CLT for fluctuations of β-ensembles with general potential
aa r X i v : . [ m a t h - ph ] F e b CLT FOR FLUCTUATIONS OF β -ENSEMBLES WITH GENERALPOTENTIAL FLORENT BEKERMAN, THOMAS LEBL´E, AND SYLVIA SERFATY
Abstract.
We prove a central limit theorem for the linear statistics of one-dimensionallog-gases, or β -ensembles. We use a method based on a change of variables which allows totreat fairly general situations, including multi-cut and, for the first time, critical cases, andgeneralizes the previously known results of Johansson, Borot-Guionnet and Shcherbina. Inthe one-cut regular case, our approach also allows to retrieve a rate of convergence as wellas previously known expansions of the free energy to arbitrary order. keywords: β -ensembles, Log Gas, Central Limit Theorem, Linear statistics. MSC classification:
Introduction
Let β > N ≥
1, we are interested in the N -point canonical Gibbs measure for a one-dimensional log-gas at the inverse temperature β , defined by(1.1) d P VN,β ( ~X N ) = 1 Z VN,β exp (cid:18) − β H VN ( ~X N ) (cid:19) d ~X N , where ~X N = ( x , . . . , x N ) is an N -tuple of points in R , and H VN ( ~X N ), defined by(1.2) H VN ( ~X N ) := X ≤ i = j ≤ N − log | x i − x j | + N X i =1 N V ( x i ) , is the energy of the system in the state ~X N , given by the sum of the pairwise repulsivelogarithmic interaction between all particles plus the effect on each particle of an externalfield 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 VN,β in the definition (1.1) is thenormalizing constant, called the partition function , and is equal to Z VN,β := ˆ R N exp (cid:18) − β H VN ( ~X N ) (cid:19) d ~X N . Such systems of particles with logarithmic repulsive interaction on the line have been exten-sively 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 particlesconverges almost surely to some deterministic probability measure on R called the equilibrium Date : Thursday 8 th February, 2018. We use β 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 (1.3) fluct N := N X i =1 δ x i − N µ V , 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(1.4) Fluct N ( ξ ) := ˆ R ξ d fluct N . The goal of this paper is to prove a Central Limit Theorem (CLT) for Fluct N ( ξ ), under someregularity assumptions on V and ξ .1.1. Assumptions.(H1) - Regularity and growth of V : The potential V is in C p ( R ) and satisfies the growthcondition(1.5) lim inf | x |→∞ V ( x )2 log | x | > . It is well-known, see e.g. [ST13], that if V satisfies (H1) with p ≥
0, then the logarithmicpotential energy functional defined on the space of probability measures by(1.6) I V ( µ ) = ˆ R × R − log | x − y | dµ ( x ) dµ ( y ) + ˆ R V ( x ) dµ ( x )has a unique global minimizer µ V , the aforementioned equilibrium measure . This measurehas a compact support that we will denote by Σ V , and µ V is characterized by the fact thatthere exists a constant c V such that the function ζ V defined by(1.7) ζ V ( x ) := ˆ − log | x − y | dµ V ( y ) + V ( x )2 − c V satisfies the Euler-Lagrange conditions(1.8) ζ V ≥ R , ζ V = 0 on Σ V . 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 . (H2) - Form of the equilibrium measure: The support Σ V of µ V is a finite union of n +1non-degenerate intervalsΣ V = [ ≤ l ≤ n [ α l, − ; α l, + ] , with α l, − < α l, + .The points α l, ± are called the endpoints of the support Σ V . For x in Σ V , we let(1.9) σ ( x ) := n Y l =0 q | x − α l, − || x − α l, + | . We assume that the equilibrium measure has a density with respect to the Lebesguemeasure on Σ V given by(1.10) µ V ( x ) = S ( x ) σ ( x ) , LT FOR FLUCTUATIONS OF β -ENSEMBLES WITH GENERAL POTENTIAL 3 where S can be written as(1.11) S ( x ) = S ( x ) m Y i =1 ( x − s i ) k i , S > V , where m ≥
0, all the points s i , called singular points , belong to Σ V and the k i arenatural integers. (H3) - Non-criticality of ζ V : The function ζ V is positive on R \ Σ V .1.2. Main result.Definition 1.1.
