CLT for fluctuations of $\beta$-ensembles with general potential

We prove a central limit theorem for the linear statistics of one-dimensional log-gases, or $\beta$-ensembles. We use a method based on a change of variables which allows to treat fairly general situations, including multi-cut and, for the first time, critical cases, and generalizes the previously known results of Johansson, Borot-Guionnet and Shcherbina. In the one-cut regular case, our approach also allows to retrieve a rate of convergence as well as previously known expansions of the free energy to arbitrary order.


Introduction
Let β > 0 be fixed. For N ≥ 1, we are interested in the N -point canonical Gibbs measure 1 for a one-dimensional log-gas at the inverse temperature β, defined by where X N = (x 1 , . . . , x N ) is an N -tuple of points in R, and H V N ( X N ), defined by is the energy of the system in the state X N , given by the sum of the pairwise repulsive logarithmic interaction between all particles plus the effect on each particle of an external field or confining potential N V whose intensity is proportional to N . We will use d X N to denote the Lebesgue measure on R N . The constant Z V N,β in the definition (1.1) is the normalizing constant, called the partition function, and is equal to Such systems of particles with logarithmic repulsive interaction on the line have been extensively studied, in particular because of their connection with random matrix theory, see [For10] for a survey. Under mild assumptions on V , it is known that the empirical measure of the particles converges almost surely to some deterministic probability measure on R called the equilibrium Date: Thursday 8 th February, 2018. 1 We use β 2 instead of β in order to match the existing literature. The first sum in (1.2), over indices i = j, is twice the physical one, but is more convenient for our analysis. measure µ V , with no simple expression in terms of V . For any N ≥ 1, let us define the fluctuation measure which is a random signed measure. For any test function ξ regular enough we define the fluctuations of the linear statistics associated to ξ as the random real variable The goal of this paper is to prove a Central Limit Theorem (CLT) for Fluct N (ξ), under some regularity assumptions on V and ξ.
(H1) -Regularity and growth of V : The potential V is in C p (R) and satisfies the growth condition It is well-known, see e.g. [ST13], that if V satisfies (H1) with p ≥ 0, then the logarithmic potential energy functional defined on the space of probability measures by (1.6) has a unique global minimizer µ V , the aforementioned equilibrium measure. This measure has a compact support that we will denote by Σ V , and µ V is characterized by the fact that there exists a constant c V such that the function ζ V defined by We will work under two additional assumptions: one deals with the possible form of µ V and the other one is a non-criticality hypothesis concerning ζ V . We assume that the equilibrium measure has a density with respect to the Lebesgue measure on Σ V given by where S can be written as where m ≥ 0, all the points s i , called singular points 2 , belong to Σ V and the k i are natural integers. (H3) -Non-criticality of ζ V : The function ζ V is positive on R \ Σ V .

