Critical Liouville measure as a limit of subcritical measures

We study how the Gaussian multiplicative chaos (GMC) measures $\mu^\gamma$ corresponding to the 2D Gaussian free field change when $\gamma$ approaches the critical parameter $2$. In particular, we show that as $\gamma\to 2^{-}$, $(2-\gamma)^{-1}\mu^\gamma$ converges in probability to $2\mu'$, where $\mu'$ is the critical GMC measure.


Introduction
Gaussian multiplicative chaos (GMC) theory aims to give a meaning to the heuristic volume form "e Γ d Leb", where Γ is some rough Gaussian field that is not defined pointwise. Such constructions first appeared for Gaussian free fields in the early 70s [HK71], where D e Γ d Leb was defined to add an exponential interaction to the underlying free field. The theory was then developed for a larger class of Gaussian fields, and named as Gaussian multiplicative chaos, by Kahane [Kah85].
GMC measures corresponding to the 2D continuum Gaussian free field (GFF), have recently become an active area of study, due to their links to the probabilistic description of 2D Liouville quantum gravity [DS11,DKRV16]. Out of convention, we will call such measures the Liouville measures 1 .
In this article, we study how the Liouville measures µ γ vary, for a fixed underlying field, when the parameter γ tends to the critical parameter γ = 2 from below. It is known that the measures µ γ change analytically in γ throughout the subcritical regime 0 ≤ γ < 2 (it follows, for example, from a more general result, Theorem 4 of [Jun16] 2 ), and also not hard to show that µ γ → 0 as γ → 2. In Conjecture 9 of [DRSV14a], the authors conjecture that (2 − γ) −1 µ γ converges to a multiple of the so-called critical Liouville measure µ ′ . The main result of this paper is the confirmation of this conjecture, and determination that the constant is equal to two (see Remark 4.3 for a discussion). More precisely, we prove that: Theorem 1.1. Let Γ be a zero boundary Gaussian free field in a domain D ⊂ C and let µ γ for γ < 2 be the associated sequence of Liouville measures (defined in Theorem 2.4) together with µ ′ the critical Liouville measure (defined in Theorem 2.6). Then as γ → 2 − we have in probability, with respect to weak convergence of measures.
The analogous result, in the setting of multiplicative cascades/branching random walk, was already known [Mad16]. Our proof strategy is to use the construction of Liouville measure as the multiplicative cascade introduced in [APS17] and to then transfer the proof from the case of cascades over to the case of the Liouville measure. Whereas we roughly follow the proof in [Mad16], in some places there are additional technicalities, and in others there are simplifications. Moreover, we strongly use the results on the Seneta-Heyde scaling of the Liouville measure proved in [APS17].
Our results are a very first step (see Section 4.1) towards taking γ → 2 − limits in the peano-sphere approach to 2D Liouville quantum gravity [DMS14], and also allow one to extend the Fyodorov-Bouchaud formula [Rem17] to the critical case.
The rest of the article is structured as follows: we start with basic definitions followed by a few preliminary lemmas; in Section 4 we prove the main result; and finally, we discuss some extensions.

Basic definitions
Since this article is intended to be a rather brief follow-up to [APS17], we keep the preliminaries to a minimum. We refer the reader to [APS17] for more detailed background on the planar Gaussian free field, its local sets and associated chaos measures.
2.1. The Gaussian free field and first passage sets. We denote by Γ a Gaussian free field with zero boundary conditions in a simply connected domain D ⊂ C. That is, Γ is a centered Gaussian process indexed by the set of smooth functions in D, with covariance given by Here G D is the Dirichlet Green's function in D. We normalise it so that as x → y, G D (x, y) ∼ log(1/|x − y|). One important characteristic of the Gaussian free field is that it satisfies a spatial Markov property. In fact, it also satisfies a strong spatial Markov property at certain stopping, or "local" sets, first studied in [SS13]: Definition 2.1 (Local sets). Consider a random triple (Γ, A, Γ A ), where Γ is a GFF in D, A is a random closed subset of D and Γ A a random distribution that can be viewed as a harmonic function, h A , when restricted to D \ A. We say that A is a local set for Γ if conditionally on A and One particularly nice class of local sets are those corresponding to the first hitting time of level a ≥ 0 of a Brownian motion. They are called first passage sets (FPS), were introduced in [ALS17], and are characterised by the following proposition: Proposition 2.2 (First passage sets). Let a ≥ 0 and Γ be a GFF in D. Then, there exists a unique local set A a of Γ such that: A a is called the "first passage set" of level a of Γ.