We introduce the so-called master operator Ξ V , which acts on C functionsby (1.12) Ξ V [ ψ ] := − ψV ′ + ˆ ψ ( · ) − ψ ( y ) · − y dµ V ( y ) . Theorem 1 (Central limit theorem for fluctuations of linear statistics) . Let ξ be a functionin C r ( R ) , assume that (H1), (H2), (H3) hold. We let k = max i =1 ,..., m k i , where the k i ’s are as in (1.11) . Assume that (1.13) p ≥ (3 k + 6) , r ≥ (2 k + 4) , where p (resp. r ) denotes the regularity of V (resp. ξ )If n ≥ , assume that ξ satisfies the n following conditions (1.14) ˆ Σ V ξ ( y ) y d σ ( y ) dy = 0 for d = 0 , . . . , n − . Moreover, if k ≥ , assume that for all i = 1 , . . . , m (1.15) ˆ Σ V ξ ( y ) − R s i ,d ξ ( y ) σ ( y )( y − s i ) d dy = 0 for d = 1 , . . . , k i , where R x,d ξ is the Taylor expansion of ξ to order d − around x given by R x,d ξ ( y ) = ξ ( x ) + ( y − x ) ξ ′ ( x ) + · · · + ( y − x ) d − ( d − ξ ( d − ( x ) . Then there exists a constant c ξ and a function ψ of class C in some open neighborhood U of Σ V such that Ξ V [ ψ ] = ξ + c ξ on U , and the fluctuation Fluct N ( ξ ) converges in law as N → ∞ to a Gaussian distribution with mean m ξ = (cid:18) − β (cid:19) ˆ ψ ′ dµ V , and variance v ξ = − β ˆ ψξ ′ dµ V . Let us emphasize that a singular point s i can be equal to an endpoint α l, ± . FLORENT BEKERMAN, THOMAS LEBL´E, AND SYLVIA SERFATY
It is proven in (4.7) that the variance v ξ has the equivalent expression(1.16) v ξ := 2 β ¨ (cid:18) ψ ( x ) − ψ ( y ) x − y (cid:19) dµ V ( x ) dµ V ( y ) + ˆ V ′′ ψ dµ V ! . 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 ex-presses 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 neighbor-hood 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 ≥ 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 ex-amples 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]. • If V is uniformly convex and smooth, then the assumptions are satisfied with n = 0, m = 0, and S smooth on Σ V , see e.g. [BdMPS95, Example 1]. • Examples of multi-cut, non-critical situations with n = 0 , , m = 0, are men-tioned in [BdMPS95, Examples 3-4]. • An example of criticality at the edge of the support is given by V ( x ) = x − x + x + x , for which the equilibrium measure, as computed in [CKI10, Example 1.2],is given by µ V ( x ) = 110 π q | x − ( − || x − | ( x − [ − , ( x ) . • An example of criticality in the bulk of the support is given by V ( x ) = x − x , forwhich the equilibrium measure, as computed in [CK06], is µ V ( x ) = 12 π q | x − ( − || x − | ( x − [ − , ( x ) . Following the terminology used in the literature [DKM +
99, KM00, CK06], we may say thatour assumptions allow the existence of singular points of type II (the density vanishes in thebulk) 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 whichthe behavior might be quite different, see [Cla08, Mo08].1.4.
Existing literature, strategy and perspectives.
Connection to previous results.
The CLT for fluctuations of linear statistics in thecontext of β -ensembles was proven in the pioneering paper [Joh98] for polynomial potentialsin the case n = 0 , m = 0, and generalized in [Shc13] to real-analytic potentials in the possiblymulti-cut, non-critical cases ( n ≥ , m = 0), where a set of n necessary and sufficient conditionson a given test function in order to satisfy the CLT is derived. If these conditions arenot fulfilled, the fluctuations are shown to exhibit oscillatory behaviour. Such results arealso a by-product of the all-orders expansion of the partition function obtained in [BG13b] LT FOR FLUCTUATIONS OF β -ENSEMBLES WITH GENERAL POTENTIAL 5 ( n = 0 , m = 0) and [BG13a] ( n ≥ , m = 0). A CLT for the fluctuations of linear statisticsfor test functions living at mesoscopic scales was recently obtained in [BL16]. Finally, a newproof of the CLT in the one-cut non-critical case was very recently given in [LLW17]. It isbased 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 simpleproof of the CLT using a change of variables argument, retrieving the results cited above. Onthe other hand, our method allows to substantially relax the assumptions on V , in particularfor the first time we are able to treat critical situations where m ≥ N ( ξ )leads to working with a new potential V + tξ (with t small), and thus to consideringthe associated perturbed equilibrium measure.(2) Following [LS16], our method then consists in finding a change of variables (or atransport map) that pushes µ V onto the perturbed equilibrium measure. In fact wedo not exactly achieve this, but rather we construct a transport map I + tψ , whichis a perturbation of identity, and consider the approximate perturbed equilibriummeasure ( I + tψ ) µ V . The map ψ is found by inverting the operator (1.12), whichis well-known in this context, it appears e.g. in [BG13b, BG13a, Shc13, BFG13]. ACLT will hold if the function ξ is (up to constants) in the image of Ξ V , leading tothe conditions (1.14)–(1.15). The change of variables approach for one-dimensionallog-gases was already used e.g. in [Shc14, BFG13], see also [GMS07, GS14] which dealwith the non-commutative context.(3) The proof then leverages on the expansion of log Z VN,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 Cram´er-Wold theorem, the result of The-orem 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 bilinearform 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 necessaryand sufficient conditions on the test function, and the non-Gaussian nature of the fluctuationsif these conditions are not satisfied is explicitly described. In the critical cases, we only statesufficient conditions (1.15) under which the CLT holds. It would be interesting to provethat these conditions are necessary, and to characterize the behavior of the fluctuations forfunctions which do not satisfy (1.15).Finally, we expect Theorem 1 to hold also at mesoscopic scales. The proof of [BL16] usesthe rigidity estimates of [BEY14] which are, to the best of our knowledge, not available tothe critical case.1.5. The one-cut noncritical case.