Main result.
Definition 1.1. We introduce the so-called master operator Ξ V , which acts on C 1 functions by Theorem 1 (Central limit theorem for fluctuations of linear statistics). Let ξ be a function in C r (R), assume that (H1), (H2), (H3) hold. We let where the k i 's are as in (1.11). Assume that (1.13) p ≥ (3k + 6), r ≥ (2k + 4), where p (resp. r) denotes the regularity of V (resp. ξ) If n ≥ 1, assume that ξ satisfies the n following conditions (1.14)ˆΣ where R x,d ξ is the Taylor expansion of ξ to order d − 1 around x given by Then there exists a constant c ξ and a function ψ of class C 3 in some open neighborhood U of Σ V such that Ξ V [ψ] = ξ 2 + c ξ on U , and the fluctuation Fluct N (ξ) converges in law as N → ∞ to a Gaussian distribution with mean It is proven in (4.7) that the variance v ξ has the equivalent expression Let us note that ψ, hence also m ξ and v ξ , can be explicitly written in terms of ξ.
1.3. Comments on the assumptions. The growth condition (1.5) is standard and expresses the fact that the logarithmic repulsion is beaten at long distance by the confinement, thus ensuring that µ V has a compact support. Together with the non-criticality assumption (H3) on ζ V , it implies that the particles of the log-gas effectively stay within some neighborhood of Σ V , up to very rare events. The case n = 0, where the support has a single connected component, is called one-cut, whereas n ≥ 1 is a multi-cut situation. If m ≥ 1, we are in a critical case.
The relationship between V and µ V is complicated in general, and we mention some examples where µ V is known to satisfy our assumptions.
• If V is real-analytic, then the assumptions are satisfied with n finite, m finite and S analytic on Σ V , see [DKM98, Theorem 1.38], [DKM + 99, Sec.1]. • If V is real-analytic, then for a "generic" V the assumptions are satisfied with n finite, m = 0 and S analytic on Σ V , see [KM00].
• An example of criticality in the bulk of the support is given by V (x) = x 4 4 − x 2 , for which the equilibrium measure, as computed in [CK06], is Following the terminology used in the literature [DKM + 99, KM00, CK06], we may say that our assumptions allow the existence of singular points of type II (the density vanishes in the bulk) and III (the density vanishes at the edge faster than a square root). Assumption (H3) rules out the possibility of singular points of type I, also called "birth of a new cut", for which the behavior might be quite different, see [Cla08,Mo08].
1.4.1. Connection to previous results. The CLT for fluctuations of linear statistics in the context of β-ensembles was proven in the pioneering paper [Joh98] for polynomial potentials in the case n = 0, m = 0, and generalized in [Shc13] to real-analytic potentials in the possibly multi-cut, non-critical cases (n ≥ 0, m = 0), where a set of n necessary and sufficient conditions on a given test function in order to satisfy the CLT is derived. If these conditions are not fulfilled, the fluctuations are shown to exhibit oscillatory behaviour. Such results are also a by-product of the all-orders expansion of the partition function obtained in [BG13b] (n = 0, m = 0) and [BG13a] (n ≥ 0, m = 0). A CLT for the fluctuations of linear statistics for test functions living at mesoscopic scales was recently obtained in [BL16]. Finally, a new proof of the CLT in the one-cut non-critical case was very recently given in [LLW17]. It is based on Stein's method and provides a rate of convergence in Wasserstein distance.
1.4.2. Motivation and strategy. Our goal is twofold: on the one hand, we provide a simple proof of the CLT using a change of variables argument, retrieving the results cited above. On the other hand, our method allows to substantially relax the assumptions on V , in particular for the first time we are able to treat critical situations where m ≥ 1. Our method, which is adapted from the one introduced in [LS16] for two-dimensional loggases, can be summarized as follows (1) We prove the CLT by showing that the Laplace transform of the fluctuations converges to the Laplace transform of the correct Gaussian law. This idea is already present in [Joh98] and many further works. Computing the Laplace transform of Fluct N (ξ) leads to working with a new potential V + tξ (with t small), and thus to considering the associated perturbed equilibrium measure.