We will need the following simple properties of the FPS, see e.g. [APS17]

2.2.
Construction of the Liouville measures. We will now briefly define the "Liouville" measures: that is, the family of multiplicative chaos measures corresponding to the 2D Gaussian free field.
In [APS17], it was shown that µ γ for γ < 2 can alternatively be constructed using the local sets of Γ. In this paper, we will use the explicit construction using first passage sets: for n ∈ N and γ ≥ 0, we define the measures M γ n (dz) := e γn CR(z, D \ A n ) γ 2 /2 dz where A n is the n-FPS of Γ, and for z ∈ D, CR(z, D \ A n ) is the conformal radius seen from z of the connected component of D \ A n containing z.
where A n is the set of connected components of D\A n and (μ D ′ ) D ′ ∈An is a sequence of (conditionally) independent Liouville measures in (D ′ ) D ′ ∈An .
Let us remark that such a simple construction of the Liouville measure was first proposed (but not proved) in [Aïd15].
2.2.2. Critical regime. For γ ≥ 2 it is known, [RV10,APS17], that the approximate measures (2.2) and (2.3) converge to the zero measure almost surely. Thus, to obtain a non-trivial limit in these cases one must renormalise differently. We concern ourselves here only with the critical case γ = 2.
It is now known that all the above approximations converge to the same (up to a constant) limiting measure. The following is a combination of results of [DRSV14a, DRSV14b, HRV15, JS17, Pow18]: Theorem 2.6. The sequence of measures D ε converge weakly in probability as ε → 0 to a limiting measure µ ′ . Furthermore, log(1/ε)µ γ=2 ε converges weakly in probability to 2 π µ ′ .
2.3. Rooted measures. One of the key techniques used to study chaos measures, is to work with certain "rooted" probability measures. This idea goes back to Peyrière [KP76]. It has been widely used in the classical "spine" theory of branching processes [BK04], as well as to study the law of the field plus a "typical" point under the Liouville measure [DS11]. We will introduce them in the setting of FPS, as employed in [APS17]. We define, for γ ≥ 0, a probability measure on the field Γ plus a distinguished point Z, by where F * n := σ(Z) ∨ σ(A n ), and setP * :=P * 2 (this is the measure we will work with most often). We make the following straightforward observations concerning the law of the random variables Γ and Z underP * γ : • The marginal law of the field Γ restricted to σ(A n ) is given by (M γ 0 (D)) −1 M γ n (D)P(dΓ) and is absolutely continuous w.r.t the law of the GFF.
• The marginal law of the point Z has density (w.r.t Lebesgue measure) ∝ CR(z, D) γ 2 /2 . • Conditionally on A n , the point Z is chosen proportionally to M γ n (dz). Slightly less immediate is the following description of the conditional law of the field given the point Z, see for example [Aru17, Lemma 2.1] for a proof of the following statement, which concerns the subcritical regime: Lemma 2.8. For γ ∈ (0, 2), one can sample fromP * γ by first sampling the point Z proportionally to CR(Z, D) γ 2 /2 , and then sampling the field according to the law of Γ + γG D (Z, ·). We writeP * γ,z for the law of Γ + γG D (z, ·).
This lemma tells us roughly that, for γ < 2, the GFF around a typical point sampled from the Liouville measure has an additional γ singularity. In the case of the FPS approximation, this is encoded in a certain random walk. More precisely, if under the conditional lawP * γ (·|Z) we set is a centred random walk, the law of whose increments do not depend on the point Z, and have mean zero and variance γ (see Remark 6.3 of [APS17]).
