits of regularized correlations and arrive at the same result as our definition.) Finally, for any f 1 , . . . , f n : R 2 → R such that the following integrand is (absolutely) integrable, we will write n i=1ψ α i ψ β i (f i ) T

FF(µ)
= dx 1 · · · dx n f 1 (x 1 ) · · · f n (x n ) . (1.11) Since S(x, y) has singularity O(1/|x − y|), for n 3, the above integrand is integrable for all bounded f i with compact support. For n = 2, this is true for f 1 and f 2 with disjoint compact support.
We emphasize that the right-hand side is the explicit polynomial in S αβ (x, y) given by (1.10) which is integrated over the test functions as in (1.11). To lighten notation, we will write the limit on left-hand side of (1.12) as n k=1 :e +i √ 4πϕ : .
In particular, since the modified Bessel function K 0 and its derivative decay exponentially, the massless sine-Gordon correlation functions decay exponentially whenever z = 0, when β = 4π. It is conjectured (but in general open) that the massless sine-Gordon model has exponential decay of correlations for all β ∈ (0, 8π) and z ∈ R \ {0}, with an explicit conjectured relation between the rate of exponential decay (mass) and the parameters of the sine-Gordon model [51]. For further discussion of this problem, see also the last paragraph of [5, p.717] and Section 1.3 below.
The exponential decay is, of course, in contrast to the well-known situation for the GFF (i.e., the case z = 0). It is an elementary computation that GFF correlations decay polynomially: and that the one-point functions exist and vanish; see, for example, the computations in Section 2.2. The free field correlations · GFF are defined as in (1.13) with z = 0. While the above results are for the charge and gradient correlation functions, as a consequence we can also construct the (probabilistic) massless sine-Gordon field itself when β = 4π and z = 0. Note that the assumption z = 0 is essential as the massless GFF on R 2 only exists up to an additive constant -not in the sense of the following theorem. Theorem 1.3. Let β = 4π and z ∈ R, z = 0. Then there exists a probability measure on S ′ (R 2 ) (not restricted to test functions with mean 0) whose expectation we denote by · SG(4π,z) with the following properties. For any f, g ∈ C ∞ c (R 2 ) with dx f = 0 = dx g, For f, g ∈ C ∞ c (R 2 ), one has ϕ(f ) SG(4π,z) = 0 and the two-point function is given by ϕ(f )ϕ(g) SG(4π,z) = R 2 dp (2π) 2f (p)ĝ(−p)Ĉ Az (p), (1.28) wheref (p) = R 2 dx f (x)e −ip·x is the Fourier transform of f , 29) and arsinh(x) = log(x + √ x 2 + 1) is the inverse hyperbolic sine. In particular, the above massless sine-Gordon field on R 2 is localized and has exponential decay of correlation: for any f ∈ C ∞ c (R 2 ), sup x,y∈R 2 (ϕ(f x ) − ϕ(f y )) 2 SG(4π,z) < ∞, (1.30) and ϕ(f x )ϕ(f y ) SG(4π,z) decays exponentially as |x − y| → ∞, (1.31) where f x (y) = f (y − x) denotes the translation of f to x ∈ R 2 .
Finally, we comment on the exclusion of the one-point functions in Theorem 1.1. While the charge one-point functions vanish in the massless free field case, the following proposition shows that they typically diverge when z = 0. The above divergence of the charge one-point functions is shown in Theorem 3.1 item (iv), in fact more generally for all β ∈ [4π, 6π). As a consequence of this and of the existence of the truncated charge correlation functions, none of the untruncated charge correlation functions involving a test function with R 2 f dx = 0 converge as ε → 0. On the other hand, since the gradient one-point functions exist, the existence of the truncated gradient correlation functions also implies that of the untruncated gradient correlation functions , (1.34) with explicit expressions given by inverting (1.6). Before discussing consequences of Theorems 1.1-1.2 and our more general analysis in their proofs, we remark on the physical interpretation of the fermionic side of the Coleman correspondence.
Remark 1.5. The Coleman correspondence can be written in terms of Dirac matrices γ µ satisfying γ µ γ ν + γ ν γ µ = −2δ µν 1. In the representation we have chosen, these are (1. 35) Thus γ 0 = iσ 2 , γ 1 = −iσ 1 , γ 5 = σ 3 , where the σ i are the Pauli matrices. In terms of these, the Dirac operator can be written as The Coleman correspondence can then be regarded as the following equivalence of the fields: The right-hand sides of (1.37)-(1.38) have the interpretation of being the chiral densities associated with the spinor field ψ, and the right-hand sides of (1.39)-(1.40) that of the vector current ψγ µ ψ (written in complex coordinates); see, for example, [23, Section 3].

Further results.
Our estimates for the sine-Gordon model together with the correlation inequalities from [25] also imply the following results for the infinite volume limit for β ∈ (0, 6π). The first theorem is for the infinite volume limit of the massless sine-Gordon field modulo constants (the 'gradient field'). Let S ′ (R 2 )/constants denote the topological dual of the (closed) subspace of integral-0 functions of the Schwartz space S(R 2 ). Theorem 1.6. Let β ∈ (0, 6π) and z ∈ R. Then for any f ∈ C ∞ c (R 2 ) with f dx = 0, the limit (1. 41) exists, and extends to the characteristic functional of a probability measure on S ′ (R 2 )/constants whose expectation we denote by · SG(β,z) . This measure is invariant under Euclidean transformations and satisfies For m > 0 fixed and z > 0, we similarly obtain the existence of the infinite volume of limit of the massive sine-Gordon field.
exists, extends to the characteristic functional of a probability measure on S ′ (R 2 ) whose expectation we denote by · SG(β,z|m) . This measure is invariant under Euclidean transformations and satisfies where the right-hand sides of the last two bounds are as in Theorem 1.6 and hold if dx f = 0.
For β = 4π, we can then deduce using the localization bound (1.30) that the m → 0 limit can be taken after the infinite volume limit, which means that the formal ϕ → ϕ + 2π √ β Z-symmetry of the massless sine-Gordon model is spontaneously broken in the infinite volume limit. Corollary 1.8. Let β = 4π and z > 0. Then for any f ∈ C ∞ c (R 2 ) (not assuming dx f = 0), whereĈ µ (p) is as in Theorem 1.2. Moreover, the limit · SG(4π,z|0 + ) := lim m→0 lim L→∞ · SG(4π,z|m,L) exists in the sense of characteristic functionals and defines a probability measure on S ′ (R 2 ) (not dividing out constants).
We expect that · SG(4π,z|0 + ) is the same as · SG(4π,z) but our arguments do not imply this.
1.3. Heuristics and previous results. The formal equivalence of the massless sine-Gordon model and the massive Thirring model was observed by Coleman in [12]. The massive Thirring model with parameters (g, µ) is formally given by a fermionic path integral with "density" Coleman observed that, order by order in a formal expansion, the massless sine-Gordon model with parameters (β, z) is related to the massive Thirring model if the parameters of the two models are related by Heuristically this prediction is not difficult to understand from the type of massless Gaussian free field and massless free fermion computations we derive in Section 2; these are versions of the identifications (1.37)- (1.40) in the elementary situation of the massless Gaussian free field and massless free fermions. Indeed, after rescaling ϕ by 4π/β, the measure of sine-Gordon model with parameters (β, z) has formal density Thus relative to the massless free fermion "measure" respectively the massless Gaussian free field measure, formally, the massive Thirring model and the sine-Gorden model are weighted by and one can see the order by order correspondence of the models with parameters (1.48), using the equivalence of the correlations of (1.37)-(1.40) with respect to the noninteracting measures.
To directly apply these identities, note that we changed the order of the Grassmann variables in (1.50) compared to (1.47) explaining the change of the sign of the quadratic term. Mathematically, this formal argument is however far from a proof. To start with, the probabilistic or analytic existence of the massless sine-Gordon model and the massive Thirring model is a nontrivial problem. Even in finite volume, the ultraviolet (short-distance) stability of both models is delicate when β 4π (for β < 4π the situation is simpler). For example, various constructions of the sine-Gordon model exist under different assumptions (see in particular [3,6,9,16,17,20,26,36,38,43]), but none of these covers all β ∈ (0, 8π) and all z ∈ R, even for interactions in a fixed finite volume. The analysis of the infinite volume limit is a different problem. For the massive Thirring model, we refer in particular to [4] and references therein.
The previous results on variants of the Coleman correspondence are the following. For β < 4π, a variant of the Coleman correspondence between the massive sine-Gordon model (i.e., m > 0 fixed) and the massive Thirring-Schwinger model (QED 2 ) was proved in [26] for z/m 2 is sufficiently small; see also [22] for a review. Also for β < 4π, but now with finite volume interaction instead of with an external mass term, a version of the Coleman correspondence was shown in [46]. For β = 4π and again with finite volume interaction, a version of the Coleman correspondence between the sine-Gordon model with small coupling constant z (depending on the volume) was proved in [15]. Finally, for β in a neighbourhood of 4π, but again in finite volume and all coupling constants sufficiently small (again depending on the volume), the Coleman correspondence was proved in [5], and accompanied by a construction of the corresponding massive Thirring model [4].
The integrability of two-dimensional conformal field theories is celebrated and well known. That non-conformal perturbations of conformal field theories are in some cases expected to remain integrable is perhaps more surprising. The sine-Gordon and massive Thirring models are such examples, and our result confirms the most fundamental (and simplest) instance of this. In the physics literature, many other exact results have been predicted by employing various techniques. For example, at the free fermion point β = 4π, exact expressions for the fractional charge two-point functions, i.e., :e i √ α 1 ϕ(x 1 ) ::e i √ α 2 ϕ(x 2 ) : SG(4π,z) with α 1 , α 2 ∈ (0, 4π), were derived in [7], the mass was determined for general β by using a mapping to the continuum limit of an inhomogeneous six-vertex model and the Bethe ansatz in [14,51], and exact expressions for the fractional charge one-point function :e i √ αϕ(x) : SG(β,z) for α ∈ (0, 4π) and general β were derived in [40] by extrapolation of exact results for β = 4π and in the asymptotics β → 0. Further references are [13,19,41], and for a review, see also [49]. All of these integrability results in infinite volume remain conjectural (except for our results at β = 4π). In finite volume, we mention the rigorous connection to the XOR Ising model at β = 2π proved in [35]. It is also well known that the sine-Gordon model is exactly related to the classical twodimensional (two-component) Coulomb gas. For this, we refer in particular to [21] and also [25] where, using this relation, many fundamental properties of the Coulomb gas have been derived when β < 4π including existence of the pressure and correlation functions, the exact equation of state for the pressure, and the exact scaling behaviour in z of the rate of exponential decay of correlations assuming its existence in a suitable sense. The latter exponential decay of correlations is in general open. For the related three-dimensional Coulomb gas, exponential decay (Debye screening) was proved for β > 0 and z both small in [8]. The methods have also been partially extended to the two-dimensional Coulomb gas in [50]. Correlation inequalities for the Coulomb gas and the sine-Gordon model as well as their applications are discussed in [24,25,44]; we make some use of these in Section 3.
Next, we mention a few related bosonization results. First, in the free field case, the bosonfermion correspondence has been extended by disorder operators [23]. Second, while the bosonization relations in this paper rely essentially on the precise asymptotics of the correlations in the continuum limit, in the massless free field case, exact discrete versions have been found as well; see in particular [18]. Some bosonization results are also expected to extend to Riemann surfaces [23]. For applications of bosonization of free fermions, see, for example, [34,Chapter 10.5].
Finally, let us emphasize that the massless sine-Gordon model is an essential example of a two-dimensional non-conformal perturbation of a CFT. For conformal field theories, a lot of recent progress has been made, in particular for the Ising model (see [10,11,33] and references therein) and for the Liouville CFT (see [31,37] and references therein). Moreover, we mention that models related to the massless Thirring model have also been studied in detail, in particular recently in the form of interacting dimers [28,29].
1.4. Notation and outline of the paper. We will write f ∈ L ∞ c (R 2 ) if f is compactly supported and essentially bounded. We write similarly is compactly supported and essentially bounded. We often write ξ = (x, σ) ∈ R 2 × {±1} and (1.52) Throughout the paper, | · | denotes the Euclidean norm, and we will often make use of the identification of R 2 and C. More precisely, we will denote the components of points x ∈ R 2 by x = (x 0 , x 1 ) and its identification with an element in C by x 1 + ix 0 . We will also repeatedly write [n] := {1, 2, . . . , n}. We write A ⊂ B to indicate that A is any subset of B (no need to be proper). The paper is structured as follows. In Section 2, we derive the Coleman correspondence in the (noninteracting) massless case z = µ = 0 and collect various estimates for Gaussian free fields and massless fermions for later use. In Section 3, we state our estimates for the sine-Gordon model and free fermions, and then prove our main theorems assuming these estimates.
The remainder of the paper is mainly devoted to proving these estimates. In Sections 4 and 5, we consider the sine-Gordon side. In particular, we construct the renormalized potential for the regularized sine-Gordon model in Section 4, and then use it, in Section 5, to prove the analyticity of the partition function of the sine-Gordon model and the convergence of the correlations functions, for any finite volume interaction. In Section 6, we prove the corresponding results on the free fermion side. Our main work here goes into the analysis of the Green's function of the Dirac operator with finite volume mass term.
In Appendix A, we collect a few (well known) operational tools for cumulants and free fermions that we use in various places throughout the paper.