In the case n = 0 and m = 0, following the transportapproach, we can obtain the convergence of the Laplace transform of fluctuations with anexplicit rate, under the assumption that ξ is very regular (we have not tried to optimize inthe regularity): FLORENT BEKERMAN, THOMAS LEBL´E, AND SYLVIA SERFATY
Theorem 2 (Rate of convergence in the one-cut noncritical case) . Under the assumptions ofTheorem 1, if in addition n = 0 , m = 0 , p ≥ and r ≥ , then we also have, for any s suchthat | s | N is small enough (1.17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) log E P VN,β [exp( s Fluct N ( ξ ))] + (cid:18) − β (cid:19) sβ ˆ ψ ′ dµ V + s β ˆ ξ ′ ψdµ V (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = sN O (cid:16) k ψ k C ( U ) + k ψ k C ( U ) + k ψ k C ( U ) + s k ψ k C + √ N k ψ k C (cid:17) . where the constant C depends only on V and β . These additional assumptions allow to avoid using the result of [LS15] on the expansionof log Z VN,β . Our transport approach also provides a functional relation on the expectationof fluctuations which allows by a boostrap procedure to recover an expansion of log Z VN,β (relative to a reference potential) to arbitrary powers of 1 /N in very regular cases, i.e theresult of [BG13b] but without the analyticity assumption. All these results are presented inAppendix A.1.6. Some notation.
We denote by
P.V. the principal value of an integral having a singu-larity at x , i.e.(1.18) P.V. ˆ f = lim ε → ˆ x − ε −∞ f + ˆ + ∞ x + ε f. If Φ is a C -diffeomorphism and µ a probability measure, we denote by Φ µ the push-forward of µ by Φ, which is by definition such that for A ⊂ R Borel,(Φ µ )( A ) := µ (Φ − ( A )) . If A ⊂ R we denote by ˚ A its interior.For k ≥
0, and U some bounded domain in R , we endow the spaces C k ( U ) with the usualnorm k ψ k C k ( U ) := k X j =0 sup x ∈ U | ψ ( j ) ( x ) | . 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 generatedby µ , defined as the map(1.19) x ∈ R h µ ( x ) = ˆ − log | x − y | dµ ( y ) . Acknowledgments:
We would like to thank Alice Guionnet for suggesting the problemand for helpful discussions.2.
Next order energy and concentration bounds
We start with the energy splitting formula of [SS15] that separates fixed leading orderterms from variable next order ones, and allows to quickly obtain first concentration bounds. Depending only on ξ . LT FOR FLUCTUATIONS OF β -ENSEMBLES WITH GENERAL POTENTIAL 7 The next-order energy.
For any probability measure µ , let us define,(2.1) F N ( ~X N , µ ) = − ¨ ( R × R ) \△ log | x − y | (cid:16) N X i =1 δ x i − µ (cid:17) ( x ) (cid:16) N X i =1 δ x i − µ (cid:17) ( y ) , where △ denotes the diagonal in R × R .We have the following splitting formula for the energy, as introduced in [SS15] (we recallthe proof in Section B.1). Lemma 2.1.
For any ~X N ∈ R N , it holds that (2.2) H VN ( ~X N ) = N I V ( µ V ) + 2 N N X i =1 ζ V ( x i ) + F N ( ~X N , µ V ) . Using this splitting formula (2.2), we may re-write P VN,β as(2.3) d P VN,β ( ~X N ) = 1 K N,β ( µ V , ζ V ) exp − β F N ( ~X N , µ V ) + 2 N N X i =1 ζ V ( x i ) !! d ~X N , with a next-order partition function K N,β ( µ V , ζ V ) defined by(2.4) K N,β ( µ V , ζ V ) := ˆ R N exp − β F N ( ~X N , µ V ) + 2 N N X i =1 ζ V ( x i ) !! d ~X N . We extend this notation to K N,β ( µ, ζ ) where µ is a probability density and ζ is a confinementpotential.In view of (2.2), we have(2.5) Z VN,β = exp (cid:18) − β I V ( µ V ) (cid:19) K N,β ( µ V , ζ V ) . Expansion of the next order partition function. If µ is a probability density, wedenote by Ent( µ ) the entropy function given by (2.6) Ent( µ ) := ˆ R µ log µ. The following asymptotic expansion is proven [LS15, Corollary 1.1] (cf. [LS15, Remark 4.3])and valid in a general multi-cut critical situation.
Lemma 2.2.