(2) Following [LS16], our method then consists in finding a change of variables (or a transport map) that pushes µ V onto the perturbed equilibrium measure. In fact we do not exactly achieve this, but rather we construct a transport map I + tψ, which is a perturbation of identity, and consider the approximate perturbed equilibrium measure (I + tψ)#µ V . The map ψ is found by inverting the operator (1.12), which is well-known in this context, it appears e.g. in [BG13b,BG13a,Shc13,BFG13]. A CLT will hold if the function ξ is (up to constants) in the image of Ξ V , leading to the conditions (1.14)-(1.15). The change of variables approach for one-dimensional log-gases was already used e.g. in [Shc14,BFG13], see also [GMS07,GS14] which deal with the non-commutative context. (3) The proof then leverages on the expansion of log Z V N,β up to order N proven in [LS15], valid in the multi-cut and critical case, and whose dependency in V is explicit enough. This step replaces the a priori bound on the correlators used e.g. in [BG13b].
1.4.3. More comments and perspectives. Using the Cramér-Wold theorem, the result of Theorem 1 extends readily to any finite family of test functions satisfying the conditions ((1.14), (1.15)): the joint law of their fluctuations converges to a Gaussian vector, using the bilinear form associated to (1.16) in order to determine the covariance.
In the multi-cut case, the CLT results of [Shc13] or [BG13a] are stated under n necessary and sufficient conditions on the test function, and the non-Gaussian nature of the fluctuations if these conditions are not satisfied is explicitly described. In the critical cases, we only state sufficient conditions (1.15) under which the CLT holds. It would be interesting to prove that these conditions are necessary, and to characterize the behavior of the fluctuations for functions which do not satisfy (1.15).
Finally, we expect Theorem 1 to hold also at mesoscopic scales. The proof of [BL16] uses the rigidity estimates of [BEY14] which are, to the best of our knowledge, not available to the critical case.
1.5. The one-cut noncritical case. In the case n = 0 and m = 0, following the transport approach, we can obtain the convergence of the Laplace transform of fluctuations with an explicit rate, under the assumption that ξ is very regular (we have not tried to optimize in the regularity): Theorem 2 (Rate of convergence in the one-cut noncritical case). Under the assumptions of Theorem 1, if in addition n = 0, m = 0, p ≥ 6 and r ≥ 18, then we also have, for any s such that |s| N is small enough 3 where the constant C depends only on V and β.
These additional assumptions allow to avoid using the result of [LS15] on the expansion of log Z V N,β . Our transport approach also provides a functional relation on the expectation of fluctuations which allows by a boostrap procedure to recover an expansion of log Z V N,β (relative to a reference potential) to arbitrary powers of 1/N in very regular cases, i.e the result of [BG13b] but without the analyticity assumption. All these results are presented in Appendix A.
1.6. Some notation. We denote by P.V. the principal value of an integral having a singularity at x 0 , i.e. (1.18) If Φ is a C 1 -diffeomorphism and µ a probability measure, we denote by Φ#µ the pushforward of µ by Φ, which is by definition such that for A ⊂ R Borel, If A ⊂ R we denote byÅ its interior. For k ≥ 0, and U some bounded domain in R, we endow the spaces C k (U ) with the usual norm If z is a complex number, we denote by R(z) (resp. I(z)) its real (resp. imaginary) part. For any probability measure µ on R we denote by h µ the logarithmic potential generated by µ, defined as the map 2.1. The next-order energy. For any probability measure µ, let us define, where △ denotes the diagonal in R × R.
We have the following splitting formula for the energy, as introduced in [SS15] (we recall the proof in Section B.1).
Lemma 2.1. For any X N ∈ R N , it holds that Using this splitting formula (2.2), we may re-write P V N,β as with a next-order partition function We extend this notation to K N,β (µ, ζ) where µ is a probability density and ζ is a confinement potential.
In view of (2.2), we have 2.2. Expansion of the next order partition function. If µ is a probability density, we denote by Ent(µ) the entropy function given by 4 (2.6) Ent(µ) :=ˆR µ log µ.
Lemma 2.2. Let µ be a probability density on R. Assume that µ has the form (1.10), (1.11) with S 0 in C 2 (Σ), and that ζ is some Lipschitz function on R satisfying Then, with the notation of (2.4) and for some C β depending only on β, we have