Free field estimates and bosonization of massless fermions
A well-understood (but essential) step in the proof of Theorem 1.1 is to verify (1.12) when z = µ = 0. Results of this flavor exist in the literature, see [23,Section 3] or [15, Section 2.2], for example, but since neither of these references provides the exact statements that we need, we will give a derivation in our set-up in this section. Along the way we will also collect estimates for the correlations of the free field that we require for the proof of the Coleman correspondence with z = 0.
2.1. Fermionic side: massless free fermion correlations. We start with computation of the correlation functions of free massless Dirac fermions whose correlation kernel S is given by (1.8) with µ = 0, i.e., S(x, y) = 1 2π In this section, the fermionic correlation functions are then defined by whenever the determinant on the right-hand side is well-defined, i.e., for all i, j ∈ [n], either x i = y j or α i = β j . The Coleman correspondence is in terms of truncated correlation functions, and importantly, we shall require the setting where x i = y i . These truncated correlation functions are defined by where the sum is over cyclic permutations π, whenever the right-hand side is well-defined, i.e., for all i, j ∈ [n], either x i = x j or α i = β j and α j = β i . These definitions are consistent with (1.9) and (1.10) but slightly more general. (This generality is required for the proof of Theorem 1.1.) We will need various identities for these determinants defining our correlation functions. These identities are conveniently seen in the representation of these determinants in terms of Grassmann integrals. We discuss the details of this representation and prove the required (well-known) properties in Appendix A. The connection between our discussion there and that here is that if we are concerned with correlation functions 1 i nψ α i (x i )ψ β i (y i ) FF(0) the matrix (K ij ) in Lemma A.2 is defined to be S α i β j (x i , y j ) off of the diagonal and on the diagonal, it is defined to be a constant real number chosen so large that K is invertible -the exact value of this constant is irrelevant (see also Remark A.3). This definition and Lemma A.2 then allow us to deduce that the properties of Lemma A.4 hold also for the correlation functions we are considering here. Based on this representation, we can also use Lemma A.5, to see that (2.3) is also consistent with (1.6), i.e., , (n 2), (2.5) when the right-hand sides exist. Thus when the untruncated correlation functions exist, they determine the truncated ones by (2.5). In view of this fact, the next lemma determines the truncated correlation functions n k=1ψ when x k = y k ′ for all k and k ′ .
Lemma 2.1. For any x 1 , . . . , x n , y 1 , . . . , y n ′ in R 2 with x k = y k ′ for all k ∈ [n] and k ′ ∈ [n ′ ], (2.7) Proof. First consider n = n ′ . Then every term in the expansion of the determinant (2.2) that defines the left-hand side must contain a factor S 11 or S 22 and hence vanish. Let us thus assume now that n = n ′ . Then, by anticommutativity (see (A.22)), .
(2.8) Since S 11 = S 22 = 0 the right-hand side factorizes (see (A.23)), and by (2.2) it is hence equal to which gives the right-hand side of the claim.
The next two lemmas then allow computing all truncated correlation functions involving also the factorsψ 2 ψ 1 andψ 1 ψ 2 .
Proof. Since the proofs of (2.10) and (2.11) are analogous, we only consider (2.10). By (2.3), when n + n ′ + q + q ′ 2, we have where we have defined (2.13) By (2.1), all terms that contain a factor S 11 or S 22 vanish. Therefore it is necessary that the number of factors ofψ 1 equals that of ψ 2 , which implies that n = n ′ if (2.12) is nonzero which we thus assume from now on. The truncated correlation function is given by replacing one of the factors S αβ ′ (u, u ′ ) in (2.12) by −S α1 (u, z)S 2β ′ (z, u ′ ) (2.15) and then summing over the choice of which factor gets replaced. Using again S 11 = S 22 = 0, the last term vanishes unless (α, β ′ ) = (2, 1), and in this case, Thus the replacement of the factor S 21 (u, u ′ ) is equivalent to multiplying it by 1 2π The possibilities for (u, u ′ ) that are compatible with the constraint (α, β ′ ) = (2, 1) are for some i, j ∈ [n] and k, l ∈ [q] with k = l. In these cases we obtain factors of, respectively, 1 2π (2. 19) In the sum over cycles in (2.12), we may restrict to cycles which give a nonvanishing contribution, and we will do this in the following. Then by symmetry, given any pair (i, j) ∈ [n] 2 , the proportion r of such cycles giving the factor S 21 (y i , x j ) is independent of (i, j); given any pair (i, k) ∈ [n]×[q], the proportion s of such cycles giving the factor S 21 (y i , z k ) is independent of (i, k) and the same as the proportion of cycles giving the factor S 21 (z k , x i ); and given any pair (k, l) ∈ [q] 2 with k = l the proportion t of cycles giving the factor S 21 (z k , z l ) is independent of (k, l). Therefore (2.14) is obtained from (2.12) by multiplication with 1/2π and Since for any cycle π with nonvanishing contribution, each of the points y i must appear once as the first argument of S αβ (and then necessarily α = 2) and each x i once as the second argument of S αβ (and then necessily β = 1) in the product in (2.12), we also see that rn + sq = 1. Thus we have recovered (2.10) in the case n + n ′ = 0. For the case n = n ′ = 0, the same argument shows that the only possibility for u, u ′ is now (u, u ′ ) = (z k , z l ), and as before, this gives a zero contribution since the sum k,l ((z k − z) −1 − (z l − z) −1 ) vanishes. This concludes the proof. Lemma 2.3. For q + q ′ 2 and any distinct z 1 , . . . , z q , w 1 , . . . , else.
2.2. Bosonic side: free field correlations. For the computation of the free field correlations, we first recall that, for ε > 0 and m > 0, our regularized GFF is the centered Gaussian field with covariance We write ν GFF(ε,m) for the (centered) Gaussian measure with this covariance. It is a basic fact that this measure is supported on smooth functions and that the covariance of the derivatives of the field is given by the derivatives of the covariance, see e.g. [39,Appendix B]. We also recall the definition Our goal is to compute the truncated correlation functions as well as smeared versions of them. The following estimates for the covariance of ϕ and its derivatives will be useful. (As before, γ is the Euler-Mascheroni constant.) Lemma 2.4. Uniformly on compact subsets of x = y ∈ R 2 , as ε → 0, Moreover, uniformly on compact sets of x = y, as ε → 0 and then m → 0,

28)
and Moreover, for any g ∈ L ∞ c (R 2 ), uniformly in compact subsets of u ∈ R 2 , and for all f, g ∈ L ∞ c (R 2 ) with disjoint supports, (2.31) The limits above also exist when ε → 0 with m > 0 fixed and have the same local uniformity.
Proof. The estimates here are largely routine, so we sketch the main ideas and leave the full details to the reader. Let us consider separately pointwise estimates and smeared estimates.
Pointwise estimates: For (2.26), we note that by definition, where Γ is the incomplete Gamma function and we used its well-known asymptotics. Similarly, for (2.27), we note that as ε → 0, where K 0 the modified Bessel function of the second kind and we used its well-known asymptotics. The proofs of (2.28) and (2.29) are similar, and make use of standard asymptotics of Bessel functions -we omit further details.
Smeared estimates: Consider next (2.30). For g ∈ L ∞ c (R 2 ) and u ∈ R 2 , we have Thus if u ∈ R 2 is in some fixed compact set, say a disc of radius r 1 and we choose that r 2 > 0 is such that supp(g) ⊂ B(0, r 2 ) (where both r 1 , r 2 are fixed in ε, m), then one readily checks via the triangle inequality that |g(u + √ sy)| g L ∞ (R 2 ) 1{s (r 1 + r 2 ) 2 /|y| 2 }. Applying this type of bound in the above integral representation, it follows that as ε, m → 0, ϕ(u)∇ϕ(g) GFF(ε,m) converges uniformly in u in a fixed compact set. On the other hand, this type of estimate can readily be used to justify the use of the dominated convergence theorem so using (2.28), we see that in fact as ε, m → 0, 36) and that this is a locally bounded function of u.
The bound (2.31) follows directly from (2.29), while (2.32) follows from (2.27) through similar estimates as above (and making use of our assumption that f = 0). This concludes our proof.
Next, we record a basic estimate for the charge correlation functions.
where the empty product i<j is interpreted as 1 if n = n ′ = 1.
By definition, the truncated correlation functions of :e ±i √ βϕ : are determined by (2.39) and (1.6). The next two lemmas give the general truncated correlations also involving factors ∂ϕ or ∂ϕ.
Lemma 2.6. Let β > 0. For n 1, q, q ′ 0, x 1 , ..., x n , z 1 , ..., z q , w 1 , ..., w q ′ ∈ R 2 distinct, and σ 1 , ..., σ n ∈ {−1, 1}, the limits exist uniformly on compact subsets of u i = u j for i = j (where the u i are an enumeration of the points x k , z j , w j ′ ), and we have Proof. By Lemma A.1, when ε, m > 0, the truncated correlation functions are given by We would like to use Girsanov's theorem (or the Cameron-Martin theorem) to get rid of the ∂ϕ and∂ϕ terms at the expense of replacing µ k by something which depends on ν j , η j ′ , z j , and w j ′ as well. We need to be slightly careful here as ∂ϕ and∂ϕ are complex valued, and Girsanov's theorem holds a priori only for real-valued Gaussian random variables.
To justify the use of Girsanov's theorem in our setting, let us assume we have some realvalued Gaussian random variables X 1 , ..., X N and (possibly complex) constants γ 1 , ..., γ N . Then by a routine combination of the dominated convergence theorem (to justify continuity), Fubini's theorem, and Morera's theorem, one finds that is an entire function. By Girsanov's theorem, we find for real λ i that (2.47) Using a similar argument as before, one checks that this also defines an entire function of the λ i , so as these entire functions agree on real values, they must be the same: also for complex λ i . Applying this to our setting (taking X k to consist of ϕ(x k ) and the real and imaginary parts of ∂ϕ(z j ) and∂ϕ(w j ′ ) -the values of γ i and λ i are chosen accordingly), we see that the expectation on the right-hand side of (2.45) equals (when the expectation is non-zero -this is true at least for small enough parameter values, and in the end, we evaluate derivatives at zero) The last term does not contribute when we take derivatives with respect to µ k so we can ignore it. Therefore, using the last identity and rewriting the result in terms of the truncated charge correlations given by (2.45) with q = q ′ = 0, Thus, carrying out the ν j and η j ′ differentiations, we obtain Using the covariance estimate (2.28) (and its complex conjugate version), we obtain (2.44) by taking ε → 0 and m → 0.
We are ultimately interested in smeared correlation functions, and there is some care to be taken on the diagonal of the pointwise correlation functions. The following result describes what happens with the truncated charge correlation functions. Lemma 2.8. For β ∈ (0, 6π) and n = 2, the truncated charge correlations are in L 1 loc ((R 2 ) n ). Namely, for any σ 1 , ..., σ n ∈ {−1, 1} and any compact set K ⊂ (R 2 ) n , (2.56) (2.57) If K and the set {x k = x k ′ for some k = k ′ } are disjoint and if the f k have disjoint supports, the statements also hold for n = 2.
The proof of this lemma is not completely straightforward from the direct definition of the truncated charge correlation functions. For example, in [15,Lemma 3], the analogous statement is only shown for β < 4π (and the need for the statement at β = 4π is circumvented there by defining the sine-Gordon model with β = 4π in terms of the limit β ↑ 4π). For us, Lemma 2.8 follows immediately as a by-product of our later analysis, and we thus postpone its proof to Section 5.4.
For the gradient fields, we have the following smeared analogue of Lemma 2.7.
Lemma 2.9. For q, q ′ 0 with q + q ′ 1 and g 1 , ..., g q , h 1 , ..., h q ′ ∈ C ∞ c (R 2 ), the limits (2.60) For (q, q ′ ) = (2, 0), (0, 2), the right-hand sides are also equal to the Cauchy principal value integrals Proof. The fact that the truncated correlation function vanishes for q +q ′ = 1 or q +q ′ 3 follows from the fact that we are dealing with centered Gaussian random variables. We thus need to only focus on the q + q ′ = 2-case. The q = 2, q ′ = 0 and q ′ = 2, q = 0 cases follow readily from (2.32) (note that ∂g i = ∂ h j = 0). For the q = q ′ = 1 case, we find again from (2.32) and integrating by parts that from which the claim follows after noting that ∂ x . For smooth test functions, it is well known that (2.61) follows by integration by parts.
With Lemma 2.8 and Lemma 2.9 in hand, we can describe the smeared free field correlation functions in the generality we need them. : Moreover, if the f j have disjoint supports or if q + q ′ 1, the claims hold also for n = 2.
Proof. In the case that n = 2 or that the f k have disjoint supports, the claim follows immediately from Lemma 2.6 and Lemma 2.8. Thus the only slightly delicate case is the claim that the supports of f 1 and f 2 need not be disjoint for n = 2 if q + q ′ 1. For this, note first that if σ 1 = σ 2 , then the charge correlation function vanishes and there is nothing to prove. For σ 1 = σ 2 , let us only prove that the limiting quantity is integrable -justifying convergence can be readily deduced with a similar argument. By (2.39), the truncated charge two-point function function is proportional to and, by Lemma 2.6, the correlation function in the claim is thus proportional to It thus sufficies to show that is locally integrable. One readily checks that since we are integrating over given compact sets, each z j -integral gives a bound of the form 1 + | log |x 1 − x 2 || and analogously for the w j ′ integrals.
Thus it suffices to check the local integrability of As we are in two dimensions, this certainly holds for β < 6π when q + q ′ 1.

Bosonization in the massless case.
That the Coleman correspondence (1.12) holds in the non-interacting case z = µ = 0 follows by matching the above computations of the correlation functions of massless free fermions and of the massless Gaussian free field, together with the following well-known identity for Cauchy-Vandermonde matrices: .

(2.69)
This allows us to prove the Coleman correspondence in the case µ = z = 0.
where A and B are as in Theorem 1.1. Moreover, if n+n ′ +q +q ′ = 2, we have the following statements: (i) for n+n ′ = 2, q +q ′ = 0, the claim holds if f ± i have disjoint supports (ii) if n + n ′ = 1 and q + q ′ = 1, the claim holds in the same generality as for n + n ′ + q + q ′ 3 (both sides vanish), and (iii) if n + n ′ = 0 and q + q ′ = 2, the claim holds either if g ± j are disjoint supports, or if the right hand side is understood as that given by Lemma 2.9.
Using this, if q + q ′ > 0 then (2.10)-(2.11) and (2.21) for the fermionic side respectively (2.44) for the bosonic side imply that, for any distinct points, The claim (along with the relevant restrictions for the n + n ′ + q + q ′ = 2-case) now follows from Lemma 2.9 (possibly using integration by parts) and Lemma 2.10.