Let µ be a probability density on R . Assume that µ has the form (1.10) , (1.11) with S in C (Σ) , and that ζ is some Lipschitz function on R satisfying ζ = 0 on Σ , ζ > on R \ Σ , ˆ R e − βNζ ( x ) dx < ∞ for N large enough . Then, with the notation of (2.4) and for some C β depending only on β , we have (2.7) log K N,β ( µ, ζ ) = β N log N + C β N − N (cid:18) − β (cid:19) Ent( µ ) + N o N (1) . Exponential moments of the energy and the fluctuations.
In this paragraph weshow that the next-order energy is typically (in a strong sense) of order at most N , and thatthe fluctuations of a function in C c ( R ) are of order at most √ N . The sign convention here differs from the usual one.
FLORENT BEKERMAN, THOMAS LEBL´E, AND SYLVIA SERFATY
Exponential moments of the next-order energy.
Lemma 2.3.
We have, for some constant C depending on β and V (2.8) (cid:12)(cid:12)(cid:12)(cid:12) log E P VN,β (cid:20) exp (cid:18) β (cid:16) F N ( ~X N , µ V ) + N log N (cid:17)(cid:19)(cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ≤ CN.
Proof.
This follows e.g. from [SS15, Theorem 6], but we can also deduce it from Lemma 2.2.We may write E P VN,β (cid:20) exp (cid:18) β F N ( ~X N , µ V ) (cid:19)(cid:21) = 1 K N,β ( µ V , ζ V ) ˆ exp − β F N ( ~X N , µ V ) − N N X i =1 ζ V ( x i ) !! d ~X N = K N, β ( µ V , ζ V ) K N,β ( µ V , ζ V ) . Taking the log and using (2.7) to expand both terms up to order N yields the result. (cid:3) The next-order energy controls the fluctuations.
The following result is a consequenceof the analysis of [SS15, PS14], we give the proof in Section B.2 for completeness. It showsthat F N controls fluct N . Proposition 2.4. If ξ is compactly supported and Lipschitz, we have, for some universalconstant C (2.9) (cid:12)(cid:12)(cid:12)(cid:12) ˆ ξ d fluct N (cid:12)(cid:12)(cid:12)(cid:12) ≤ k ξ ′ k L ∞ + ( k ξ ′ k L + k ξ k L ) (cid:16) F N ( ~X N , µ V ) + N log N + C ( k µ V k L ∞ + 1) N (cid:17) / . Combining this result with Lemma 2.8 and using H¨older’s inequality, we deduce the fol-lowing concentration result, improving on the previous concentration estimates in √ N log N of [BG13b, MdMS14]. Corollary 2.5 (Exponential moments of the fluctuations) . For any ξ compactly supportedand Lipschitz function, if k ξ k H ( R ) is small enough depending on β , we have (2.10) log E P VN,β [exp (Fluct N ( ξ ))] ≤ C √ N (cid:16) k ξ ′ k L ( R ) + k ξ k L ( R ) ) (cid:17) + C k ξ ′ k L ∞ ( R ) 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 / norm of ξ , but we do not pursue this goal here.2.3.3. Confinement bound.
We will also need the following bound on the confinement. Thisis a well-known fact, an easy proof can for instance be given by following that of Lemma 3.3of [LS16].
Lemma 2.6.
For any fixed open neighborhood U of Σ , P VN,β (cid:16) ~X N ∈ U N (cid:17) ≥ − exp( − cN ) where c > depends on U and β . LT FOR FLUCTUATIONS OF β -ENSEMBLES WITH GENERAL POTENTIAL 9 Lemma 2.6 is the only place where we use the non-degeneracy assumption (H3) on thenext-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 thetransported measure φ t µ 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 isnatural to search for φ t such that the quantity ˆ − log | φ t ( x ) − φ t ( y ) | dµ ( y ) + 12 V t ( φ t ( x ))is close to a constant. This is directly related to inverting the operator Ξ V of (1.12), and wewill see that this choice allows to cancel out some crucial terms later.3.1. Preliminaries.Lemma 3.1.
We have the following • The function S of (1.11) is in C p − − k (Σ V ) . • There exists an open neighborhood U of Σ V and a positive function M in C p − − k ( U \ ˚Σ V ) such that (3.1) ζ ′ V ( x ) = M ( x ) σ ( x ) m Y i =1 ( x − s i ) k i . In particular, (3.1) quantifies how fast ζ ′ V vanishes near an endpoint of the support. Wepostpone 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 in C ( U ).We define a map F from [ − , × B to C ( U ) by setting φ := Id + ψ and(3.2) F ( t, ψ ) := ˆ − log | φ ( · ) − φ ( y ) | dµ V ( y ) + 12 V t ◦ φ ( · ) , Lemma 3.2.