Exponential moments of the energy and the fluctuations.
In this paragraph we show that the next-order energy is typically (in a strong sense) of order at most N , and that the fluctuations of a function in C 1 c (R) are of order at most √ N .

Exponential moments of the next-order energy.
Lemma 2.3. We have, for some constant C depending on β and V Proof. This follows e.g. from [SS15, Theorem 6], but we can also deduce it from Lemma 2.2. We may write Taking the log and using (2.7) to expand both terms up to order N yields the result.
Combining this result with Lemma 2.8 and using Hölder's inequality, we deduce the following concentration result, improving on the previous concentration estimates in Corollary 2.5 (Exponential moments of the fluctuations). For any ξ compactly supported and Lipschitz function, if ξ H 1 (R) is small enough depending on β, we have where C depends on β and V .
In view of the CLT result, one would expect to find concentration bounds in terms of the H 1/2 norm of ξ, but we do not pursue this goal here.

Confinement bound.
We will also need the following bound on the confinement. This is a well-known fact, an easy proof can for instance be given by following that of Lemma 3.3 of [LS16].

Lemma 2.6. For any fixed open neighborhood
Lemma 2.6 is the only place where we use the non-degeneracy assumption (H3) on the next-order confinement term ζ V .
3. Inverting the operator and defining the approximate transport The goal of this section is to find transport maps φ t for t small enough such that the transported measure φ t #µ 0 approximates the equilibrium measure associated to V t := V + tξ. Since the equilibrium measures are characterized by (1.7) with equality on the support, it is natural to search for φ t such that the quantitŷ is close to a constant. This is directly related to inverting the operator Ξ V of (1.12), and we will see that this choice allows to cancel out some crucial terms later.

Preliminaries.
Lemma 3.1. We have the following In particular, (3.1) quantifies how fast ζ ′ V vanishes near an endpoint of the support. We postpone the proof to Section B.3.
3.2. The approximate equilibrium measure equation. In the following, we let • U be an open neighborhood of Σ V such that (3.1) holds.
• B be the open ball of radius 1 2 in C 2 (U ). We define a map F from [−1, 1] × B to C 1 (U ) by setting φ := Id + ψ and

Lemma 3.2. The map F takes values in C 1 (U ) and has continuous partial derivatives in both variables. Moreover there exists C depending only on
where Ξ V is as in (1.12).
The proof is postponed to Section B.4.

Inverting the operator.
Lemma 3.3. Let ψ be defined by for some constant C depending only on V , and there exists a constant c ξ such that The proof of Lemma 3.3 is postponed to Section B.5. In view of our assumptions, ψ is in C 3 (U ) and we may extend it to R in such a way that it is in C 3 (R) with compact support.
3.4. Transport and approximate equilibrium measure. We let ψ be the function defined in Lemma 3.3, and c ξ be such that We let • We let ψ t be given by ψ t := tψ.
• We let φ t be the transport, defined by φ t := Id + ψ t .
• We letζ t be the approximate confining termζ t := ζ V • φ −1 t . Finally, we let τ t be defined by Lemma 3.5. Under our assumptions, the following holds Proof. The first two points are straightforward, the bound (3.9) follows from (3.3) and the definitions.
In the sequel, we will use the fact that the result of Lemma 2.6 allows us to assume that the points of X N all belong to the neighborhood U for t small enough, except for an event of exponentially small probability.

Study of the Laplace transform
We now follow the standard approach of reexpressing the Laplace transform of fluctuations in terms of a ratio of partition functions, and combine it with the change of variables approach, in the following central computation. βN , and assume that |t| ≤ t max . We have The Error term satisfies, for any fixed u To prove this result, we will use some auxiliary computations, whose proof is in Appendix B.

Lemma 4.2. For any bounded continuous function h we have
For ψ defined in Lemma 3.3, we have Proof of Proposition 4.1. By definition of Fluct N and in view of (2.3) we have (4.8) Let us now make the change of variables x i = φ t (y i ) with φ t = Id + tψ where ψ is the map given in Lemma 3.3. We obtain Let us now focus on the exponent in the right-hand side. First, since ψ, hence φ t , is C 1 we may reinsert the diagonal terms and write Expanding around N µ V , we may next write where T 0 , T 1 , T 2 are as follows Next, we note that the T 2 term is independent of the configuration, and we Taylor expand it as t → 0 using that φ t = Id + tψ. We may write that with ε t (x, y) L ∞ (R×R) ≤ C ψ 3 C 1 and expand all other terms to find (4.16) Applying (4.6) to ψ and using (4.7), we find We turn next to the T 1 term, which can be rewritten 5 in view of (3.8) as (4.18) with τ t C 1 (U ) ≤ Ct 2 ψ 2 C 2 (U ) as in (3.9). Thus, using Corollary 2.5 we get for any fixed u For the T 0 term, we use (4.15) to write (4.20) with ε C 2 (R×R) ≤ C ψ 2 C 3 . Applying the result of Proposition 2.4 twice and using (2.8) we find that for any fixed u and |t| ≤ t max , Combining (4.9), (4.11), (4.17), (4.18), (4.20), we obtain that . Combining the estimates (4.19), (4.21) and using the Cauchy-Schwarz inequality, we see The following lemma shows that we can treat´log φ ′ t dfluct N in the right-hand side of (4.2) as an error term.
Proof. It follows from applying Corollary 2.5 to the map log φ ′ t . Next, we deal with the term A[ X N , ψ] in (4.2).