Estimates for the sine-Gordon model and free fermions; proof of main theorems
In this section, we record our main estimates for sine-Gordon correlation functions as well as those for free fermions with a finite volume mass term. The proofs of these estimates are presented in the remainder of the paper. Assuming these estimates, we then give our proofs of the theorems of Section 1 in this section. The intuition for Theorems 1.1-1.2 is as outlined in Section 1.3. Namely, in view of the Coleman correspondence when z = µ = 0, i.e., Corollary 2.11, the sine-Gordon measure which is formally obtained from the GFF measure by weighting it by should correspond to the massless free fermion "Grassmann measure" weighted by Our estimates stated in this section provide the required analyticity and convergence to make this correspondence rigorously. Our main innovation here is that our estimates hold for all z in a complex neighborhood of the entire real axis (not just a neighborhood of the origin) and for all L > 0, and that we control the infinite volume limit L → ∞. The main analyticity results for the sine-Gordon model stated in this section do not cause additional difficulties for general β ∈ (0, 6π), so we state them in this generality. Together with well-known correlation inequalities they then imply the remaining results stated in Section 1.

3.1.
The sine-Gordon model and estimates for its correlation functions. To state our estimates for the sine-Gordon model, we begin with the precise definition of our regularization of the continuum, finite volume, massless sine-Gordon model. For ε, m > 0, we define the probability measure ν GFF(ε,m) of the regularized GFF as in Section 2.2 and recall that ν GFF(ε,m) is supported on C ∞ (R 2 ). We then take as a regularization of the sine-Gordon model the probability measure where Λ ⊂ R 2 is a compact set, β ∈ (0, 6π), z ∈ R, and Z is the partition function -a normalization constant. We will also write · GFF(ε,m) for integration with respect to ν GFF(ε,m) and · SG(β,z|ε,m,Λ) for integration with respect to ν SG(β,z|ε,m,Λ) . For Λ = Λ L = {x ∈ R 2 : |x| L} we of course recover our definition of · SG(β,z|ε,m,L) in (1.1), but we allow more general Λ here because this allows us to obtain the Euclidean invariance of the infinite volume limits in Theorems 1.6 and 1.7. Let us comment briefly on some of the restrictions we have imposed here. As mentioned earlier, the continuum sine-Gordon model is interesting for β ∈ (0, 8π). While we are mainly interested in proving the Coleman correspondence for β = 4π, the sine-Gordon estimates we prove hold for all β ∈ (0, 6π), so we present the results in this generality. The regime β ∈ [6π, 8π) is also interesting, but would require finer estimates. For β ∈ (0, 4π), the sine-Gordon measure is absolutely continuous with respect to the GFF when Λ is compact. The free fermion point, β = 4π, is precisely where this fails.
We now state our main result about the sine-Gordon correlation functions that are important for the Coleman correspondence.
exists and is finite.
(ii) Under the assumptions of item (i), the function has an analytic continuation into a Λ-dependent neighborhood of the real axis. Moreover, it is even in z when n = 0.
(iii) Under the assumptions of item (i), for any l 0, where Γ is the incomplete gamma function, and for β = 4π, By essentially the same proof, we also obtain the following existence of the ϕ field.
Theorem 3.2. Let β ∈ (0, 6π), z ∈ R, m ∈ (0, ∞), and Λ ⊂ R 2 compact. Then for any f ∈ C ∞ c (R 2 ) and w ∈ C, the limit exists and is entire in w. If also f dx = 0, then the limit also exists and is an even function of z and an entire function of w.
Before turning to fermions, we comment here on a few facts the reader might want to keep in mind concerning these theorems.
First of all, we recall from (1.6) and the discussion following it that the product notation in the truncated correlation functions n i=1 X i T is to be interpreted as X 1 ; X 2 ; . . . ; X n T , and that correspondingly, in item (iii), terms involving powers should be interpreted as where there are l copies of Y .
Next we mention that by Lemma 2.10 and our assumptions on n, q, q ′ the derivatives in item (iii) are indeed finite as they should be.
Finally we mention that in the literature, there certainly exist some results that are similar to parts of this theorem -see in particular [5,15,17,38]. What we believe is truly new, and critical to our proof of the Coleman correspondence, is that we are able to treat all values of z ∈ R and prove analyticity in a neighborhood of the real axis -not just in a neighborhood of the origin.
We now turn to describing what we need to know about free massive fermions with a finite volume mass term.
3.2. Free fermion estimates. As discussed at the beginning of Section 3, we will establish the equivalence of the sine-Gordon measure with finite volume interaction with that of Dirac fermions with a finite volume mass term. We will here choose Λ = Λ L to be a disk of radius L > 0 centered at the origin. We again take the pragmatic approach of defining the free fermion model with a finite volume mass term, formally represented by the fermionic path integral with weight (3.2), directly through its correlation functions. Namely, given a corresponding propagator S µ1 Λ L constructed in Theorem 3.3 below, the correlation functions are defined by formulas like (1.9) and (1.10) but now with S µ1 Λ L instead of S. In particular, given n 3, f 1 , ..., f n ∈ L ∞ c (R 2 ) and α 1 , β 1 , ..., α n , β n ∈ {1, 2}, the smeared truncated correlation functions are defined by (3.12) where we sum over cyclic permutations π -as we see in our proof of Theorem 3.3, this is finite for n 3. For n = 2, the same definition applies to f 1 and f 2 with disjoint compact supports. For n = 2 and f 1 and f 2 with overlapping supports the above integral is no longer necessarily finite and we will instead consider the two-point function with the singularity subtracted, i.e., with S 0 is given by the right-hand side of (2.1). This is formally equal to Therefore the existence of the propagator S µ1 Λ L and some of its basic properties are our main result concerning such models -this is summarized in the following theorem. Here recall our definition of the Dirac operator / ∂ from (1.7).
Theorem 3.3. For each µ ∈ R and L > 0, the Dirac operator with finite volume mass term, such that given n 3, f 1 , ..., f n ∈ L ∞ c (Λ L ) and α 1 , β 1 , ..., α n , β n ∈ {1, 2}, the smeared truncated correlation functions (3.12) satisfy the following properties: has an analytic continuation into an L-dependent neighborhood of the real axis. In particular, the smeared truncated correlation function is finite. . (3.18) On the right hand side, the correlation functions with index FF(µ) are defined by the propagator (1.8) of Dirac fermions with infinite volume mass term µ.
For n = 2, the same statements remain true if f 1 and f 2 have disjoint compact supports, or if the truncated two-point function is replaced by (3.13) in (i) and on the left-hand sides of (ii) and (iii), and analogously on the right-hand side of (iii).
We again comment on some issues regarding this theorem.
First of all, one could readily formulate a non-smeared version of this result as well, but for the proof of Theorem 1.1, the smeared versions of the correlation functions are the relevant ones.
Secondly, in item (ii), the correct way to understand the term on the right hand side is that one expands the power, uses multilinearity, and (1.9). Moreover, the fact that the right hand side is finite follows from the last statement in Corollary 2.11.
Finally, we mention that given that this theorem is essentially about controlling a finite volume approximation to massive free fermions, we expect that at least parts of this result is well known to some experts. Unfortunately we were not unable to find a suitable reference for the results we need. Proof of Theorem 1.1. It suffices to show that the correlation functions of the massless sine-Gordon model with an interaction term supported in Λ L at β = 4π and those of free Dirac fermions with a mass term supported in Λ L agree for all z ∈ R (and corresponding µ = Az) and all L < ∞. Indeed, by Theorem 3.3 item (iii), the smeared truncated correlation functions of free Dirac fermions with a Λ L mass term converge as L → ∞ to their infinite volume versions, and hence, the identification in finite volume implies that the sine-Gordon correlation functions converge to the same limit.
For the equivalence in finite volume, let us write O B k for one of the quantities :e ±i √ 4πϕ :, ∂ϕ, or ∂ϕ on the sine-Gordon side, and write O F k for the corresponding one on the fermionic side -the correspondence being the one given by the statement of Theorem 1.1. Thus :e i √ 4πϕ : corresponds to Aψ 1 ψ 1 , :e −i √ 4πϕ : corresponds to Aψ 2 ψ 2 , ∂ϕ corresponds to iBψ 2 ψ 1 , and∂ϕ corresponds to iBψ 1 ψ 2 . We also let f j be compactly supported and either essentially bounded or smooth (depending on whether it is a charge or gradient observable that is acting on it) that O B k and O F k act on. Let us first focus on the case where n+n ′ +q+q ′ 3 and let us not assume that the supports of the test functions are disjoint. To see that the truncated correlation functions agree for all z ∈ R when L < ∞, we use that both are analytic in z respectively µ in a complex neighbourhood of the real axis, by Theorem 3.1 item (ii) and Theorem 3.3 item (i). By unique analytic continuation, it therefore suffices to verify that they agree in a complex neighbourhood of z = µ = 0. This in turn holds if the truncated correlation functions agree at z = 0 and all z-derivatives at z = 0 agree. That they agree for z = 0 is Corollary 2.11. On the fermionic side, the µ-derivatives at µ = 0 are given by Theorem 3.3 item (ii) as On the sine-Gordon side, the z-derivatives at z = 0 are given by Theorem 3.1 item (iii) as That these are equal when β = 4π again follows from Corollary 2.11. The same argument is valid for n + n ′ = q + q ′ = 1. Moreover, if we further assume that f k have disjoint supports, then the same argument works also for general n + n ′ + q + q ′ = 2.
Proof of Theorem 1.2. By Theorem 3.1, the relevant finite volume ε, m → 0 limits exist on the sine-Gordon side. Moreover, by Lemma 2.9, it suffices to show that The claim then follows from the result about the GFF two-point function from Lemma 2.9. The proof of (3.21) and (3.22) is analogous to that of Theorem 1.1, as follows. To be concrete, we focus on the proof of (3.21); the other one is analogous. By Theorem 3.3, item (iii), it suffices to show that

(3.23)
For z = 0, this claim is trivial as both sides vanish then. Theorems 3.1 and and the special n = 2 case of Theorem 3.3 now again imply that both sides are analytic in z and that their derivatives are identical, using Corollary 2.11.
3.4. Proof of Theorems 1.6 and 1.7. For the proofs of the results stated in Section 1.2, we need the following correlation inequalities from [25].
First note that the ϕ → −ϕ symmetry implies e iϕ(f ) SG(β,z|ε,m,Λ) = cos(ϕ(f )) SG(β,z|ε,m,Λ) . For z > 0, it then follows from [25, Corollary 3.2] that, as a function of m > 0 and z > 0 and the set Λ, Indeed, by rescaling ϕ by √ β, in the notation of [25, Section 3], one has and [25,Corollary 3.2] states that if ρ 1 ρ 2 and C 2 C 1 then The monotonicity (3.24) is immediate from this. As a particular case of (3.24) we get the following infrared bound: for any f ∈ C ∞ c (R 2 ), m, z > 0 and Λ, we have Proof of Theorems 1.6 and 1.7. Since the proofs of Theorems 1.6 and 1.7 are essentially identical, we focus on the first theorem and leave the modifications for the second theorem to the reader. By Theorem 3.2, for any f ∈ C ∞ c (R 2 ) with f dx = 0, the limit exists and is invariant under z → −z. Thus without loss of generality we can and will assume z > 0. By (3.24), it follows that e iϕ(f ) SG(β,z|Λ) is monotone in Λ, and thus converges as Λ ↑ R 2 to a limit which we denote by e iϕ(f ) SG(β,z) . The limit is trivially bounded above by 1 and the map f → e iϕ(f ) SG(β,z) satisfies the following continuity estimate: for any g ∈ C ∞ c (R 2 ) with dx g = 0, Indeed, for ε, m > 0 and Λ finite, the analogue of the first inequality is immediate, and the second inequality follows from (3.24). The claimed inequality then follows by taking the limits in ε, m, Λ. In particular, if functions g k ∈ C ∞ c (R 2 ) with dx g k = 0 converge to 0 in the topology of S(R 2 ), the right-hand side of (3.30) converges to 0. Since C ∞ c (R 2 ) is dense in S(R 2 ) (and likewise for the subspaces of functions which integrate to 0), it follows that e iϕ(f ) SG(β,z) extends to a continuous functional on S ′ (R 2 )/constants (the topological dual space of the closed subspace of integral-0 functions in S(R 2 )). Minlos's theorem then implies that e iϕ(f ) SG(β,z) is the characteristic functional of a probability measure on S ′ (R 2 )/constants.
That the limit is Euclidean invariant is a standard argument that follows from the Euclidean invariance of the GFF and the monotonicity of e iϕ(f ) SG(β,z|Λ) in Λ for any increasing family of sets, see, e.g., [48,Section VIII.6].