The map F takes values in C ( U ) and has continuous partial derivatives in bothvariables. Moreover there exists C depending only on V such that for all ( t, ψ ) in [ − , × B we have (3.3) (cid:13)(cid:13)(cid:13)(cid:13) F ( t, ψ ) − F (0 , − t ξ + Ξ V [ ψ ] (cid:13)(cid:13)(cid:13)(cid:13) C ( U ) ≤ Ct k ψ k C ( U ) , 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 ψ ( x ) = − π S ( x ) (cid:18) ˆ Σ ξ ( y ) − ξ ( x ) σ ( y )( y − x ) dy (cid:19) for x in Σ V , (3.4) ψ ( x ) = ˆ ψ ( y ) x − y dµ V ( y ) + ξ ( x )2 + c ξ ˆ x − y dµ V ( y ) − V ′ ( x ) for x ∈ U \ Σ V , (3.5) then ψ is in C l ( U ) with l = ( p − − k ) ∧ ( r − − k ) and (3.6) k ψ k C l ( U ) ≤ C k ξ k C r ( R ) for some constant C depending only on V , and there exists a constant c ξ such that Ξ V [ ψ ] = ξ c ξ in U, with Ξ V as in (1.12) . The proof of Lemma 3.3 is postponed to Section B.5. In view of our assumptions, ψ is in C ( U ) and we may extend it to R in such a way that it is in C ( R ) with compact support.3.4. Transport and approximate equilibrium measure.
We let ψ be the function de-fined in Lemma 3.3, and c ξ be such thatΞ V [ ψ ] = ξ c ξ on U. We let(3.7) t max := (cid:16) k ψ k C ( U ) (cid:17) − , Definition 3.4.
For t ∈ [ − t max , t max ] , • We let ψ t be given by ψ t := tψ. • We let ˜ c t := tc ξ . • We let φ t be the transport, defined by φ t := Id + ψ t . • We let ˜ µ t be the approximate equilibrium measure, defined by ˜ µ t := φ t µ V . • We let ˜ ζ t be the approximate confining term ˜ ζ t := ζ V ◦ φ − t .Finally, we let τ t be defined by (3.8) τ t := F ( t, ψ t ) − F (0 , − ˜ c t . Lemma 3.5.
Under our assumptions, the following holds • The map ψ t satisfies Ξ V [ ψ t ] = t ξ + ˜ c t . • The map φ t is a C -diffeomorphism which coincides with the identity outside a com-pact support independent of t ∈ [ − t max , t max ] . • The error τ t is a O ( t ) , more precisely (3.9) k τ t k C ( U ) ≤ Ct k ψ k C ( U ) . LT FOR FLUCTUATIONS OF β -ENSEMBLES WITH GENERAL POTENTIAL 11 Proof.
The first two points are straightforward, the bound (3.9) follows from (3.3) and thedefinitions. (cid:3)
In the sequel, we will use the fact that the result of Lemma 2.6 allows us to assume thatthe points of ~X N all belong to the neighborhood U for t small enough, except for an event ofexponentially small probability.4. Study of the Laplace transform
We now follow the standard approach of reexpressing the Laplace transform of fluctuationsin terms of a ratio of partition functions, and combine it with the change of variables approach,in the following central computation.4.1.
Expansion of the Laplace transform of the fluctuations.Proposition 4.1.
Let s be in R , let t := − sβN , and assume that | t | ≤ t max . We have (4.1) E P VN,β [exp ( s Fluct N ( ξ ))] = exp (cid:18) − sN ˆ ξdµ V (cid:19) Z V t N,β Z VN,β and (4.2) E P VN,β [exp ( s Fluct N ( ξ ))]= exp ( Const ) E P VN,β (cid:18) exp (cid:18) t β A [ ~X N , ψ ] + (cid:18) − β (cid:19) ˆ log φ ′ t d fluct N + Error (cid:19)(cid:19) where we define
Const = − β N t ˆ ξ ′ ψdµ V + tN (cid:18) − β (cid:19) ˆ ψ ′ dµ V , (4.3) A [ ~X N , ψ ] = ¨ R × R ψ ( x ) − ψ ( y ) x − y d fluct N ( x ) d fluct N ( y ) . (4.4) The
Error term satisfies, for any fixed u (4.5) (cid:12)(cid:12)(cid:12)(cid:12) log E P VN,β (exp( u Error )) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C u (cid:16) t N √ N k ψ k C + t N k ψ k C (cid:17) . 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 (4.6) ¨ h ( x ) − h ( y ) x − y dµ V ( x ) dµ V ( y ) = ˆ V ′ ( x ) h ( x ) dµ V ( x ) . For ψ defined in Lemma 3.3, we have (4.7) ˆ ξ ′ ψdµ V = − ¨ (cid:18) ψ ( x ) − ψ ( y ) x − y (cid:19) dµ V ( x ) dµ V ( y ) − ˆ V ′′ ψ dµ V . Proof of Proposition 4.1.