4.2.
First control on the anisotropy term. With Proposition 4.1 at hand, the only thing that remains to elucidate is the behavior of the exponential moments of A[ X N , ψ], which we call the anisotropy. In particular we will show that these are o(1).
Using concentration bounds, more precisely applying Proposition 2.4 twice together with (2.8), we obtain a first bound Lemma 4.4. For |t| ≤ t max we have Proof. Let us write where we let It is clear that Using Proposition 2.4 twice, we can thus write In view of (2.8) and (4.28), we deduce that This shows that the exponential moments of the anisotropy yield bounded terms.
In view of (3.6), we can bound ψ C n ≤ ξ C 2k+1+n for any n, hence we obtain We may re-write the right-hand side as a less sharp but simpler bound.
Corollary 4.5. Under the assumptions of Theorem 1 we have for any s such that 2|s| where C depends only on β and V .
The estimate (4.30) shows that fluctuations of a smooth enough test function are typically of order 1, which is an improvement on the a priori bound (2.10) but does not yield a CLT. Let us observe that the only error term of order 1 comes from (4.29), which was derived by treating A[ X N , ψ] as a fluctuation and using the a priori bound.
In the one-cut, non-critical case, this argument can be bootstrapped, as described in Appendix A: roughly speaking we use the new control (4.30) instead of (2.10) to estimate the exponential moments of A[ X N , ψ], and improve (4.29) by a factor N . The contribution of A[ X N , ψ] in (4.2) becomes of lower order and Proposition 4.1 yields the desired convergence of Laplace transforms. This is a standard technique, see e.g. the recursion of [BG13b], and can be implemented in the one-cut, non-critical case because the operator Ξ V is invertible. In the multi-cut or critical cases, however, we only know how to invert the operator Ξ V under the extra conditions on the test function.
We then use a different way to show that the exponential moments of A are in fact smaller than (4.29), by leveraging on the expansion of log Z V N,β of [LS15] quoted in Lemma 2.2. Indeed, comparing (4.1) to (4.2), we observe that the expansion of log Z Vt N,β −log Z V N,β provides another way of evaluating the exponential moments of A. More precisely, we will use the expansion of log K N,β (μ t ,ζ t ) − log K N,β (µ V , ζ V ) whereμ t is the approximate equilibrium measure obtained by pushing forward µ V by Id + tψ.

5.
Smallness of the anisotropy term and proof of Theorem 1 5.1. Comparison of partition functions by transport.
N,β be the probability measure

Lemma 5.2 (Comparison of energies). Assume ψ ∈ C 3 (R). For any X
We may then recognize the term T 0 in (4.14) and use (4.20) and Proposition 2.4 to conclude.

Lemma 5.3 (Comparison of partition functions). We have, for any t small enough
with error terms bounded by Proof. Starting from (2.4), by a change of variables and in view of (5.2), we may write where the Error 1 term is bounded as in (5.4). We may finally write with an Error 2 term as in (5.5), since this term is the same as the one arising in (4.13). Finally, since by definition φ t #µ V =μ t we may observe that φ ′ t = µ Ṽ µt•φt and thus This yields (5.3).