Proof of Theorem 1.3 and Corollary 1.8.
Proof of Theorem 1.3. The main step of the proof will be to show that (1.28) holds for functions with integral 0, i.e., for all To this end, let us first further assume that there are In this case, we note that, by integrating by parts, and we find that, by Theorem 1.2, the ε, m → 0, L → ∞ limits exist and To express the right-hand side as in (1.28)-(1.29), let us first look at the g 1 , g 2 -term -the remaining terms are similar. Recalling that 1 2π K 0 (A|z||x − y|) is the covariance of the massive free field, we have the following Fourier space representation of K 0 : where the integral is understood either in principal value sense or in the sense of distributions. Thus with the conventionf (p) = R 2 f (x)e −ip·x dp for the Fourier transform and 2∂ = ip, Theorem 1.2 and a routine calculation shows that (with integrals understood in a principal value sense) TheĈ on the right-hand side can be computed as follows. Going into polar coordinates, scaling the radial variable, and translating the angular variable shows that (3.37) where µ p = A|z|/|p|. To evaluate the t-integral through the residue theorem, we note that out of the two poles for η = e −it , only the minus-one is inside the unit disk, and we thus find A straightforward (but slightly tedious) calculation shows that the last integrand can be written as from which we see after another slightly tedious calculation thatĈ A|z| (p) equals Finally, an elementary calculation shows that from which we can deduce (1.29) with another routine calculation. We see in particular from this thatĈ A|z| is bounded for |z| > 0. A similar calculation shows that with the sameĈ A|z| . Thus taking complex conjugates of these identities, we find that for our f i given by which is precisely the claim for the f i which can be represented this way. Finally, to extend the statement to arbitrary we note that such f i can be written as in (3.32) but with g i and h i in S(R 2 ), by Taylor expandinĝ f i . Thus it remains to extend our argument to Schwartz functions. For this, given f i ∈ S(R 2 ) satisfying f i = 0, let g i , h i ∈ S(R 2 ) be such that we have the representation (3.32). Let us take χ ∈ C ∞ c (R 2 ) non-negative, bounded by 1, supported in Λ 2R = {x ∈ R 2 : |x| < 2R}, equal to one in Λ R , and with gradient bounded as a function of R. Then write g i = χg i + (1 − χ)g i and similarly for h i . We then have where in Σ 1 , we have kept only the χg i , χh i -terms, while in Σ 2 we have at least one (1 − χ)g i or Using the initial part of this proof and a routine dominated convergence argument, we see that when we let ε → 0, m → 0, L → ∞, and finally R → ∞, Σ 1 converges to our target -namely (3.44) (which is perfectly well defined for f i ∈ S(R 2 )). Thus we need to show that Σ 2 tends to zero in the same limit. For this, using (3.28) and routine Cauchy-Schwarz arguments shows that (in the ε → 0, m → 0, L → ∞-limit) we end up estimating e.g. quantities of the form (3.46) By dominated convergence, this tends to zero as R → ∞, and one finds that Σ 2 tends to zero in our limit. This shows that (3.44) is true also for f 1 , f 2 ∈ S(R 2 ) satisfying f i = 0. The localization bound (1.30) now follows easily by observing that, du (f x (u) − f y (u)) = 0 so by (1.28) for integral 0 test functions (for which we have now established (1.28)), the left-hand side of (1.30) is given by This is uniformly bounded sinceĈ µ (p) is bounded for µ = 0. Finally, we construct the required probability measure · SG(4π,z) on S ′ (R 2 ). In Theorem 1.6, we have already constructed such a measure on S ′ (R 2 )/constants, i.e., for test functions f ∈ S(R 2 ) with dx f = 0. Using the uniform bound onĈ µ for µ = 0 we can extend this measure to all test functions in S(R 2 ) as follows. Let γ N (x) = (2πN ) −1 e −|x| 2 /(2N ) be the density of the twodimensional Gaussian probability measure of variance N and Fourier transformγ is well defined by Theorem 1.6. For µ = 0, we will show that it is a Cauchy sequence in N , as a consequence of the boundedness ofĈ µ . Indeed, That this is indeed the characteristic functional of a probability measure on S ′ (R 2 ) again follows from Minlos' theorem and the continuity of f → e iϕ(f ) SG(4π,z) which follows from the boundedness ofĈ µ by an argument analogous to the above Cauchy sequence argument. This argument also shows that the covariance is given by (3.51) The exponential decay (1.31) now follows (see e.g. [45, Theorem IX.14]) from the fact thatĈ µ (p) is uniformly bounded and that, as one readily checks from (1.29), it has an analytic continuation into a strip |Im(p 0 )|, |Im(p 1 )| < η for some η > 0 (proportional to |µ|).
For the proof of Corollary 1.8, we need the following observation from [25] adapted to our setting.
Proof. The argument is as in the proof of [25,Theorem 4.4]. Indeed, by Theorem 1.7, the measure · SG(β,z|m) is translation invariant and satisfies, for (3.53) in the distributional sense. Indeed, this follows from Thus the Riemann-Lebesgue lemma implies that as claimed.
Proof of Corollary 1.8. By (3.52), for m > 0, By monotonicity in m and L due to (3.24), the limits m → 0 and L → ∞ exist in both orders and, if dx f = 0, In conclusion, we get that For β = 4π, the right-hand side is finite by Theorem 1.3. In fact, since |f | 2Ĉ µ is integrable for µ = 0, by the Riemann-Lebesgue lemma, it is equal to lim sup as claimed.
The proof of the existence of the infinite volume measure as m → 0 is now exactly as in the proof of Theorems 1.6 and 1.7, only using the now proved bound (1.46) instead of the last bound in (3.30) for the continuity of the characteristic functional. 4 The sine-Gordon model: the renormalized potential One of our main tools in the proof of Theorem 3.1 are estimates for a renormalized version of the sine-Gordon potential, and we turn to studying it now.
which we refer to as the microscopic (sine-Gordon) potential. In terms of this microscopic potential, we introduce the following generalized partition function that can be seen as a generating function for charge correlation functions: where the GFF expectation is over ϕ. For ζ = −z1 Λ this would just be the partition function of the (regularized) sine-Gordon model. Our analysis of the generating function Z(ζ|ε, m) relies on a convenient decomposition of the regularized free field GFF(ε, m). More precisely, we define for any t, m > 0 and x, y ∈ R 2 with are covariances, so the fact that the sum of two independent Gaussian processes is a Gaussian process whose covariance is the sum of the covariances of the two processes implies that we can in fact write (4.2) as where we have defined the renormalized potential v t by and have written E c m 2 for the expectation with respect to the law of the Gaussian process with covariance c m 2 t − c m 2 ε 2 and the last integral is over η. The analysis of the ε, m → 0 behavior of the generating function Z(ζ|ε, m) can thus be rephrased in terms of ε, m → 0 asymptotics of the renormalized potential v t (ζ, ·|ε, m). Note that as ζ is complex, only e −vt(ζ,ϕ|ε,m) is a priori well defined, but we will see in this section that for any ) and t small enough, its logarithm v t (ζ, ϕ|ε, m) is also well-defined. Moreover, the goal of this section is to prove bounds for v t (ζ, ϕ|ε, m) that are uniform in ε > 0 and m > 0. Our analysis follows the approach of [9] as presented in [2, Section 3]. The ε → 0 and m → 0 limits will be studied in Section 5.
To control v t we will show in this section that the following expansion is convergent and agrees with v t (ϕ, ζ|ε, m) for ζ ∈ L ∞ c (R 2 × {−1, 1}) and suitable t: where the coefficients v n t are determined recursively as follows. For t > ε 2 and ξ ∈ R 2 × {−1, 1}, and for n 2 and ξ j = (x j , σ j ) ∈ R 2 × {−1, 1}, v n t (ξ 1 , ..., ξ n |ε, m) = For controlling the expansion (4.6), we introduce the following norms for f : (R 2 ×{−1, 1}) n → C: (4.11) The goal of the rest of this section is to prove the following proposition. for some constant C β depending only on β.
This allows us to identify the expansion (4.6) with the renormalized potential as follows.
where C β is the constant from Proposition 4.1, the sums and integrals in the expansion (4.6) converge absolutely and equal v t (ζ, ϕ|ε, m) defined in (4.5): Proof. Throughout the proof, we fix ε, m > 0 and ζ ∈ L ∞ (R 2 × {−1, 1}) with support in a compact set Λ × {−1, 1} ⊂ R 2 × {−1, 1}, and we will always assume that t ∈ (ε 2 , t 0 ) where t 0 is the supremum over t > ε 2 such that (4.14) holds. Then, for n 3, 1 n! (R 2 ×{−1,1}) n dξ 1 · · · dξ n ζ(ξ 1 ) · · · ζ(ξ n ) v n t (ξ 1 , . . . , ξ n |ε, m)e i where we used n n /n! e n . The n = 1, 2 terms are trivially bounded with ε, m-dependent constants when t > ε 2 , uniformly in ϕ : R 2 → R, by the definitions (4.7)-(4.8). For t < t 0 , it follows that the sum over n in (4.6) converges geometrically, again uniformly in ϕ : R 2 → R. We denote this sum by a t (ϕ) and note that a t (ϕ) only depends on ϕ| Λ , so that we can consider ϕ → a t (ϕ) as a function a t : C(Λ) → R. We will denote the supremum norm on C(Λ) by · below. From the geometric convergence, and similar estimates for higher derivatives, we then see that a t : C(Λ) → R is actually smooth, i.e., Frechet differentiable to any order, for t ∈ (ε 2 , t 0 ). Its first two derivatives are given by where f 1 , f 2 ∈ C(Λ). As in (4.16), it follows that ∇a t (ϕ, ·) L ∞ (R 2 ) and Hess a t (ϕ, ·, ·) L 1 L ∞ (R 2 ×R 2 ) are bounded independently of ϕ and, since ζ has support in Λ, it is also clear that ∇a t (ϕ, ·) has support in Λ and that Hess a t (ϕ, ·, ·) has support in Λ 2 . Defining it then follows from (4.8) that, for t ∈ (ε 2 , t 0 ), Let h t (ϕ) = e −at(ϕ) . Then by the chain rule, h t is also twice Frechet differentiable with (using similar notation as above) and hence We will show that e −vt satisfies this same heat equation (with the same initial data at t = ε 2 ) and argue that the solution must be unique, so v t = a t , which will then yield the proof.
The Laplacian ∆ċ t h t can alternatively be expressed as follows. Since Λ is bounded, assume that Λ ⊂ [−L, L] 2 . Let χ be a smooth function with χ(t) = 1 for t 4L and χ(t) = 0 for t 8L. We then choose a torus Λ ′ of period 16L and setċ ′ t (x) = n∈Z 2ċ m 2 t (x + 16Ln)χ(|x + 16Ln|). Thusċ ′ t is a smooth (periodic) function on Λ ′ , and we note thatċ ′ t (x) =ċ m 2 t (x) if |x| 4L. Thus if we regard Λ as a subset of Λ ′ (by embedding it into a fundamental domain centered at 0 in the obvious way), we haveċ ′ t (x − y) =ċ m 2 t (x − y), for x, y ∈ Λ. In particular, there areλ t,k 0 decaying rapidly in k for each t > 0 such thaṫ where (f k ) is the real orthonormal Fourier basis of L 2 (Λ ′ ) consisting of sin and cos functions, so in particular satisfying f k C. For a general function g ∈ C 2 b (C(Λ)) and t > 0 we can now define By Fubini (whose application is justified by rapid convergence of all sums and integrals), this definition is consistent with (4.20). Let Π N ϕ be the L 2 (Λ ′ ) projection of ϕ| Λ ′ to Fourier modes k N . For any N , the above implies that h N t (ϕ) = h t (Π N ϕ) satisfies the finite dimensional heat equation Next we will verify that g t (ϕ) = e −vt(ϕ) defined in (4.5) also satisfies the heat equation ∂ t g t = 1 2 ∆ċ t g t with the same initial condition g ε 2 = h ε 2 . To see this, first observe that the definition of g t (ϕ) in (4.5) only depends on η| Λ . The Gaussian field η| Λ has covariance c m 2 t − c m 2 ε 2 | Λ×Λ and can be realized in terms of independent standard Gaussian random variables (X k ) k∈N as whereλ k,t above is the t-derivative of these λ k,t . (This follows from the fact that λ k,t = (f k , c ′ t f k ) and the differentibility of c ′ t in t.) From this representation we again see that g N t (ϕ) = g t (Π N ϕ) satisfies By the standard uniqueness of bounded solutions to such equations (finite dimensional heat equations), we conclude that h N t (ϕ) = g N t (ϕ) for all t ∈ (ε 2 , t 0 ) and N ∈ N. It remains to conclude that this implies that g t (ϕ) = h t (ϕ) for all smooth ϕ. Indeed, Π N ϕ − ϕ → 0 for any smooth ϕ and since both g t and h t are continuous in ϕ ∈ C(Λ), thus h N t (ϕ) = h t (Π N ϕ) → h t (ϕ) as N → ∞ and analogously g N t (ϕ) → g t (ϕ).  so it is sufficient to focus on the regime |x| < √ t. Here (using 4πs 4πs . (4.34) Recalling that we are assuming that m 2 t 1, this concludes the proof.
The next estimate is slightly more involved.
which is invariant under permutations of the coordinates, independent of ε, m, and in the notation (4.11), satisfies for some constant C β depending only on β.
Proof. The proof is slightly lengthy and we split it in two parts.
Case 1 -σ 1 = σ 2 : Let us first consider σ 1 = σ 2 and bound the quantity |u for the line segment from x 1 to x 2 . We then have by the mean value theorem (recalling that t m −2 ) To bound the gradient, we use that for any α > 0, there exists A(α) (depending only on α) such that We used here the estimate that there exists a A(α) such that xe − x 2 4 A(α)e −αx for all x > 0. From the triangle inequality we find that, for u ∈ [x 1 , This leads to the following bound: for any α > 0 (4.40) The second term in our statement we bound with the following estimate which is a consequence of Lemma 4.3 (recall that where the constant is universal. Combining our estimates, we see that for each α > 0, there exist A(α), A(α) (depending only on α and possibly different from our previous A(α)) such that where we made use of the estimate that for some A(α), e −x 2 A(α)e −4αx for x > 0 and that for for some universal constant C and we can then define F t (at least in the case σ 1 = σ 2 ) to be the function obtained by symmetrizing the above function with respect to the variables x i . Note in particular that this is independent of ε, m.
To control F t 3 , let us in all of our terms (coming from symmetrization) shift x 2 and x 3 by x 1 so we are left with the estimate for some universal constant C. By a change of integration variables, the x-integral is some universal constant times t while the y-integral is some constant depending on β times r (note that the singularity at the origin is integrable precisely for β < 6π). Thus for some constant C β (depending only on β) which was the claim.
Case 2 -σ 1 = σ 2 : For σ 1 = σ 2 , we simply write while for the exponential, we have by Lemma 4.3 (for some universal constant C) Combining the estimates, we have (for some possibly different universal constant) The relevant function F t is again obtained by symmetrizing with respect to x 1 , x 2 , x 3 .
To estimate the norm, we can again get rid of x 1 by a shift of the integration variables. One is left with the estimate now for universal constants C, C. This concludes the proof.
The final estimate we shall need involves four points.
Case 1 -σ 1 = σ 2 and σ 3 = σ 4 : Let us begin by considering the case σ 1 = σ 2 and σ 3 = σ 4 . Arguing as in (4.37), but noting that now we are dealing with a kind of second order difference, we find bounds in terms of the second order derivative of the heat kernel. Using again an elementary estimate bounding x 2 e −x 2 + e −x 2 in terms of e −αx times a constant depending only on α > 0, we find that for each α > 0 there exists a constant A(α) (depending only on α) such that i∈{1,2},j∈{3,4}u m 2 Instead of the bound (4.39), we now use the fact (again following from the triangle inequality) that which yields for our choice of σ's that for each α > 0, there exists A(α) such that i∈{1,2},j∈{3,4}u (4.54) The exponentials we estimate as in (4.41) and arguing as in the proof of Lemma 4.4 (choosing α ′ and α in a similar way etc.), we arrive at the bound i∈{1,2},j∈{3,4}u Symmetrizing with respect to the x i yields our function G t . Its norm can be estimated with similar scaling arguments as in the proof of Lemma 4.4 and we find which was the claim.