By definition of Fluct N and in view of (2.3) we have(4.8) E P VN,β [exp ( s Fluct N ( ξ ))] = e − sN ´ ξdµ V Z VN,β ˆ R N exp − β H N ( ~X N ) + N t N X i =1 ξ ( x i ) !! d ~X N and (4.1) immediately follows by definition of V t = V + tξ . Let us now make the change of variables x i = φ t ( y i ) with φ t = Id + tψ where ψ is the mapgiven in Lemma 3.3. We obtain(4.9) e sN ´ ξdµ V E P VN,β [exp ( s Fluct N ( ξ ))]= 1 Z VN,β ˆ exp − β − X i = j log | φ t ( x i ) − φ t ( x j ) | + N N X i =1 ( V + tξ )( φ t ( x i )) + N X i =1 log φ ′ t ( x i ) d ~X N = E P VN,β exp − β − X i = j log | φ t ( x i ) − φ t ( x j ) || x i − x j | + N N X i =1 ( V t ( φ t ( x i )) − V ( x i )) − β N X i =1 log φ ′ t ( x i ) . Let us now focus on the exponent in the right-hand side. First, since ψ , hence φ t , is C wemay reinsert the diagonal terms and write(4.10) − X i = j log | φ t ( x i ) − φ t ( x j ) || x i − x j | + N N X i =1 ( V t ( φ t ( x i )) − V ( x i )) − β N X i =1 log φ ′ t ( x i )= − X i,j log | φ t ( x i ) − φ t ( x j ) || x i − x j | + N N X i =1 ( V t ( φ t ( x i )) − V ( x i )) + (cid:18) − β (cid:19) N X i =1 log φ ′ t ( x i ) . Expanding around
N µ V , we may next write(4.11) − X i,j log | φ t ( x i ) − φ t ( x j ) || x i − x j | + N N X i =1 ( V t ( φ t ( x i )) − V ( x i ))+ (cid:18) − β (cid:19) N X i =1 log φ ′ t ( x i ) = T + T + T where T , T , T are as follows T = − N ¨ log | φ t ( x ) − φ t ( y ) || x − y | dµ V ( x ) dµ V ( y ) + N ˆ ( V t ◦ φ t − V ) dµ V (4.12) + N (cid:18) − β (cid:19) ˆ log φ ′ t dµ V T = − N ¨ log | φ t ( x ) − φ t ( y ) || x − y | dµ V ( x ) d fluct N ( y ) + N ˆ ( V t ◦ φ t − V ) d fluct N (4.13) + (cid:18) − β (cid:19) ˆ log φ ′ t d fluct N T = − ¨ log | φ t ( x ) − φ t ( y ) || x − y | d fluct N ( x ) d fluct N ( y ) . (4.14)Next, we note that the T term is independent of the configuration, and we Taylor expandit as t → φ t = Id + tψ . We may write that(4.15) log | φ t ( x ) − φ t ( y ) || x − y | = t ψ ( x ) − ψ ( y ) x − y − t (cid:18) ψ ( x ) − ψ ( y ) x − y (cid:19) + t ε t ( x, y ) LT FOR FLUCTUATIONS OF β -ENSEMBLES WITH GENERAL POTENTIAL 13 with k ε t ( x, y ) k L ∞ ( R × R ) ≤ C k ψ k C and expand all other terms to find(4.16) T N = − t ¨ ψ ( x ) − ψ ( y ) x − y dµ V ( x ) dµ V ( y ) + t ¨ (cid:18) ψ ( x ) − ψ ( y ) x − y (cid:19) dµ V ( x ) dµ V ( y )+ t ˆ V ′ ψ dµ V + t ˆ V ′′ ψ dµ V + t ˆ ξdµ V + t ˆ ξ ′ ψdµ V + tN (cid:18) − β (cid:19) ˆ ψ ′ dµ V + O t k ψ k C + t k ψ k L ∞ k ξ k C + t N k ψ k C ! . Applying (4.6) to ψ and using (4.7), we find(4.17) T = N t ˆ ξdµ V + 12 N t ˆ ξ ′ ψdµ V + tN (cid:18) − β (cid:19) ˆ ψ ′ dµ V + O (cid:16) t N k ψ k C + t N k ψ k L ∞ k ξ k C + t N k ψ k C (cid:17) . We turn next to the T term, which can be rewritten in view of (3.8) as(4.18) T = ˆ (cid:18) N τ t + (cid:18) − β (cid:19) log φ ′ t (cid:19) d fluct N = Fluct N [2 N τ t ] + ˆ (cid:18) − β (cid:19) log φ ′ t d fluct N . with k τ t k C ( U ) ≤ Ct k ψ k C ( U ) as in (3.9). Thus, using Corollary 2.5 we get for any fixed u (4.19) (cid:12)(cid:12)(cid:12)(cid:12) log E P VN,β [exp ( u Fluct N [2 N τ t ])] (cid:12)(cid:12)(cid:12)(cid:12) ≤ C u t N √ N k ψ k C . For the T term, we use (4.15) to write(4.20) T = − t A [ ~X N , ψ ] + t ˆ ε ( x, y ) d fluct N ( x ) d fluct N ( y )with k ε k C ( R × R ) ≤ C k ψ k C . Applying the result of