In fact for us, the most important case is when γ = 2, which is not included in Lemma 2.8. However, we still know (see [APS17, Section 6]) that underP * =P 2 and conditionally on the point Z, the random walk S n := S 2 n is as described in the previous paragraph, with γ = 2.

Preliminary lemmas
Here, we collect a few slightly technical preliminary lemmas. One may safely skip this section in the first reading, and only return to them when they appear in the main proofs.
3.1. Uniform control of Liouville moments. We will need some control on the moments of the subcritical Liouville measures. In the following we write µ γ D for the γ-Liouville measure defined from a zero boundary Gaussian free field on D as in Theorem 2.4.
Lemma 3.1. Let p = p(γ) := 1 + 2−γ 2 . There exists a universal K > 0, such that for any f : C → [0, 1], any γ ∈ (1, 2), and any simply connected domain D ⊆ D, if µ γ D is the limiting measure associated with a zero boundary GFF Γ on D: The key ingredient is a uniform control on (p − 1)-th moments under the rooted measure, with p as above.
Lemma 3.2. There exists a universal constant C > 0 such that for all γ ∈ (0, 2) and p = p(γ) := 1 + 2−γ 2 < 2 we have (recalling the definition ofÊ γ,z from Lemma 2.8) Proof of Lemma 3.2. First, by Lemma 2.8, we can write Consider the radial decomposition of Γ, i.e. write (in the sense of distributions) Γ(z) as B |z| + Γ ∢ (z), where B r has the law of a standard Brownian motion when parametrized by − log r and Γ ∢ is a log-correlated Gaussian field, whose circle-averages around the origin are zero (see e.g. [DMS14]). Writing for the angular GMC measure, we have By first conditioning on B |z| , using the fact that E dµ γ Γ ∢ (z) = CR(z, D) γ 2 /2 dz and Jensen's inequality, we can bound the RHS further by a constant times . Now for n ∈ N, consider the decomposition of the radial part into intervals of the form Denote by R n the corresponding annulus, and observe that: • B |rn| is a zero-mean Gaussian of variance − log |r n |; • the maximum of the process B s − B rn (that is a time-changed Brownian motion) over the interval s ∈ [r n+1 , r n ] is a sub-Gaussian of mean bounded by an absolute constant times (p − 1) −1 and of variance bounded by an absolute constant times (p − 1) −2 ; Thus, there exist a universal constant K > 0 such that Note that 2 − γ 2 + γ 3 /4 is strictly positive for γ ∈ [0, 2). Thus by the sub-additivity of x p−1 , we have that Finally, as (2 − γ 2 + γ 3 /4)(p − 1) −1 = γ + 2 − γ 2 /2 > 2 for all γ ∈ [0, 2], we can conclude.
Let us now prove Lemma 3.1.
Proof of Lemma 3.1. As p < 4/γ 2 , standard theory of GMC [Kah85] guarantees that E[(µ γ D (D)) p ] is finite for any γ ∈ (0, 2). Now, by Jensen's inequality we have , which by definition of the measureÊ * γ is also equal to D CR(w, D) γ 2 /2 dw timeŝ Since the marginal density of Z is proportional to CR(Z, D) γ 2 /2 , by conditioning on Z the LHS of (3.2) can be bounded by Now, let B z be the ball of radius CR(z, D)/8 around z so that B z ⊂ D. Then, we have that . Since for y ∈ D\B z , G D (z, y) is bounded above by some universal constant (by conformal invariance of the Green's function and the distortion theorem [Law08, Theorem 3.23]), the second term in (3.3) can be bounded by a universal constant times where in the second inequality we use that CR(z, D) 2 ≤ 10Area(D).