Proof of Proposition 4.1.
We begin our proof of Proposition 4.1 with the remark that (4.7) implies (using x > 0, 1 − e −x x and that m 2 t 1) for a constant C β depending only on β. This verifies the bound in Proposition 4.1 for n = 1. We will verify the claimed bound explicitly also for n = 3 and n = 4, but prove the rest of it by induction. We will make use of the following explicit form of the n = 2 term: Indeed, this equalitity follows from a straightforward calculation (using (4.7) and (4.8)). This allows us to prove Proposition 4.1 in the special case n = 3.
Proof. From the definitions of v n t in (4.7) and (4.8) and the expression for v 2 t from (4.60), a straightforward calculation shows that, for any ξ 1 , ξ 2 , ξ 3 , We claim that we have the following estimate: for t > s, for any σ 1 , σ 2 , σ 3 and x 1 , x 2 , x 3 and for some universal constant C > 0, (4.63) Indeed, the worst case scenario is when σ 1 = σ 2 = σ 3 and |x 1 − x 2 | |x 1 − x 3 |, |x 2 − x 3 | (or the same with a permutation of indices). By the triangle inequality, at least one of |x 1 − x 3 | and |x 2 − x 3 | is greater than 1 2 |x 1 − x 2 |. Thus we have from Lemma 4.3, where the implied constant is universal. Going through the various cases (|x| < √ t, √ t |x| < 2 √ t, and |x| 2 √ t), one readily checks that the infimum is − 1 2π log 2 and we have the bound (4.63). Now making use of Lemma 4.4, (4.63), and (4.59), we find that for a constant C β (depending only on β) which is independent of ε, m as required. Finally, by Lemma 4.4, there is another constant C β depending only on β such that which was precisely the claim.
We now turn to v 4 t . Lemma 4.7. For β ∈ (0, 6π) and t > 0 there exists a function h 4 t , which is independent of m and ε such that for 0 < ε 2 < t < m −2 , 4 for a constant C β depending only on β.
Here we can use (4.59) and Lemma 4.6 along with the remark that w m 2 t − w m 2 s 0 (sinceċ m 2 r is a covariance), to get the simple upper bound 1 2 which is the contribution to h 4 t from the |I 1 |, |I 2 | = 2-case. Note that using (4.59) and Lemma 4.6, one can check readily that the · 4 -norm of this quantity is bounded by (for some constants C β , C β depending only on β) which is precisely of the required form (note that the integral here is convergent since β < 6π). It remains to control the |I 1 | = |I 2 | = 2-case. A typical term that one encounters in the sum is of the form The last exponential term can again be dropped by positive definiteness ofċ m 2 r , so using Lemma 4.5 and (4.59), we see that for some C β depending only on β, such terms can be bounded by where G s is as in Lemma 4.5. Summing over the other contributions shows that all of the |I 1 | = |I 2 |-terms can be bounded by such quantities. Combining this with the |I 1 |, |I 2 | = 2 case gives the definition of h 4 t . Moreover, we note from Lemma 4.5 that for some constants C β , C β depending only on β. Again, β < 6π played an important role here.
Combined with the estimate from the previous case, we see that h 4 t 4 C β t −1 t 4(1− β 8π ) as required. This concludes the proof.
We turn now to the proof of the general case.
Proof of Proposition 4.1. As mentioned already, the proof is by induction. For propagating the induction, we find it convenient to prove the claim in a slightly different form. More precisely, we will prove the existence of functions h n t (independent of ε, m) for which | v n t (·|ε, m)| h n t and for some C β depending only on β and some universal constant C > 0 Let us now as our induction hypothesis assume that for some n 5, the estimate (4.73) holds for all k n − 1 with k = 2. As mentioned, this has been verified for n = 5. To advance the induction, we plug the hypothesis into (4.8), and need to be slightly careful about the contributions from |I 1 | = 2 or |I 2 | = 2. Let us consider the terms in (4.8) with |I 1 | = 2 and |I 2 | = 2 first. In (4.8), it will be sufficient to just drop the w m 2 t − w m 2 s -term (which, as before, is allowed due to the positive definiteness oḟ c m 2 r ). Then one readily checks (from (4.8) and our induction hypothesis) that the |I 1 |, |I 2 | = 2contribution can be bounded by h n,1 t (ξ 1 , . . . , ξ n ) := β 8π Note that this is indeed independent of ε, m as required. Using the fact that R 2 dx e − |x| 2 4t 4πt = 1 and our induction hypothesis, we find for the norm of this the bound h n,1 t n β 2 where in the last equality we made use of the identity n−1 k=1 n k k k−1 (n − k) n−k−1 = 2(n − 1)n n−2 . (This identity has the following combinatorial interpretation. The number of trees on [n] is n n−2 . Thus 2(n − 1)n n−2 represents the number of trees on [n] together with a choice of a directed edge. Such trees rooted by a directed edge can also be obtained by connected two disjoint vertex rooted trees with k and n − k vertices by an edge connecting their roots.) Now for n 5 and β ∈ (0, 6π), 0 β 2 2(n−1) n(1− β 8π )−1 is bounded by a universal constant, so possibly increasing C β verifies that the bound (4.73) holds for the contribution coming from |I 1 |, |I 2 | = 2.
Let us now turn to the case where |I 1 | = 2 or |I 2 | = 2. We again drop the w m 2 t − w m 2 s -term from the exponential by positive definiteness. In terms of the notation of Lemma 4.4, we find (using the lemma and (4.60)) that the contribution from the |I 1 | = 2 or |I 2 | = 2 case can be bounded by  for some constant C β depending only on β. For the norm of this, we readily find from Lemma 4.4 and our induction hypothesis that (for some possibly different C β , still depending only on β) The ratio here is again bounded by a universal constant, so possibly increasing C β (to account for this universal constant and C β ) then yields the bound we are after.
In particular, choosing h n t = h n,1 t +h n,2 t gives the required function and concludes the proof.

The sine-Gordon model: the partition and correlation functions
The goal of this section is to prove Theorem 3.1, which is our main statement about the correlation functions of the sine-Gordon model. As already suggested in the previous section, a central tool in our proof of Theorem 3.1 is a suitable generating function for the correlation functions. To reiterate, the generating function we consider is (as in (4.2)) for ζ ∈ L ∞ c (R 2 × {−1, 1}) given by with ξ = (x, σ) and dξ = σ∈{±1} R 2 dx as before. Of course, ζ(ξ) = −z1 Λ (x) is admissible and Z(ζ|ε, m) then reduces to the normalization constant in (3.3). In general, note that we allow complex valued functions ζ, and that Z(ζ|ε, m) is then not necessarily a normalizing constant for a positive measure. The purpose of introducing Z(ζ|ε, m) is that by choosing ζ to depend on suitable external parameters, we can obtain (smeared) sine-Gordon correlation functions from logarithmic derivatives of Z(ζ|ε, m) with respect to these parameters. Thus if we can control Z(ζ|ε, m) in the ε, m → 0 limit, we can also control the correlation functions. A significant part of our analysis will rely on properties of the free field correlation functions studied in Section 2.2. Particularly important for us will be charge correlation functions. Their importance can be seen, for example, from the fact that since :e i √ βϕ(x) : ε is a bounded random variable for any ε > 0 one finds (for more details, see Lemma 5.5) .

(5.2)
It turns out that for β 4π (so in particular for β = 4π), Z(ζ|ε, m) does not converge as ε → 0. Heuristic evidence for this can be seen from Lemma 2.5 combined with the expansion (5.2): one expects to have a divergence already at order k = 2 in the expansion since is not integrable for β 4π. It turns out that for β ∈ [4π, 6π), this is in a sense the only type of divergence that occurs and a non-trivial limit can be obtained once Z is multiplied by an explicit counterterm. This counterterm and the limit theorem for the partition function are most conveniently expressed in terms of truncated free field correlation functions which we again recall from Section 2.2.
The counterterm is then defined as follows: for .

(5.4)
We then define our renormalized partition function as It follows from Lemma 2.5 that and since this is non-integrable for β 4π, our counterterm at least has a chance to cure the divergence of the partition function. This is indeed true, in that Z turns out to have a finite limit for β < 6π, and thus in particular for β = 4π which is the case we are interested in. For β 6π further counterterms, which turn out to involve higher order truncated correlation functions, would be required, see [6,17,43].
Before stating our result about the convergence of Z(ζ|ε, m), we recall from Lemma 2.8 that, while the truncated charge two-point function is not integrable, all higher order charge correlation functions are integrable. With this notation and fact in hand, we are in a position to state our main result about Z(ζ|ε, m). We prove this in Section 5.4 and then deduce Theorem 3.1 in Section 5.6.  (iv) Finally if ζ α ∈ L ∞ c (R 2 × {−1, 1}) depends on some complex parameters α ∈ C N and ζ α (·|ε, m) ∈ L ∞ c (R 2 × {−1, 1}) depends also on ε, m > 0 and these complex parameters α in such a way that for some K ⊂ C N compact and the convergence is uniform in α ∈ K. An analogous statement holds for m ∈ (0, ∞) fixed.
As a preliminary remark, we note that by rescaling space it suffices to prove the statements for fixed m > 0 in this theorem only for m ∈ (0, 1); we will henceforth assume this.
Before we turn to the actual proofs, we need to recall some basic facts about regularity and extrema of Gaussian processes.

Preliminaries -regularity and extrema of Gaussian processes.
In this section, we record some basic facts we need to know about the regularization of the GFF to a scale √ t which is of order one, namely we look at the Gaussian process with law ν GFF( √ t,m) -in particular in the m → 0 limit. Given (4.4), this will be useful to control the renormalized partition function. The main fact we will prove in this section is the following.
for some constant C p,t,Λ < ∞ which is independent of m.
First of all, using arguments based on Kolmogorov-Chentsov-type results (see e.g. [39, Appendix B]), one can check that the smoothness of c m 2 ∞ −c m 2 t (recall the notation (4.3)) implies that we can regard ϕ as a smooth function, and ∇ϕ is a centered Gaussian process with covariance To estimate the exponential moments in Lemma 5.2, we rely on two classical theorems about Gaussian processes. The first one is Dudley's theorem (see e.g. [1, Theorem 1.3.3]) which states that if for a centered real-valued Gaussian process X on say a compact metric space T we define a new (pseudo) metric by setting d X (t, s) = E[(X(t) − X(s)) 2 ], then where C is a universal constant, and N X (ε) is the minimal number of (closed) d X -radius ε balls required to cover T . The second result we need is the Borell-TIS inequality (see e.g. [1, Theorem 2.1.1]), which states that in the same setting as Dudley's theorem, if X is further assumed to be almost surely bounded on T , and if σ 2 T := sup t∈T EX(t) 2 , then for all u > 0 With these tools, we can prove our claim about ∇ϕ.
Proof of Lemma 5.2. First of all, we note that by a simple Cauchy-Schwarz argument, it is enough for us to prove the claim for ∂ 0 ϕ L ∞ (Λ) (or ∂ 1 ϕ L ∞ (Λ) as they both have the same distribution) instead of ∇ϕ L ∞ (Λ) . Then noting that we see that is enough to consider only sup x∈Λ ∂ 0 ϕ(x) instead of sup x∈Λ |∂ 0 ϕ(x)|. In this setup we can use Dudley's theorem and Borell-TIS.
To apply Dudley's theorem, we note that ) for a constant C t independent of m. Thus we have {y ∈ Λ : d ϕ 1 (x, y) ε} ⊃ {y ∈ Λ : C t |x − y| ε}. So the number of d ∂ 0 ϕ -radius ε balls it takes to cover Λ is less than the number of Euclidean radius ε/C t -balls it takes to cover Λ. It thus follows from Dudley's theorem, (5.12), In particular, we have for say u > 2 C t,Λ Thus we find from the Borell-TIS inequality (5.13) that, for u > 2 C t,Λ , and which yields the desired claim for the exponential moments of sup x∈Λ ∂ 0 ϕ(x) and by our preliminary considerations, also for ∇ϕ L ∞ (Λ) . This concludes the proof.

Uniform bounds for the renormalized partition function.
We are now ready to turn to analysis of the partition function. We begin with proving bounds for Z.
(i) For any fixed M > 0, Before we turn to the proof, we record a simple estimate that we will have use for in the proof and also later on.
Lemma 5.4. For each t > 0 and β ∈ (0, 6π), there exists a function g t ∈ L 1 loc (R 2 × R 2 ) (namely for any compact K ⊂ R 2 × R 2 , K dx dy |g t (x, y)| < ∞) which is independent of ε, m ∈ (0, 1) such that for all ξ 1 , ξ 2 ∈ R 2 × {−1, 1}, with A(ξ 1 , ξ 2 |ε, m) from (5.4) and v 2 t (ξ 1 , ξ 2 |ε, m) from (4.60), Proof. By the definitions in (5.4) and (4.60), we have Note first of all, that the 1 ε 2 -integral is common to both terms and bounded for m < 1, so that we can ignore it. Moreover, the contribution of the 1's is also uniformly bounded in m (for any fixed t), so that we can also ignore them. Finally if σ 1 = σ 2 , then also the e −βσ 1 σ 2 (··· ) -terms are uniformly bounded, so there is nothing to prove in this case. The remaining question is to prove the required estimate for Using repeatedly the estimate |1−e −x | x for x > 0 and Lemma 4.3, along with some elementary considerations, we find that, for some universal constant C, Note that this function is locally integrable for β < 6π (in fact for β < 8π), so we are done.
With this in hand, we can turn to the proof of Proposition 5.3. The second factor in (5.27) is bounded (uniformly in ε, m, ζ) by Lemma 5.4, our assumption that ζ(·, ±1) has support in Λ, and our assumption ζ L ∞ (R 2 ×{−1,1}) M , so we only need to consider the first factor. For this, Proposition 4.1 yields .