Conclusion: proof of Theorem 1.
Proof of Theorem 1. Combining (4.2) and (4.24) for t = − 2s βN (where s is independent of N ) and (5.9), together with the Cauchy-Schwarz inequality, we find Letting N → ∞, we obtain, and the rate of convergence is uniform for s in a compact set of R.
Thus the Laplace transform of Fluct N (ξ) converges (uniformly on compact sets) to that of a Gaussian of mean m ξ and variance v ξ , which implies convergence in law and proves the main theorem.
Appendix A. The one-cut regular case In the one-cut noncritical case, every regular enough function is in the range of the operator Ξ, so that the map ψ can always be built. This allows to bootstrap the approach used for proving Theorem 1. In this appendix, we expand on how we can proceed in this simpler setting without refering to the result of [LS15] but assuming more regularity of ξ, and retrieve the findings of [BG13b] (but without assuming analyticity), as well as a rate of convergence for the Laplace transform of the fluctuations.
A.1. The bootstrap argument. We will consider the whole family P where K N,β (μ t ,ζ t ) is as in (2.4). We will also emphasize the t dependence by writing and using similarly the notation Fluct (t) and A (t) . Let us first explain the main computational point for the bootstrap argument. Differentiating (4.2) with respect to t and using (4.5), we obtain Note that here all the error terms in (4.2) have disappeared because they were in factor of t 2 . Also this is true as well for all t ∈ [−t max , t max ], i.e.
We may in addition write that This provides a functional equation which gives the expectation of the fluctuation in terms of a constant term plus a lower order expectation of another fluctuation and the A term (which itself can be written as a fluctuation, as noted below), allowing to expand it in powers of 1/N recursively.
A.2. Improved control on the fluctuations. Assuming from now on that n = 0 and m = 0 so that every regular function is in the range of Ξ V , sinceμ t is the push forward of µ V by a regular map, it is also one-cut, thus all the results proved thus far remain true for P (t) N,β and for any regular enough test function ξ. Thanks to this, we can upgrade the control of exponential moments given in Corollary 4.5 into the control of a weak norm of Fluct where C depends only on V .
Proof. The proof is inspired by [AKM17], in particular we start from [AKM17, Prop. D.1] which states that On the other hand we may easily check that, letting ξ x,r := Φ(r, x − ·), we have Applying the result of Corollary 4.5 to ξ x,r gives us a control on the second moment of Fluct Inserting into (A.6) and (A.7), we are led to Since U is bounded, the right-hand side can be bounded by C´1 0 r α−1 (1 + r −27/2 ) dr, which converges if α > 27/2.
A.3. Proof of Theorem 2. First, by (5.6) and in view of Lemma 5.2, we may write Similarly, we have for all t Indeed,μ t has the same regularity as µ V .
For any test function φ(x, y) we may writë and so by the result of Lemma A.1, we find We may return to (4.26) and, using (A.10), write that On the other hand, by differentiating (4.30) applied with ξ = d dt log φ ′ t , we have Inserting (5.8) and (A.11) and (A.12), (A.2) into (A.9), and integrating between 0 and t = −2s/N β, we obtain (A.13) Comparing (A.13) with (5.3), we obtain Using the bounds of (5.4)-(5.5) and the Cauchy-Schwarz inequality, we deduce that This can be inserted in place of (5.9) into (4.2) yields Taking α = 14, this proves Theorem 2.
A.4. Iteration and expansion of the partition function to arbitrary order. Let V, W be two C ∞ potentials, such that the associated equilibrium measures µ V , µ W satisfy our assumptions with n = 0, m = 0. In this section, we explain how to iterate the procedure described above to obtain a relative expansion of the partition function, namely an expansion of log Z W N,β − log Z V N,β to any order of 1/N . Up to applying an affine transformation to one of the gases, whose effect on the partition function is easy to compute, we may assume that µ V and µ W have the same support Σ, which is a line segment.
Since V, W are C ∞ and µ V , µ W have the same support and a density of the same form (1.10) which is C ∞ on the interior of Σ, the optimal transportation map (or monotone rearrangement) φ from µ V to µ W is C ∞ on Σ and can be extended as a C ∞ function with compact support on R. We let ψ := φ − Id, which is smooth, and for t ∈ [0, 1] the map φ t := Id + tψ is a C ∞ -diffeomorphism, by the properties of optimal transport. We letμ t := φ t #µ V as before.