For the first, we apply the scaling and translation map φ : w → 8(w − z)/ CR(z, D), which sends B z to D and, by the Koebe 1/4-Theorem, D to φ(D) ⊃ 2D. Letting µ γ φ(D) denote the Liouville measure associated to a GFF in φ(D), by scaling invariance of the GFF and the Green's function we thus see that the first term of (3.3) is less than or equal to a universal constant times This, using again that CR(z, D) 2 ≤ 10Area(D) and Kahane's convexity inequality [Kah85] then S n (Z) − S 0 (Z) is a simple random walk, whose distribution does not actually depend on the point Z, and whose increments have mean zero and variance equal to 2 [APS17, Lemma 6.1, Remark 6.3]. Additionally note that S n+1 − S n is always greater than −2, because the conformal radius is strictly decreasing in n. One can also extract easily from this proof that (S n − S 0 ) has exponential moments.
We will later need the following lemma controlling the exponential moments of the conditioned walk: Lemma 3.3. Fix C > 1 and a deterministic sequence C n → C. Then there exists c(C) > 0 and n 0 (C) > 0 such that for any p, n ≥ n 0 This lemma is a direct consequence of a more general lemma, by checking that all the conditions hold for our random walk S n . This is the analogue of [Mad16, Lemma A.2], but we give a different proof.
3.3. First passage set seen from the root. For a later technical argument we will also need to have some control on the geometry of first passage sets with respect to the marked point Z under the rooted measuresP * γ . The following lemma comes from [Aïd15].
Proof. This statement is part of [Aïd15, Lemma 2.3(iii)]. Uniformity of c, c ′ in γ ∈ (1, 2) is not explicitly stated in this lemma, but it comes directly from the proof.

Proof of Theorem 1.1
By conformal invariance of the Gaussian free field, we may take D = D. As mentioned in the introduction, the proof follows closely the strategy in [Mad16, Proof of Theorem 1.1]. However, the presentation is self-contained and some technical details differ.
By a standard argument (e.g. see [APS17,Remark 4.3]), Theorem 1.1 follows, once we show that in probability, as γ ր 2. This in turn follows from a diagonal argument: we will define below an approximation speed n(C, γ) such that on the one hand the level n approximations of µ γ converge to 2µ ′ , and on the other hand the error of the approximations w.r.t µ γ go to zero. These steps are separated into two lemmas: where the limit is in P-probability.
It turns out that the right choice of n(C, γ) is given by and that it is also necessary to include the dependence on the extra parameter C. Before going to the proof, let us try to briefly discuss this choice.
Let us first consider (2 − γ) −1 M γ n (D) to see which choices of approximation level n = n(γ) could possibly give us the right limit. Notice that for γ < 2 we can write Now, we know from Theorem 2.7 that we have to multiply the measures M 2 n (dz) by √ n in order to converge to a multiple of the critical measure. Thus, forgetting about the first terms in the integrand, it seems that in order to obtain a non-trivial limit we should pick n(γ) ∝ (2 − γ) −2 . So, let us consider n(γ) = n(C, γ) = ⌊(C/(2 − γ)) 2 ⌋ for some C > 0. In the proof of Lemma 4.1, we will see that as γ → 2 the measures (2 − γ) −1 M γ n converge to c 1 (C) × µ ′ for some C−dependent constant c 1 (C). This hints that for any fixed C, the error introduced when approximating (2−γ) −1 µ γ by (2 − γ) −1 M γ n with n = n(γ, C) does not go to 0 as γ → 2. Taking the extra limit C → ∞ allows us to control this error. • For the derivative martingale, however, lim n→∞ D − n = 0, whereas lim n→∞ D + n = µ ′ (see [APS17] for an explanation; for the same reason that M 2 n → 0 as n → ∞ we see that the limit of D n is supported only where S n is large). Thus, µ ′ is only the limit of the derivatives of M + n as n → ∞. • Finally, µ ′ is the limit of both the derivatives of M γ,+ and M γ,− , i.e.