(5.28)
Since n n /n! e n , by our choice of t in (5.26), the n = 2 sum in the first term on the right-hand side above depends only on β, Λ, M (in particular, it does not depend on ε, m, ζ), so it only remains to control the n = 2 contribution, i.e., the second term on the right-hand side. For this, we note from (4.60) that the σ 1 = σ 2 -contribution is uniformly bounded by a quantity independent of ε, m, ζ, so again using (4.60) and the elementary estimate .
can be bounded by a quantity depending only on Λ, t, β, while on the other hand, recalling (4.59) (and that we chose in addition to (5.26) for a constant C β depending only on β. In summary, for some constants C β,Λ,M and p β,Λ,M independent of ε, m, ζ. Thus Lemma 5.2 shows that the expectation part of (5.27) is bounded by a quantity independent of m, ε, ζ.
Proof of Proposition 5.3 (ii). Consider first ζ ∈ L ∞ c (R 2 × {−1, 1}) with say ζ L ∞ (R 2 ×{−1,1}) = 1 and let us look at the function z → Z(zζ|ε, m). Since for z = 0, Z(zζ|ε, m) = 1, it will turn out to be sufficient to bound the derivative of Z(zζ|ε, m) uniformly in some neighborhood of the origin -then taking a small enough neighborhood (independent of ε, m), we can bound the distance to zero in this neighborhood. Translating this into a statement about ζ with small enough L ∞ -norm will follow from a scaling argument.
For the derivative, note that for ε, m > 0, the relevant random variables are deterministically bounded, so Z(zζ|ε, m) is an entire function. Thus for any compact K ′ ⊂ B(0, R) and z ∈ K ′ , we have by Cauchy's integral formula, (5.32) By Proposition 5.3 (i), we can bound the numerator uniformly in ε, m, ζ (recall that we normalized ζ L ∞ (R 2 ×{−1,1}) = 1) and we can assume |w − z| to be uniformly bounded from below, so we see that also the derivative is uniformly bounded in compact subsets. Thus we see that there exists some δ > 0 (independent of ε, m, ζ) for which we have |Z(zζ|ε, m)| > 0. (5.33) By scaling, we note that this can be translated into precisely the claim of the proposition.
Proof of Proposition 5.3 (iii). Note from (4.6) that under the condition ζ(x, 1) = ζ(x, −1), the renormalized potential v t (ζ, ·|ε, m) is real. Thus making the same choices as in the proof of Proposition 5.3 (i), we can write We can now argue exactly as in the proof of Proposition 5.3 (i): the sum of the n = 2-terms is deterministically bounded uniformly in ε, m, ζ. Similarly, Lemma 5.4 lets us deduce that that the A + v 2 t (·|ε, m) term can be bounded from below by a uniform constant. It remains to argue that for some constant C independent of ε, m, ζ. As in the proof of Proposition 5.3, this follows from Lemma 5.2, though now combined with Jensen's inequality (used in the form E 1 X 1 EX for a positive random variable X). This concludes the proof.
Before turning to the proof of Theorem 5.1, we introduce some notation and make some preliminary remarks about the renormalized partition function as an analytic function.

Expansion of the renormalized partition function.
Our proof of convergence of the renormalized partition function and entirety of the limit will go through analyzing the series expansion of z → Z(zζ|ε, m) and making use of the estimates from Proposition 4.1 and Proposition 5.3. For this purpose, we introduce some notation for the series expansion of z → Z(zζ|ε, m). We formulate this as the following lemma. Proof. Let us recall from (5.1) and (5.5) that As mentioned in the proof of Proposition 5.3, for each ε, m > 0, the expectation is an entire function of z since is a bounded random variable. The second factor in (5.40) is trivially an entire function of z (for any fixed ε, m > 0). Thus we see that indeed Z(zζ|ε, m) is entire. For the expansion coefficients, by series expanding both terms (and interchanging the order of summation/integration and expectation which is justified by the boundedness of the relevant random variables and a routine Fubini argument), we find .
The claim about the representation of M n (ζ|ε, m) now follows by relabeling our integration variables (write n = k + 2j and ξ ′ l ′ = ξ 2j+l ′ ) and then symmetrizing in the arguments. Finally for the proof of the bound (5.39), note that as Z(zζ|ε, m) is entire, Cauchy's integral formula implies that, for any R > 0, The claim now follows by choosing R = δ −1 and (by Proposition 5.3) In the course of our proof of convergence of Z(ζ|ε, m), we will have use for an alternative representation for M n (ζ|ε, m) in terms of the renormalized potential. To control the kernel M in terms of this alternative representation, we record the following simple fact. · · · ∂ ∂δ n M n (δ 1 f 1 + · · · + δ n f n |ε, m) = (R 2 ×{−1,1}) n dξ 1 · · · dξ n f 1 (ξ 1 ) · · · f n (ξ n ) M(ξ 1 , . . . , ξ n |ε, m) Proof. Uniqueness can be seen, for example, by choosing f i to be of the form in the x variables and to be a Kronecker δ-in the σ-variable. This shows that any two functions M satisfying this relation have the same Fourier series in an arbitrary square, so by continuity they must be the same in this square. Since the square was arbitrary, they must be the same in all of R 2 . To see that M actually satisfies this relation, write Say the δ n -derivative can hit any of the n terms in the j product and produce a factor of f n (ξ j ) for some j. The δ n−1 -derivative can hit any of the n − 1 remaining terms in the j product (and produce a factor of f n−1 (ξ j ′ ) for some j ′ = j) etc. We see that . . , ξ n |ε, m), (5.48) where in the last step we simply relabeled our integration variables.
As a final ingredient before we turn to proving convergence of the renormalized partition function, we will construct an integrable upper bound on | M| by representing M n (ζ|ε, m) in terms of the renormalized partition function. Before stating the bound, we state the representation of M n (ζ|ε, m) in terms of the renormalized potential v t (ζ, ·|ε, m).
..,n k n−2j: Note that F 1 is entire and non-vanishing, so by Lemma 5.5, also F 2 is entire and we have for any fixed R > 0 By Fubini and the proof of Proposition 5.3 (more precisely, Fubini is readily justified by controlling the n = 2 terms with Proposition 4.1 and the n = 2 term with Lemma 5.2), we find that if R is chosen to satisfy (5.26), then . (5.54) Moreover, again by Proposition 4.1, the z-integrand is an entire (random) function and we find with the interpretation that if k = 0, then F To conclude this section, we now use this representation to prove that M(ξ 1 , . . . , ξ n |ε, m) has an integrable upper bound which is independent of ε, m (allowing the use of dominated convergence).
Lemma 5.8. For β ∈ (0, 6π), n 1, and t > 0 there exists a function g t ∈ L 1 loc ((R 2 × {−1, 1}) n ), independent of ε, m, such that for 0 < ε 2 < t < m −2 , | M(ξ 1 , . . . , ξ n |ε, m)| g t (ξ 1 , . . . , ξ n ). (5.57) Proof. By Lemma 5.7, we see that | M| can be bounded by sum of products of terms of the form with 2 = n ′ n, and (5.60) where we have written ξ, ξ ′ , ξ to indicate that these variables are subsets (depending on the term in the sum) of the actual integration variables ξ i of M. Note that in each product of the G i , the factors depend on disjoint sets of the ξ i . Thus it is enough to prove the corresponding integrability bounds for the G i terms separately.
For G 1 , this is simply Lemma 5.4. For G 2 , this follows from Proposition 4.1. Finally, for G 3 , we note that if we have σ 2l−1 = σ 2l , we can bound |e i √ β( σ 2l−1 ϕ( x 2l−1 )+ σ 2l ϕ( x 2l )) − 1| 2 and from (4.60), one readily checks the uniform bound It remains to control the quantities with σ 2l−1 = σ 2l . For these we note (as in the proof of Proposition 5.3), that for x 2l−1 , x 2l in a given compact set Λ ⊂ R 2 , we find from (4.60) that . m, so using that |x − y| 1− β 2π is locally integrable (for β < 6π), one gets a locally integrable upper bound which is independent of ε, m also for G 3 . This concludes the proof.

Convergence of the renormalized partition function.
We now turn to the convergence of Z(ζ|ε, m) as ε, m → 0. With the uniform bounds from Section 5.3 in place, the main step is to show that M(ξ 1 , . . . , ξ n |ε, m) has a pointwise limit as ε, m → 0. This is the content of the following lemma. With this result in hand, we can prove Theorem 5.1.

(5.73)
Carrying out the t-derivatives, we see that and the convergence and local integrability, for n 3, follow immediately from Lemma 5.9 respectively Lemma 5.8.
It remains to consider the n = 1 and n = 2 cases. The n = 1 case is trivial, while for n = 2 the statements are straightforward to check due to the assumptions that K is supported away from the diagonal respectively that the test functions have disjoint support; we omit further details.
5.6. Analysis of the sine-Gordon correlation functions. We are finally in a position to prove our main result concerning the sine-Gordon correlation functions, i.e., prove Theorem 3.1.
Proof of Theorem 3.1, (i)-(iii) for n > 2. For n > 2, only Z(ζ µ,ν,η,z,Λ (·|ε, m)|ε, m) in (5.80) plays a role for the correlation functions -the other terms vanish when we take logarithmic derivatives and set the various parameters to zero. Indeed, log Z(ζ µ,ν,η,z,Λ (·|ε, m)|ε, m). and that the right-hand side is entire in µ k , ν j , η j ′ , z. Moreover, by Theorem 5.1 item (iii), we know that Z(ζ 0,0,0,z,Λ ) > 0 for z ∈ R, so we see that there exists some complex neighborhood of the origin N ⊂ C and some neighborhood of the real axis R ⊂ N ′ ⊂ C such that log Z(ζ µ,ν,η,z,Λ (·|ε, m)|ε, m) → log Z(ζ µ,ν,η,z,Λ ) uniformly in µ k , ν j , η j ′ ∈ N and z in a compact subset of N ′ , and that the limit is analytic in this domain. This implies that also the µ, ν, η derivatives of this logarithm evaluated at zero converge and are analytic in z ∈ N ′ . We have thus proven item (i) and item (ii) of Theorem 3.1 for n > 2. Let us turn to item (iii). Again, since we know that log Z(ζ µ,ν,η,z,Λ (·|ε, m)|ε, m) converges uniformly (and is analytic in a suitable domain), we know that also its derivatives converge. In particular, going back in our argument, our remaining task is to evaluate : for l 0. The claim for item (iii) for n > 2 now follows from Lemma 2.10.
Since n = 1 is relevant also to item (iv), let us next consider n = 0, and then conclude the proof with the case n = 1.
The Z-term can be treated as before and the last three terms converge by Lemma 2.9. For the z 2 -term, we can argue exactly as in the proof of Lemma 2.10 and conclude that also in this case, the correlation functions converge and define analytic functions of z. Thus we have proven items (i) and (ii) of Theorem 3.1 in the case n = 0. Item (iii) is verified in the same way as for n > 2. Finally, to see that the limit of right-hand side is symmetric in z, note that ζ 0,ν,η,z is proportional to z (since n = 0) and that Z is even in ζ in the limit ε → 0 and m → 0.
Proof of Theorem 3.1, (i)-(iii) for n = 1. We finally consider n = 1. Let us first look at the situation where q + q ′ 1 where we have log Z(ζ µ,ν,η,z,Λ (·|ε, m)|ε, m) Finiteness and convergence of this quantity is again argued analogously as in the proof of Lemma 2.10, so we have the proof of items (i) and (ii) also in the n = 1 case. Item (iii) follows by the same argument as before.
Proof of Theorem 3.1, (iv). The only thing that remains is thus item (iv). For this, we find with similar reasoning as before (recall we chose f to be supported in Λ) f (x 1 ).
The first term once again has a finite limit as ε, m → 0, but as we now prove, the second term blows up. For the second term, if τ = σ 1 , then everything is bounded, but for τ = σ 1 , the leading order behavior (in ε) is given by (making use of asymptotics e.g. from Lemma 2.4) from which we see that the leading order asymptotics (as ε, m → 0) are given by , and we see that which also concludes the proof of item (iv) for β > 4π. For β = 4π, on the other hand, we obtain a logarithmic singularity from the long range behavior of the u integral and one finds for the relevant asymptotics This concludes the proof of item (iv) for β = 4π as well, and also the proof of the theorem.

5.7.
Existence of ϕ field. Finally, we prove Theorem 3.2. Since the proof is essentially identical to that of Theorem 3.1, we will be somewhat brief. which we may again write using Girsanov's theorem as We only consider the limit ε → 0 and m → 0; the argument for ε → 0 with m > 0 fixed is analogous. Thus let f ∈ L ∞ c (R 2 ) with f dx = 0. Then ϕ(f ) 2 GFF(ε,m) converges as ε, m → 0 by Lemma 2.4. Moreover, again using Lemma 2.4, we have that ζ w,z,Λ (ξ|ε, m) → ζ w,z,Λ (ξ) uniformly in ξ ∈ R 2 × {±1} and uniformly on compact sets of z, w, and thus Z(ζ w,z,Λ (·|ε, m)|ε, m) → Z(ζ w,z,Λ ) and the limit is entire in z, w. As in the proof of Theorem 3.1, the same is true for the other terms.