We can integrate (A.9) to obtain The integral on the right-hand side is of order 1, and we claim that the terms in the integral can actually be computed and expanded up to an error O(1/N ) using the previous lemma. This is clear for the term E P (t) can on the other hand be deduced from the knowledge of the covariance structure of the fluctuations. Let F denote the Fourier transform. In view of (4.26), using the identity and the Fourier inversion formula we may write On the other hand, let ϕ s,λ be the map associated to e isλ· by Lemma 3.3. Separating the real part and the imaginary part we may use the results of the previous subsection to e isλ· and obtain By polarization of the expression for the variance (see (1.16)) and linearity Letting N → ∞, we may then find the expansion up to O(1/N ) of E P (t) . Inserting it into the integral gives a relative expansion to order 1/N of the (logarithm of the) partition function log K N,β . This procedure can then be iterated to yield a relative expansion to arbitrary order of 1/N as desired. Proof. Denoting △ the diagonal in R × R we may write We now recall that ζ V was defined in (1.7), and that ζ V = 0 in Σ V . With the help of this we may rewrite the medium line in the right-hand side of (B.1) as The last equalities are due to the facts that ζ V vanishes on the support of µ V and that fluct N has a total mass 0 since µ V is a probability measure. We may also notice that since µ V is absolutely continuous with respect to the Lebesgue measure, we may include the diagonal back into the domain of integration. By that same argument, one may recognize in the first line of the right-hand side of (B.1) the quantity N 2 I V (µ V ).
B.2. Proof of Proposition 2.4. We follow the energy approach introduced in [SS15,PS14], which views the energy as a Coulomb interaction in the plane, after embedding the real line in the plane. We view R as identified with R × {0} ⊂ R 2 = {(x, y), x ∈ R, y ∈ R}. Let us denote by δ R the uniform measure on R × {0}, i.e. such that for any smooth ϕ(x, y) (with Given (x 1 , . . . , x N ) in R N , we identify them with the points (x 1 , 0), . . . , (x N , 0) in R 2 . For a fixed X N and a given probability density µ we introduce the electric potential H µ N by Next, we define versions of this potential which are truncated hence regular near the point charges. For that let δ (η) x denote the uniform measure of mass 1 on ∂B(x, η) (where B denotes an Euclidean ball in R 2 ). We define H µ N,η in R 2 by These potentials make sense as functions in R 2 and are harmonic outside of the real axis.
Lemma B.1. For any probability density µ, X N in R N and η in (0, 1), we have Proof. First we notice that´R 2 |∇H N,η | 2 is a convergent integral and that Indeed, we may choose R large enough so that all the points of X N are contained in the ball B R = B(0, R). By Green's formula and (B.4), we have In view of the decay of H N and ∇H N , the boundary integral tends to 0 as R → ∞, and so we may writeˆR and thus (B.6) holds. We may next write We have used the fact that for any as follows from a direct computation of Newton's theorem.
Let us now observe that´− log |x − y|δ (η) x i (y), the potential generated by δ (η) x i is equal tó − log |x − y|δ x i outside of B(x i , η), and smaller otherwise. Since its Laplacian is −2πδ (η) x i , a negative measure, this is also a superharmonic function, so by the maximum principle, its value at a point x j is larger or equal to its average on a sphere centered at x j . Moreover, outside B(x i , η) it is a harmonic function, so its values are equal to its averages. We deduce from these considerations, and reversing the roles of i and j, that for each i = j, We may also obviously writê We conclude that the second term in the right-hand side of (B.8) is nonpositive, equal to 0 if all the balls are disjoint, and bounded below by i =j log |x i − x j |1 |x i −x j |≤2η . Finally, by the above considerations, since´− log |x− y|δ (η) x i coincides with´− log |x− y|δ x i outside B(x i , η), we may rewrite the last term in the right-hand side of (B.8) as this last term is bounded by 2 µ L ∞ N 2 η. Combining with all the above results yields the proof.
It is easy to check that for any (x, y), andχ is supported in an horizontal stripe of width 1. Letting #I denote the number of balls B(x i , η) intersecting the support of ξ, we have (with where we have bounded #I by N in the last inequality. In view of (B.4), we also have (B.12) Combining (B.10), (B.11) and (B.12), we obtain B.3. Proof of Lemma 3.1.
Proof. Since µ V minimizes the logarithmic potential energy (1.6), for any bounded continuous function h, (4.6) holds. Of course, an identity like (4.6) extends to complex-valued functions, and applying it to h = 1 z−· for some fixed z ∈ C \ Σ V leads to (B.14) and L is defined by Solving (B.14) for G yields As is well-known, since µ V is continuous on Σ V , the quantity − 1 π I(G(x + iε)) converges towards the density µ V (x) as ε → 0 + , hence we have for x in Σ V This proves that µ V has regularity C p−2 at any point where it does not vanish. Assuming the form (1.11) for S, we also deduce that the function S 0 has regularity at least C p−3−2k on Σ V .
Applying (B.17) on R \ Σ, we obtain and the left-hand side is equal to ζ ′ (x). Using (1.11), (B.18) and the fact that V is regular, we may find a neighborhood U small enough such that ζ ′ does not vanish on U \ Σ V and on which we can write ζ ′ as in (3.1).