Indeed, this follows from a direct calculation, after observing that in the proof of Lemma 4.1 when considering M γ,+ or M γ,− , respectively. In other words, the factor 2 originates from the fact that taking the limit as n → ∞ and taking the derivative in γ do not commute: when one first takes the derivative and then the limit n → ∞, the contribution of ∂ γ | γ=2 M γ,− disappears.
Proof of Lemma 4.1. From now on, we work for simplicity in the case O = D and also write n = n(C, γ) to try and keep notations compact.
Our main input is [APS17, Theorem 6.7], which says that for any positive, continuous bounded function F on D as n → ∞ in probability. Here l(z, n) := log CR −1 (z, D \ A n ), S n (z) := −2n + 4l(z, n) and R 1 has the law of a Brownian meander at time 1. We will aim to write (2 − γ) −1 M γ n in a similar form. First, by the definition of n = n(C, γ) in (4.1), we can write where by o(1) we mean a deterministic function of γ (possibly depending on C) that converges to 0 as γ → 2 and that comes from the fact that C 2 (2 − γ) −2 may not be an integer. Substituting now S n (z) := −2n + 4l(z, n), we further rewrite this as This already looks very much like (4.3); however there are some error terms, and moreover, the function x → e Cx/2 is not bounded. To get around this, we truncate the exponential and control the error. For fixed p > 0, approximating the indicator functions 1 {x≤p} by continuous functions 3 , it follows from (4.3) that Sn(z) √ n 1 { Sn(z) √ n ≤p} dz converges in probability as γ → 2 − (and therefore n → ∞) to Since the o(1) is deterministic, the same then also holds for √ n D n (D) D e 2n−2l(z,n) e C 2 (1+o(1)) But now for any fixed C > 0, we have and one can verify by hand that E e mR 1 ∼ √ 2πme m 2 /2 as m → ∞ (see for example [Mad16, above equation (4.6)]). Therefore, since we know from Theorem 2.7 that D n (D) → µ ′ (D) almost surely as n → ∞, we can conclude that Sn (z) √ n 1 { Sn(z) √ n ≤p} dz converges to 2µ ′ (D) in probability, as n → ∞ and then p → ∞.
Thus, it remains to show that for fixed C, converges to 0 in probability as γ → 2 − and then p → ∞. Fix ǫ > 0, and recall the definition of the event E η (n, z) = {S k (z) ≥ −2η , 0 ≤ k ≤ n}. Let C η = ∩ n,z E η (n, z). We now bound (4.6) P √ n D e 2n−2l(z,n) e C 2 Sn(z) √ n (1+o(1)) 1 { Sn √ n >p} dz > ǫ by the sum of P C c η and By the Markov inequality and the fact that C η ⊂ ∩ z E η (n, z), the second term is less than Moreover, by definition of the lawP * , we see that the expectation in (4.7) is equal to a deterministic constant timesÊ * e C 2

Sn(Z)
√ n (1+o(1)) 1 { Sn(Z) √ n >p} 1 Eη(n,Z) , which by Lemma 3.3 is less than or equal tô for n large enough and for some c(C) > 0. Using that Z is chosen proportionally to CR(Z, D) 2 underP * we can deduce that for every η > 1 and n large enough, (4.7) is less than a deterministic constant times η e −p/4 /ǫ for every fixed p. Thus, we can bound (4.6) by a deterministic constant times P C c η + η ǫ e −p/4 . By [APS17, proof of Proposition 6.4] P(C η ) → 1 as η → ∞ and thus by choosing first η large, we can make the first term as small as we wish. Then, uniformly in large n by choosing p large, we can also make the second term arbitrarily small. From here the claim follows.
Proof of Lemma 4.2. Again we write n = n(γ, C), and assume that O = D. We now use the decomposition (2.4), and further separate each component D ′ ∈ A n into two parts: the points z around which the area of the disk B(z, d(z, D ′ )) is comparable to Area(D ′ ), and the points where Area(D ′ ) is much bigger. More precisely, define where by D n (z) we denote the component D ′ ∈ A n containing z and take p = 1 + (2 − γ)/2 as in Lemma 3.1. The reason for choosing this comparison will be clear from the proof.