Estimates for free fermions with finite volume mass
In this section, we prove Theorem 3.3. Most of our work goes into the construction and analysis of the fundamental solution (Green's function) of the Dirac operator with a finite volume mass term. We state these estimates in Section 6.1, then deduce Theorem 3.3 in Section 6.2, and finally prove the estimates stated in Section 6.1 in the remainder of Section 6.
6.1. Statement of estimates on the Green's function. Recall that we are considering Λ L = {x = (x 0 , x 1 ) ∈ R 2 : |x| L}, and that we identify R 2 with C. We are interested in the Dirac operator . For each y ∈ int(Λ L ) = {z ∈ C : |z| < L}, we are looking for a continuous function S µ1 Λ L (·, y) : Our results for this function S µ1 Λ L are summarized in the following theorem. In the statement, we also use S 0 and S µ to denote the explicit infinite volume Dirac Green's function (1.8): S 0 (x, y) = 1 2π where K 0 is the 0'th modified Bessel function of the second kind. It is well known that S 0 and S µ really are the fundamental solutions of i / ∂ and i / ∂ +µ on R 2 and it also follows from the well-known asymptotics of K 0 that S µ (x, y) → S 0 (x, y) as µ → 0 when x = y. For a matrix S, we will denote by |S| a submultiplicative matrix norm of S. Theorem 6.1. For each L 1, y ∈ int(Λ L ) and µ ∈ R, there exists a continuous function S µ1 Λ L (·, y) : C \ {y} → C 2×2 that satisfies (6.2) and has the following properties for some polynomial P = P (L, |µ|) (which does not depend on any of the arguments below).
(i) For all x, y ∈ int(Λ L ), x = y, and L 1, we have (with S 0 as in (6.3)) For each x, y ∈ int(Λ L ) with x = y, the function µ → S µ1 Λ L (x, y) has an analytic continuation into some L-dependent neighborhood of the real axis, this analytic continuation also satisfies the estimate (6.5), and . (6.9) 6.2. Proof of Theorem 3.3. For Theorem 3.3, our function S µ1 Λ L is of course the Green's function of Theorem 6.1. We will denote the components of the 2 × 2 matrix S µ1 Λ L (x, y) by S µ1 Λ L ;ij (x, y) where i, j ∈ {1, 2}. We also recall the definition of the truncated correlation functions from (3.12), as well as the truncated two-point functions with singularity subtracted from (3.13). Let us begin with the proof of item (i) of Theorem 3.3. We formulate this as the following lemma.
Lemma 6.2. For n 3 and f 1 , has an analytic continuation to an L-dependent neighborhood of R (with S µ1 Λ L = S 0 for µ = 0). For n = 2, the same holds if f 1 and f 2 have disjoint compact supports or if the truncated two-point function is replaced by (3.13).
Proof. From Theorem 6.1 item (i), which implies |S µ1 Λ (x, y)| P (L, |µ|)/|x − y| for x, y ∈ Λ L , and the representation (3.12), we see that the smeared truncated correlation functions exist for all µ ∈ R and n 3 (the claim about what happens at µ = 0 following from Theorem 6.1 item (iii)). Here we used that for, any compact K ⊂ R 2 , For n = 2, in the same way, the truncated two-point function with subtracted singularity (3.13) exists; or, alternatively, if f 1 and f 2 have disjoint compact supports, the truncated two-point function also exists trivially. Moreover, Theorem 6.1 item (iii) (in particular, the analogue of (6.5) for complex µ) allows us to construct a candidate for the analytic continuation of the truncated correlation functions (with subtracted singularity for n = 2). More precisely, we define the candidate by the formula (3.12) though now using the analytic continuation of the Green's function provided by Theorem 6.1 item (iii). Using Theorem 6.1 item (iii) (or more precisely, the bound analogous to (6.5) for complex µ), a routine dominated convergence argument shows that this candidate for the analytic continuation is continuous in µ (in this L-dependent neighborhood of the real axis). By Morera's theorem, it remains to prove that for any closed loop γ (in our L-dependent neighborhood of the real axis), we have where we have used the · FF(µ1 Λ L ) -notation for our candidate for the analytic continuation. Now using the analogue of (6.5) provided by Theorem 6.1 item (iii), one can use Fubini to translate this into a contour integral over suitable products of S µ1 Λ L at distinct points. By Theorem 6.1 item (iii) and Cauchy's integral theorem, this contour integral vanishes, and we are done.
We next turn to item (ii) of Theorem 3.3 which we formulate as the following lemma.
. (6.13) For n = 2, the same holds if f 1 and f 2 have disjoint compact supports or if the truncated two-point function on the left-hand side is replaced by (3.13).
Before the proof, let us just mention that this derivative is finite by the massless correspondence Corollary 2.11 and Lemma 2.10, which implies that the corresponding bosonic correlation functions are integrable, and thus these smeared correlation functions exist.
Proof. First assume that n 3. We begin by noting that due to Theorem 6.1 item (iv) (interchanging the order of integration and differentiation follows from a routine Cauchy-integral formula/Fubini argument utilizing Lemma 6.2 and Theorem 6.1) (1) ). (6.14) Note that (π, j) ∈ C n ×[n] defines a cyclic permutation σ ∈ C n+1 in terms of which the right-hand is (1) ). (6.15) In particular, this implies (recalling (1.10)) that .

(6.16)
Setting µ = 0 (note that this uses Theorem 6.1 item (iii)), we find that which is the claim when l = 1. The case of general l follows by induction. For n = 2, assuming that the truncated two-point function is replaced by (3.13) (or alternatively that f 1 and f 2 have disjoint compact supports), we note that the argument is completely analoguous. The subtracted singularity ensures the integrability of the left-hand sides, but does not contribute to the derivatives.
The final statement of Theorem 3.3 is the following lemma. Recall that, on the right hand side, the correlation functions are given by (smeared versions of) (1.10) now with the propagator (1.8) (with infinite volume mass term). .

(6.18)
For n = 2, the same holds if f 1 and f 2 have disjoint compact supports or if the truncated two-point function on the left-hand side is replaced by (3.13) and analogously on the right-hand side.
Proof. This is immediate from the uniform convergence of Theorem 6.1 item (v). (The modification for n = 2 is again only used to guarantee integrability.) Combining these lemmas yields the proof of Theorem 3.3, so we are done.