Proposition 2.4 twice and using (2.8) we find that for any fixed u and | t | ≤ t max ,(4.21) (cid:12)(cid:12)(cid:12)(cid:12) log E P N,β (cid:20) exp (cid:18) ut ˆ ε ( x, y ) d fluct N ( x ) d fluct N ( y ) (cid:19)(cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C u t N k ψ k C . Combining (4.9), (4.11), (4.17), (4.18), (4.20), we obtain that(4.22) exp (cid:18) sN ˆ ξdµ V (cid:19) E P VN,β [exp ( s Fluct N ( ξ ))]= exp (cid:18) − β (cid:18) N t ˆ ξdµ V + 12 N t ˆ ξ ′ ψdµ V (cid:19) + tN (cid:18) − β (cid:19) ˆ ψ ′ dµ V (cid:19) E P VN,β (cid:18) exp (cid:18) t β A [ ~X N , ψ ] + (cid:18) − β (cid:19) ˆ log φ ′ t d fluct N + Error (cid:19)(cid:19) , This uses crucially the fact that ψ is chosen to satisfy Ξ V ( ψ ) = ξ + c ξ . with(4.23) Error = − β N [2 N τ t ] − β t ˆ ε ( x, y ) d fluct N ( x ) d fluct N ( y )+ O ( t N k ψ k C + t N k ψ k L ∞ k ξ k C + t N k ψ k C ) . Combining the estimates (4.19), (4.21) and using the Cauchy-Schwarz inequality, we see (cid:12)(cid:12)(cid:12)(cid:12) log E P VN,β (exp( u Error )) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C u (cid:16) t N √ N k ψ k C + t N k ψ k C + t N k ψ k L ∞ k ξ k C + t N k ψ k C + t N k ψ k C (cid:17) , which we may simplify as (cid:12)(cid:12)(cid:12)(cid:12) log E P VN,β (exp( u Error )) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C u (cid:16) t N √ N k ψ k C + t N k ψ k L ∞ k ξ k C + t N k ψ k C (cid:17) . (cid:3) The following lemma shows that we can treat ´ log φ ′ t d fluct N in the right-hand side of (4.2)as an error term. Lemma 4.3.
For any fixed u , we have (4.24) (cid:12)(cid:12)(cid:12)(cid:12) log E P VN,β (cid:20) exp (cid:18) u (cid:18) − β (cid:19) ˆ log φ ′ t d fluct N (cid:19)(cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C u t √ N k ψ k C . Proof.
It follows from applying Corollary 2.5 to the map log φ ′ t . (cid:3) 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 thingthat remains to elucidate is the behavior of the exponential moments of A [ ~X N , ψ ], which wecall 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 (4.25) (cid:12)(cid:12)(cid:12)(cid:12) log E P VN,β (exp( − βt A [ ~X N , ψ ])) (cid:12)(cid:12)(cid:12)(cid:12) ≤ CtN k ψ k C ( U ) . Proof.
Let us write(4.26) A [ ~X N , ψ ] = ˆ g ( x ) d fluct N ( x ) , where we let(4.27) g ( x ) := ˆ ˆ ψ ( x, y ) d fluct N ( y ) , ˆ ψ ( x, y ) := ψ ( x ) − ψ ( y ) x − y . It is clear that(4.28) k ˆ ψ k C ( U × U ) ≤ k ψ k C ( U ) . Using Proposition 2.4 twice, we can thus write k∇ g k L ∞ ≤ (cid:12)(cid:12)(cid:12)(cid:12) ˆ ∇ x ˆ ψ ( x, y ) d fluct N ( y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k∇ x ∇ y ˆ ψ k L ∞ (cid:16) F N ( ~X N , µ V ) + N log N + CN (cid:17) LT FOR FLUCTUATIONS OF β -ENSEMBLES WITH GENERAL POTENTIAL 15 and | A [ ~X N , ψ ] | = (cid:12)(cid:12)(cid:12)(cid:12) ˆ g ( x ) d fluct N ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k∇ g k L ∞ (cid:16) F N ( ~X N , µ V ) + N log N + CN (cid:17) ≤ C k ˆ ψ k C ( U × U ) (cid:16) F N ( ~X N , µ V ) + N log N + CN (cid:17) . In view of (2.8) and (4.28), we deduce that(4.29) (cid:12)(cid:12)(cid:12) log E P N,β h − βt A [ ~X N , ψ ] i(cid:12)(cid:12)(cid:12) ≤ CtN k ψ k C ( U ) . (cid:3) This shows that the exponential moments of the anisotropy yield bounded terms.4.3.
Intermediary conclusion on the Laplace transform.