B.4. Proof of Lemma 3.2.
Proof. We first prove that the image of F is indeed contained in C 1 (U ).
For (t, ψ) = (0, 0), we have indeed F(0, 0) = ζ V + c and ζ V is in C 1 (R) by the regularity assumptions on V . We may also write and since ψ C 2 (U ) ≤ 1/2, the second and third terms are also in C 1 (U ). Next, we compute the partial derivatives of F at a fixed point (t 0 , ψ 0 ) ∈ [−1, 1] × B. It is easy to see that ∂F and the map (t 0 , ψ 0 ) → ξ • φ 0 is indeed continuous. The Fréchet derivative of F with respect to the second variable can be computed as follows where ε t 0 ,ψ 0 (ψ 1 ) is given by By differentiating twice inside the integral we get the bound with a constant depending on V . It implies that and we can check that this expression is also continuous in (t 0 , ψ 0 ). In particular, we may observe that Finally, we prove the bound (3.3). For any fixed (t, ψ) ∈ [−1, 1] × B, we write with φ s = Id + sψ. It is straightforward to check that To control the second term inside the integral we write and we obtain ∂F ∂ψ (st,sψ) [ψ] − ∂F ∂ψ (0,0) [ψ] We now use that In the second and the fourth line, we used Leibniz formula . In the last line we used that s(ψ(·) − ψ(y))/(· − y) is uniformely bounded by 1/2 in C 2 (U ) so its composition with the function x → 1/(1 + x) is bounded in C 2 (U ). We conclude by checking that where the integral is now a definite Riemann integral. From (B.26) we deduce that the map ψ 0 σ is of class C r−1 inΣ V and extends readily to a C r−1 function on Σ V . For d = 0, . . . , r − 1 and for x ∈ Σ V , we compute that In particular, if conditions (1.15) hold, in view of Lemma 3.1 the map extends to a function of class (p − 3 − 2k) ∧ (r − 1 − k), hence C 2 on Σ V , and in view of (B.24) it satisfies Ξ V [ψ] = ξ 2 + c ξ on Σ V . Now, we define ψ outside Σ V . By definition, for x outside Σ V , the equation can be written as and thus the choice (3.5) ensures that Ξ V [ψ] = 1 2 ξ +c ξ . Moreover, ψ is clearly of class C r∧(p−1) on R \ Σ V . It remains to check that ψ has the desired regularity at the endpoints of Σ V . Let us considerψ an extension of ψ in C l with l := (p − 3 − 2k) ∧ (r − 1 − k), which coincides with ψ on Σ V (given for instance by a Taylor expansion at the endpoints). As ψ andψ are equal on the support we can rewrite (3.5) aś ψ(y) .

B.6. Proof of Lemma 4.2.
Proof. The first item is a consequence of the fact that µ V minimizes the logarithmic potential energy (1.6) and hence as is well-known´− log | · −y| dµ V (y) + 1 2 V is constant on the support of µ V . Differentiating this and integrating against hdµ V gives the result. For the second relation, by definition of ψ we have and thus Integrating both sides against ψµ V yieldŝ Using (4.6) for the second term we obtain We may then combine the first two terms in the right-hand side to obtain (4.7).