Facts about the Laplacian Green's function and eigenfunctions in a disk.
Our proof of Theorem 6.1 relies on relating S µ1 Λ L to the the Green's function of the Laplacian in the disk as well as expansions in terms of the eigenfunctions of this Laplacian. We begin by collecting some well known facts about these. First, we recall that is the Green's function for the (positive) Laplacian with zero Dirichlet boundary conditions: In (6.19) we wrote x = x 1 − ix 0 for x = x 1 + ix 0 , while in (6.21) we wrote ∆ x for the Laplacian acting on the x variable. We also recall that the eigenfunctions of −∆ on Λ L (with zero boundary conditions) can be written explicitly in terms of Bessel functions and Fourier modes. More precisely, if for n 0, J n is the n'th Bessel function of the first kind and for k 1, j n,k is the k'th positive zero of J n (recall that J n (0) = 0 for n > 0, so we do not count this zero), then for n ∈ Z and k 1 e n,k (x) = 1 √ πLJ |n|+1 (j |n|,k ) J |n| (j |n|,k r L )e inθ (6.22) are the eigenfunctions of −∆ on Λ L (with zero boundary conditions), normalized so that they form an orthonormal basis of L 2 (Λ L ). Here we have written x = re iθ . In particular, Λ L dx e n,k (x)e m,l (x) = δ n,m δ k,l . (6.23) To simplify notation, we set j n,k = j |n|,k for n < 0. The eigenvalue associated to e n,k is then In terms of the eigenfunctions and eigenvalues, the Laplacian Green's function is n,k e n,k (x)e n,k (y), (6.25) understood in the sense that for g ∈ L 2 (Λ L ), which we can write as g = n∈Z ∞ k=1 g n,k e n,k (with convergence in L 2 (Λ L ) since e n,k form an orthonormal basis of L 2 (Λ L )), we have n,k g n,k e n,k (x), (6.26) again with convergence in L 2 (Λ L ).
In what follows, we will also need pointwise estimates for series expansions in terms of e n,k . For this, we will need some bounds for e n,k . We will also have use for another family of functions (which are also Laplacian eigenfunctions, but with different boundary conditions): for n, k 1, we define f n,k (x) = −2 L j n−1,k∂ e n−1,k (x). (6.27) The following lemma collects the properties of the e n,k and f n,k we need. The stated estimates are not all optimal, but sufficient for our purposes. We write ∇ p g for the vector of all combinations of p derivatives of g and ∇ p g L ∞ (K) for the maximum of the L ∞ (K) norm of all combinations of p derivatives of g. The eigenfunctions satisfy (for some universal constant C) Moreover, for any p, q 0, any compact K ⊂ int(Λ L ), and any f ∈ C ∞ c (int(Λ L )), there are constants C p,K,L and C p,q,f,L such that Proof. The bounds on the j n,k follow, for example, from [32,Theorem 3] and the main result of [42]. The bounds (6.29) on the eigenfunctions e n,k and their derivatives ∇e n,k follow, for example, from [30, Theorem 1] and [47, Corollary 1.1] (as well as scaling by L). The claim for f n,k L ∞ (Λ L ) in (6.30) follows directly from the definition of f n,k in (6.27) combined with the gradient estimate from (6.29).
For the bound on the gradient of f n,k in (6.30), we note that since 2∂f n,k = − L j n−1,k ∆e n−1,k = j n−1,k L e n−1,k , we have from (6.29) that ∂f n,k L ∞ (Λ L ) C j 2 n−1,k L 2 , so that we only need to control ∂ f n,k L ∞ (Λ L ) . For this purpose, using the eigenfunction property (6.24), we see from (6.26) that where in the last step we integrated by parts and made use of the fact that e n−1,k vanishes on the boundary. Thus we find for some universal constant C > 0 Since x ∈ Λ L , for the first integral we readily get the bound for a universal constant C. For the second integral, one finds on the other hand by rotational invariance that The last integral here is simply some finite constant. Putting everything together and using (6.29) (and (6.28) to deduce that j 2 n−1,k j 3 n−1,k ), we see that for some universal constant C > 0, which leads to the claim, as we discussed before. The bounds (6.31) on the higher derivatives in the interior are a standard consequence of elliptic regularity theory for the Laplace operator. For example, one may apply [27, (4.19)] iteratively.
To see the decay of (6.32), by integrating by parts, it suffices to check this for p = 0 and for e n,k only. In this case, that e n,k is a Laplace eigenfunction and integration by parts show that, for any q, which gives the claimed bound.
6.4. The building blocks of the Dirac Green's function -I. We next introduce the key building blocks of our construction of the Dirac Green's function with a finite volume mass term. We begin with the following function which is the projection of the Laplacian Green's function to non-positive Fourier modes related to the x-variable. More precisely, for x, y ∈ Λ L , let n,k e −n,k (x)e −n,k (y), (6.39) where convergence is understood in L 2 (Λ L × Λ L ). We then define inductively, for j 1, the functions That the derivatives indeed exist is a consequence of the explicit formulas we will derive below. These show that, for y ∈ Λ L , the funtions E 1 (x, y) and F 1 (x, y) are defined pointwise for x = y, and that E j (x, y) and F j (x, y) with j > 1 are defined pointwise for all x, y ∈ Λ L . We also note that, by (6.25), n,k j e −n,k (x)e −n,k (y), (6.42) as an element of L 2 (Λ L × Λ L ). Based on the definition of F j and f n,k , one then expects that also This is indeed true, and we prove it in Lemma 6.8 (for j 3). We begin calculating E 1 and F 1 using (6.19) in the next lemma.
Lemma 6.6. For (almost every) x, y ∈ int(Λ L ) with x = y, 45) where the branches of the logarithms are understood to be given by the series expansion of log(1+z) for |z| < 1, and, for |x| = |y| with x = y, E 1 and F 1 are defined by continuity. In particular, Proof. Let us write E 1 for the right hand side of the claim. Using (6.19), we see that Going into polar coordinates, one can readily check from this (since there are only strictly positive Fourier modes when one expands the logarithms) that for n 0 and k 1 Similarly one finds in polar coordinates that for n > 0 and k 1 (again since there are only non-positive Fourier modes in the expansion of the logarithms) From these two facts, one finds immediately that E 1 = E 1 . For the claim for F 1 , note that from the identity for E 1 , for y, u ∈ int(Λ L ), Moreover, we have for x, u ∈ int(Λ L ), (6.51) In polar coordinates, one readily checks that the second term on the right-hand side is orthogonal to ∂ y E 1 (u, y) (when integrated over u). One then finds The claim that the values of E 1 (x, y) and F 1 (x, y) for |x| = |y| with x = y are given by continuity follows by noting that (6.40) and (6.41) are continuous away from the diagonal. Finally, (6.46) is a direct computation. This concludes the proof. Lemma 6.7. There exists a polynomial P = P (L) such that for L 1 and all x, y ∈ int(Λ L ) with x = y, Since |y| > |x|, we have |y| 1 2 |y − x|, so we conclude that for some (possibly different) C > 0 that Similarly, |x| < |y| implies |L 2 − xy| = |y|| L 2 |y| 2 y − x| |y||y − x| 1 2 |y − x| 2 , which leads to Similar reasoning readily proves the analogous bound for F 1 . For ∂ x E 1 and ∂ x F 2 , we obtain the required bound by noting that these derivatives are explicitly given by 1/(4π(x − y)) or 0 by differentiating (6.44) and (6.45).
In order to bound E 2 , we start from the definition Using the bound for E 1 and that |G Λ L (x, u)| C(1 + | log |x − u||), we readily see that E 2 is uniformly bounded by a polynomial in L. For the derivative, using that |∂ x G Λ L (x, u)| P (L)/|x − u|, it similarly follows from the above bound for E 1 that where the two polynomials P (L) can be different. Again the bounds for F 2 are similar.
We next show that the E j and F j are bounded for j 3.
Lemma 6.8. For j 3, E j and F j are given by the series (6.42) and (6.43), which converge uniformly in Λ L × Λ L .
Proof. For E j , we already saw that it agrees with the series in an L 2 -sense. Let us now argue that the series converge uniformly and the E j series can be differentiated termwise (which implies that F j will be given by the corresponding series.) This follows immediately from applying the bounds (6.29), (6.30), and (6.28) in the series representations (6.42) and (6.43).
Next we note that the E j and F j are smooth when tested against a smooth test function that is compactly supported in Λ L . Lemma 6.9. For any j 1 and f ∈ C ∞ c (int(Λ L )), ). (6.60) Proof. By (6.32) and (6.31), it follows that for any p > 0, f ∈ C ∞ c (int(Λ L )), and K ⊂ int(Λ L ), there are constants C p,f,K,L such that sup y∈K ∇ p y dx f (x) e −n,k (x)e −n,k (y) C p,f,K,L j −p n,k , (6.61) and analogously for such expressions with e −n,k replaced by f n,k . From this, the claim follows again by differentiating the series term by term.
As a final property of the functions E j and F j , we record the following recursion properties.
Lemma 6.10. For j, k 1 and x, y ∈ int(Λ L ), we have Proof. The claim for E j+k follows immediately from continuity and the representation (6.42) which implies that the two functions are the same as elements of L 2 (Λ L × Λ L ) (and thus in particular for almost every x, y ∈ int(Λ L )).
For F j+k , we integrate by parts (note that∂ x E j+1 (x, u) vanishes for u ∈ ∂Λ L by (6.40) and the fact that E j (v, u) vanishes for u ∈ ∂Λ L -which follows e.g. from the explicit representation of E 1 from Lemma 6.6 and (6.40)) and find where we used the fact that −∆ x E j+1 (x, y) = E j (x, y) by (6.40) and the first claim of this lemma. For the final claim, the fact that first integral vanishes follows immediately from the remark that considering E j (x, u) and F k (u, y) in polar coordinates for u, E j has only non-positive Fourier modes while F k has only strictly positive Fourier modes, so the claim follows from Fourier orthogonality. The vanishing of the second integral follows by a similar argument.
6.5. The building blocks of the Dirac Green's function -II. The functions E j and F j constructed above will turn out to be responsible for the singular behavior in our Dirac Green's function. To understand the behavior in µ, we introduce the following functions: for m 1, let n,k e −n,k (x)e −n,k (y) and (6.67) A priori, it may not be clear in what sense these series converge, but we now describe the basic facts we will need about these functions -including regularity.
(iii) There exists a polynomial P m = P m (L, |µ|), which does not depend on x, y, such that Proof. For continuous differentiability, let us first consider R 11 . Using (6.29) and (6.30), we find that, for some C(L, µ) > 0, By (6.28), this series is convergent for m 3, so standard results concerning uniform convergent series (involving continuity and differentiability) yields continuous differentiability. For R 21 , we point out that, by (6.27) and (6.24), 2∂e −n,k = − j n,k L f n+1,k , 2∂f n,k = − L j n−1,k ∆e n−1,k = j n−1,k L e n−1,k . (6.70) Thus the same argument making use of (6.29), (6.30), and (6.28) implies the continuous differentiability. (Now we end up with the series ∞ n=0 ∞ k=1 j −2(m+1)+5 n,k which is still convergent for m 3.) We now turn to statement (i). This follows immediately from our preceding argument for continuous differentiability as it allows us to differentiate term by term.
For (ii), we note that again by our continuous differentiability argument, we can differentiate term by term. We find (using (6.27)) that n,k e −n,k (x)e −n,k (y) n−1,k f n,k (x)f n,k (y) (6.71) and − 2∂ x R 21 (x, y)−µR 11 (x, y) n,k L 2 e −n,k (x)e −n,k (y) n,k m e −n,k (x)e −n,k (y) Recalling (6.42) and Lemma 6.8, this concludes the proof of claim (ii). For (iii), we will prove the claim for R 21 -the proof for R 11 being similar. Using (6.29) and (6.30), we have for some constant C > 0 (independent of m, µ, L) that n,k (6.73) and the claim follows from (6.28).
Finally, to prove (iv), let f ∈ C ∞ c (Λ L ). Then (6.61) holds and an analogous bound holds with e −n,k replaced by f n,k or derivatives of these. Substituting this into the definition of R m;ij , we see that all y-derivatives of series that defines dxf (x)R m;ij (x, y) converge, and thus that dxf (x)R m;ij (x, y) is smooth in y. This concludes the proof.
6.6. The proof of Theorem 6.1. In this section, we first define our Dirac Green's function, then prove it satisfies the bounds stated in items (i) and (ii) of Theorem 6.1, then prove item (iii) of the theorem, namely analyticity in a neighborhood of the real axis, item (iv) of the theorema kind of resolvent identity -and finally prove convergence as L → ∞. We split this section into parts where these tasks are carried out.
6.6.1. Constructing the Green's function. We begin by defining the function that will be our S µ1 Λ L in Theorem 6.1, and then prove that it satisfies (6.2). For this definition, recall first the key building blocks E j , F j , R m;11 and R m;12 from (6.40), (6.41), (6.66), and (6.67).
Definition 6.12. For µ ∈ R, L 1, and y ∈ int(Λ L ) and x ∈ Λ L , let and for y ∈ int(Λ L ) but x = re iθ ∈ Λ c L , define S µ1 Λ L (Le iφ , y). (6.75) In particular, note that for x ∈ Λ c L , S µ1 Λ L (x, y) is the harmonic extension of S µ1 Λ (·, y)| ∂Λ L . We now show that this is indeed a Green's function for the problem we are considering. Proposition 6.13. S µ1 Λ L defined in Definition 6.12 satisfies (6.2).
Proof. Our goal is to show that S µ1 Λ L (·, y) as defined in Definition 6.12 vanishes at infinity and that for each f ∈ C ∞ c (R 2 ) and y ∈ int(Λ L ), (6.76) Writing out DS µ1 L from (6.2) explicitly, we have for x, y ∈ int(Λ L ), (6.77) We can thus focus on the first column. Let us first consider the 21-entry. Using Definition 6.12 and Lemma 6.11, we find immediately that For the 11-entry, we have similarly using Definition 6.12 and Lemma 6.11, − µ 6 (E 3 (x, y) + F 3 (x, y)). (6.79) Note that for l 1, we have from (6.40) and (6.41) −∆ x (E l+1 (x, y) + F l+1 (x, y)) = E l (x, y) + F l (x, y). (6.80) Thus we see that there are cancellations in the sum and we have in fact 2∂ x S µ1 Λ L ;21 (x, y) + µS µ1 Λ L ;11 (x, y) = −∆ x (E 1 (x, y) + F 1 (x, y)). (6.81) By Lemma 6.6, we see that As the latter term here is harmonic (or actually anti-analytic) in int(Λ L ), we have (6.83) Let us consider now the case x ∈ Λ c L . Note that for x ∈ Λ c L and y ∈ int(Λ L ), to prove (6.2) we need to show that 2∂ x S µ1 Λ L ;21 (x, y) 2∂ x S µ1 Λ L ;11 (x, y) 2∂ x S µ1 Λ L ;11 (x, y) 2∂ x S µ1 Λ L ;21 (x, y) = 0 (6.84) and that x → S µ1 Λ L (x, y) vanishes at infinity. For x ∈ Λ c L , we note that S µ1 Λ L (x, y) is defined as the harmonic extension of S µ1 Λ L (·, y)| ∂Λ L (where the boundary values are understood as being given by a limit from int(Λ L )). Thus our goal is equivalent to S µ1 Λ L ;21 (·, y)| ∂Λ L having only (strictly) negative Fourier modes and S µ1 Λ L ;11 (·, y)| ∂Λ L only (strictly) positive Fourier modes.
Recalling that e −n,k vanishes on ∂Λ L while f n,k (Le iθ ) is proportional to e inθ , we see from (6.66), that R 3;11 | ∂Λ L has only positive Fourier modes. Similarly E j (·, y) vanishes on ∂Λ L while F j (·, y)| ∂Λ L has only positive Fourier modes (this follows from (6.41) since E j+1 has only negative Fourier modes), so we see indeed that S µ1 Λ L ;11 (·, y)| ∂Λ L has only positive Fourier modes. Thus S µ1 Λ L ;11 (·, y) is of the correct form. The argument for S µ1 Λ L ;21 (·, y) is similar, but makes use of the fact that ∂e −n+1,k ∝ f n,k and ∂f n,k ∝ ∆e n−1,k ∝ e n−1,k -we omit the details. We conclude that, for x ∈ Λ c L and y ∈ int(Λ L ), DS µ1 Λ L (x, y) = 0. (6.85) The claim (6.76) now follows by splitting the integral over R 2 into that over Λ L and Λ c L , integrating by parts -the boundary terms cancel due to continuity of x → S µ1 Λ L (x, y) across the boundary -and combining (6.78), (6.83), and (6.85). Vanishing at infinity follows from the fact that the entries of S µ1 Λ L (·, y)| ∂Λ L had only strictly positive or negative Fourier modes (and the corresponding entries were given by either antiholomorphic or holomorphic continuation of these boundary values) so S µ1 Λ L (x, y) decays at worst like |x| −1 as x → ∞.
We now turn to proving that S µ1 Λ L satisfies the bounds we are after.
6.6.3. Proof of Theorem 6.1 item (iii): Analyticity in µ. The goal of this section is to prove item (iii) of Theorem 6.1, which is implied by the following proposition.
The expansion in terms of E l and F l in a neighborhood of the origin follows readily from similar arguments and the definition of R 11 and R 21 along with (6.42) and Lemma 6.8.
Our next goal is to establish a type of resolvent identity for S 1 Λ L .
6.6.4. Proof of Theorem 6.1 item (iv): A resolvent identity. The goal of this section is to prove item (iv) in Theorem 6.1, namely the following result.
The first integrals can again be evaluated directly by taking the x-derivative outside from under the integral and using Lemma 6.10. For the second integrals, we treat various terms in different ways: the E-E term we integrate by parts and note as before that −∂ u E l+1 (x, u) = ∂ x F l+1 (x, u), (6.106) which by Lemma 6.10 leads to a term which integrates to zero.
Finally we turn to convergence as L → ∞.
6.6.5. Proof of Theorem 6.1 item (v): the L → ∞ limit. In this section, we prove item (v) of Theorem 6.1. We state this separately as the following proposition.
For the proof of Proposition 6.17, we will need the following result which can also be interpreted as a resolvent identity. Lemma 6.18. For x, y ∈ int(Λ L ), x = y, S µ1 Λ L (x, y) − S µ (x, y) = µ Λ c L du S µ (x, u)S µ1 Λ L (u, y). (6.111) Proof. It is of course sufficient for us to prove that for any f, g ∈ C ∞ c (int(Λ L )) with disjoint supports, Using the disjointness of the supports of f and g, Proposition 6.14 for S µ1 Λ L , and routine asymptotics of Bessel functions for S µ , we see that the integrands here are L 1 -functions. By Fubini, we can thus perform the integrals in any order we wish. We now claim that, on the left-hand side of (6.112), y → g(y) Λ L dx f (x)S µ1 Λ L (x, y) ∈ C ∞ c (int(Λ L )), x → f (x) Λ L dy g(y)S µ (x, y) ∈ C ∞ c (int(Λ L )). (6.113) The fact that these functions have compact support follows from f and g having compact support. The smoothness of the S µ1 Λ L -term follows from Lemma 6.11 item (iv) and (6.60). The smoothness of the S µ -term follows immediately from the explicit expression of S µ which is smooth off the diagonal. By definition of the Green's functions, (i / ∂ x +µ)S µ (x, u) = δ(x−u) and (i / ∂ x +µ1 Λ L (x))S µ1 Λ L (x, u) = δ(x − u). Since S µ (x, u) is a function of x − u, also (−i / ∂ u + µ)S µ (x, u) = δ(x − u). Thus the above smoothness (and integration by parts) implies that we have Λ L dy g(y) S µ1 Λ L (u, y) Λ L dy g(y)S µ1 Λ L (u, y) (6.114) which is the right-hand side of (6.111).
We now turn to the proof of the final claim of Theorem 6.1.
Putting together the propositions from this section also concludes our proof of Theorem 6.1, and thus that of Theorem 3.3.

A Truncated and free fermion correlations
In this appendix, we collect some well-known properties of truncated correlations (joint cumulants) and free fermion correlations.
A.1. Truncated correlations. For arbitrary random variables A i , the truncated correlations are defined by A 1 · · · A n T = ∂ n ∂t 1 . . . ∂t n t=0 log e n i=1 t i A i (A.1) when the right-hand side exists. For N ∈ N and t = (t 1 , . . . , t N ), it is often convenient to define the tilted measure with expectation · t by when these expressions exist. For 1 n N − 1, it then follows from (A.1) that 3) The next lemma shows that the definition (A.1) is consistent with (1.6).
Lemma A.1. Assume that A 1 , . . . , A n are random variables. Then A i · · · A n T = A 1 · · · A n − P ∈Pn j i∈P j Proof. It suffices to show the claim with · replaced by · t where t = (t 1 , . . . , t N ) and n N . This is clear for n = 1. To advance the induction, note that as needed.
A.2. Grassmann integrals. Let ∧ 2N be the exterior algebra (Grassmann algebra) on 2N generatorsψ 1 , ψ 1 , . . . ,ψ N , ψ N over C. The bars only have notational meaning here and for notational simplicity we drop the ∧ from the product notation, e.g.,ψ i ∧ ψ j ≡ψ i ψ j . Thus elements F ∈ ∧ 2N are noncommutative polynomials in the generators of degree at most 2N . An element F ∈ ∧ 2N is called even if it is a linear combination of even monomials (i.e., ones with an even number of factors of the generators). Let ∂ψ j and ∂ ψ j be the antiderivations on ∧ 2N defined by ∂ψ j (ψ j F ) = F, ∂ψ j F = 0 (A. 6) for any (noncommutative) monomial F ∈ ∧ 2N that does not contain a factorψ j , and analogously for the ∂ ψ j . For any F ∈ ∧ 2N the Grassmann integral of F is then defined by d ψ dψ F := ∂ ψ ∂ψ F := ∂ ψ N ∂ψ N · · · ∂ ψ 1 ∂ψ 1 F. (A.7) Note that the right-hand side is a scalar. For any even elements A 1 , . . . , A n of ∧ 2N and any smooth function g ∈ C ∞ (R n ), we define an element g(A 1 , . . . , A n ) ∈ ∧ 2N by the truncation of the formal Taylor expansion of g of at order 2N . For example, using the above definitions, we write, for any N × N matrix M , 2). For finitely many points x 1 , . . . , x n , y 1 , . . . , y n one may indeed apply this lemma to the matrix defined by K ij = S α i β j (x i , y j ) for i = j and K ii = C for a sufficiently large constant C such that K is invertible. Moreover, if F, G ∈ ∧ 2N are monomials such that for every factorψ i in F and every factor ψ j in G one has K ij = 0 and for every factor ψ i in F and every factorψ j in G one also has K ij = 0 then F G = F G . (A.23) As in (A.1), for even elements A i ∈ ∧ 2N the truncated correlations are defined by The next lemma gives equivalent characterizations of the truncated correlation functions.
Lemma A.5. Assume that A 1 , . . . , A n are even elements of ∧ 2N . Then Moreover, if the A i are of the form A i =ψ α i ψ β i , then Proof. The proof of (A.25) is identical to that of Lemma A.1. To see (A.26), we assume by induction that the identity holds for every invertible matrix K. If n = 1, this claim is which is true by Lemma A.2. To advance the induction, for t sufficiently small, set K(t) = (1 + i t i K1 β i α i ) −1 K where 1 ij is the matrix with value 1 in entry ij and 0 in all other entries, and define · t as in (A.21) with K(t) instead of K. Since 28) this definition is consistent with · t is defined as in (A.2), i.e., Also note that ∂ ∂t j K(t) = −K(t)1 β j α j K(t) (A. 30) as follows from