We now bound |(M γ n (D) − µ γ (D))/(2 − γ)| by the sum of (4.8) D ′ ∈An D ′ 1 A c z,γ e γn− γ 2 2 l(z,n) dz + e γnμ γ D ′ (dz) 2 − γ and (4.9) D ′ ∈An D ′ 1 Az,γ (e γn− γ 2 2 l(z,n) dz − e γnμ γ D ′ (dz)) 2 − γ . Thus, where the limits are in probability. Thanks to Lemma 4.1, this is bounded by 4.1. Extensions. In this section, we will shortly discuss how our results can be extended to the boundary Liouville measure and to the case of the Liouville measure for the Neumann GFF. Our results can be also easily extended to the case of quantum surfaces like quantum wedges, quantum disks or quantum spheres introduced in [She10, DMS14], but this will be discussed elsewhere for the brevity of this note [AP18]. Given our aim of leaving this a short note, we will not define any of the terms in detail, but rather refer to [Ber15] for the Neumann GFF and to [DS11], for the boundary Liouville measure.  [DS11], for discussion of the boundary Liouville measure. The most important application is the extension of the Fyodorov-Bouchaud formula [Rem17] to the critical case. So for clarity, let us see how our results can be extended to this particular case, where the underlying Gaussian field is defined on the unit circle, with covariance −2 log ||x−y|| 2 . In fact, it is easier to generalize our argument first to the case of boundary measures associated with a Neumann-Dirichlet GFF (on the "Neumann" part of the boundary), and then to conclude the result for the circular boundary measure above, by absolute continuity. So let us discuss the case of the Neumann-Dirichlet GFF. It was already explained in [APS17], Section 5, how to extend our construction of the Liouville measure using FPS to this boundary measure. However, this was only done in the subcritical regime.
The first step in adapting the proof therefore, is to provide a construction of the critical boundary measure using the boundary equivalent of the first passage sets. These boundary-FPS are discussed in Section 5 of [APS17] and their behaviour is completely analoguous to the normal FPS. In fact, via the boundary-FPS, the proofs in Section 6 of [APS17] will work essentially word-for-word to prove that one can construct a critical boundary measure using the derivative martingale and using a Seneta-Heyde scaling, and that these constructions agree (up to explicit constants) with the critical boundary measure as constructed using semi-circle averages of the field 4 . One only needs to replace the relevant definitions for sets, conformal radius etc, exactly as done in Section 5 of [APS17] for the subcritical case. For clarity, we also list here the external inputs to Section 6, and how they extend to the critical case: • Lemmas 2.3 and 3.5 from the article [Aïd15]. One can check that these also hold for the boundary-loops; the arguments are based on the iterative nature of conformal loop ensembles and their conformal invariance, both of which hold for the boundary loop ensembles. • Theorem 1.1 from the article [Pow18], which says that the critical Liouville measure for a Dirichlet GFF in the bulk ([DKRV16, JS17]) can equivalently be constructed using the "derivative martingale" defined via circle averages of the field. The proof in [Pow18] directly adapts to the setting of boundary measures for the Neumann-Dirichlet GFF. Indeed, the argument is based around certain changes of measure for the Brownian motions arising from circle averages of the Dirichlet GFF in the bulk, and when one instead considers semicircle averages of the Neumann-Dirichlet GFF on the boundary, these remain Brownian motions. The only change is that they have speed 2. Thus, one obtains that the boundary derivative martingale defined using semi-circle averages of the Neumann-Dirichlet GFF gives an equivalent construction of the critical boundary measure defined in [HRV15, Theorem 4.1]. • Proposition 3.6 from [Pow18]. This states that certain cut-off versions of the (bulk) derivative martingale are uniformly integrable. For the same reason as above the proof extends directly to give the equivalent result for the boundary derivative martingale. The second, and final, step is to adapt the proof of the current article to the Dirichlet-Neumann case. Again, this goes through word-for-word when one replaces the relevant definitions appropriately.