Inserting into the results ofProposition 4.1 and Lemma 4.3 we obtain the following (with t = − sβN ) (cid:12)(cid:12)(cid:12)(cid:12) log E P VN,β [exp( s Fluct N ( ξ )] (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:16) s ( k ψ k C + k ψ k C ) + s k ψ k L ∞ k ξ k C (cid:17) + C s √ N k ψ k C + s N k ψ k L ∞ k ξ k C + s N k ψ k C + s √ N k ψ k C ! . In view of (3.6), we can bound k ψ k C n ≤ k ξ k C k +1+ n for any n , hence we obtain (cid:12)(cid:12)(cid:12)(cid:12) log E P VN,β [exp ( s Fluct N ( ξ ))] (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:16) s ( k ξ k C k +2 + k ξ k C k +4 ) + s k ξ k C k +1 k ξ k C (cid:17) + C s √ N k ξ k C k +4 + s N k ξ k C k +1 k ξ k C + s N k ξ k C k +2 + s √ N k ξ k C k +3 ! . 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 | s | βN ≤ t max (4.30) (cid:12)(cid:12)(cid:12)(cid:12) log E P VN,β [exp( s Fluct N ( ξ ))] (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( s + s ) (cid:16) k ξ k C k +4 + k ξ k C k +4 (cid:17) where C depends only on β and V . The estimate (4.30) shows that fluctuations of a smooth enough test function are typicallyof 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 bytreating 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 Ap-pendix A: roughly speaking we use the new control (4.30) instead of (2.10) to estimate theexponential 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 convergenceof Laplace transforms. This is a standard technique, see e.g. the recursion of [BG13b], andcan be implemented in the one-cut, non-critical case because the operator Ξ V is invertible. Inthe multi-cut or critical cases, however, we only know how to invert the operator Ξ V underthe extra conditions on the test function. We then use a different way to show that the exponential moments of A are in fact smallerthan (4.29), by leveraging on the expansion of log Z VN,β of [LS15] quoted in Lemma 2.2. Indeed,comparing (4.1) to (4.2), we observe that the expansion of log Z V t N,β − log Z VN,β provides anotherway of evaluating the exponential moments of A . More precisely, we will use the expansion oflog K N,β (˜ µ t , ˜ ζ t ) − log K N,β ( µ V , ζ V ) where ˜ µ t is the approximate equilibrium measure obtainedby pushing forward µ V by Id + tψ .5. Smallness of the anisotropy term and proof of Theorem 1
Comparison of partition functions by transport.Definition 5.1.
For t ∈ [ − t max , t max ] , where t max is as in (3.7) we let P ( t ) N,β be the probabilitymeasure (5.1) d P ( t ) N,β ( ~X N ) = 1 K N,β (˜ µ t , ˜ ζ t ) exp − β F N ( ~X N , ˜ µ t ) + 2 N N X i =1 ˜ ζ t ( x i ) !! d ~X N , where K N,β (˜ µ t , ˜ ζ t ) is as in (2.4) . Lemma 5.2 (Comparison of energies) . Assume ψ ∈ C ( R ) . For any ~X N ∈ U N , letting Φ t ( ~X N ) = ( φ t ( x ) , · · · , φ t ( x N )) , we have (5.2) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F N (Φ t ( ~X N ) , ˜ µ t ) − F N ( ~X N , µ V ) − N X i =1 log φ ′ t ( x i ) + t A [ ~X N , ψ ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Ct (cid:16) F N ( ~X N , µ V ) + N log N (cid:17) k ψ k C . Proof.
Since by definition ˜ µ t = φ t µ we may write F N (Φ t ( ~X N ) , ˜ µ t ) − F N ( ~X N , µ V )= − ¨ R × R \△ log | x − y | (cid:16) N X i =1 δ φ t ( x i ) − N ˜ µ t (cid:17) ( x ) (cid:16) N X i =1 δ φ t ( x i ) − N ˜ µ t (cid:17) ( y )+ ¨ R × R \△ log | x − y | d fluct N ( x ) d fluct N ( y )= − ¨ R × R \△ log | φ t ( x ) − φ t ( y ) || x − y | d fluct N ( x ) d fluct N ( y )= − ¨ R × R log | φ t ( x ) − φ t ( y ) || x − y | d fluct N ( x ) d fluct N ( y ) + N X i =1 log φ ′ t ( x i ) . We may then recognize the term T in (4.14) and use (4.20) and Proposition 2.4 to conclude. (cid:3) Lemma 5.3 (Comparison of partition functions) . We have, for any t small enough (5.3) K N,β (˜ µ t , ˜ ζ t ) K N,β ( µ V , ζ V ) = exp (cid:18) N (cid:18) − β (cid:19) (Ent( µ V ) − Ent( ˜ µ t )) (cid:19) E P (0) N,β (cid:18) exp (cid:18) β t A [ ~X N , ψ ] + Error ( ~X N ) + Error ( ~X N ) (cid:19)(cid:19) , LT FOR FLUCTUATIONS OF β -ENSEMBLES WITH GENERAL POTENTIAL 17 with error terms bounded by | log E P (0) N,β [exp( − Error ( ~X N ))] | ≤ Ct N k ψ k C , (5.4) | log E P (0) N,β [exp( − Error ( ~X N ))] | ≤ Ct √ N k ψ k C . (5.5) Proof.
Starting from (2.4), by a change of variables and in view of (5.2), we may write K N,β (˜ µ t , ˜ ζ t ) = ˆ exp − β (cid:16) F N (Φ t ( ~X N ) , ˜ µ t ) + 2 N N X i =1 ˜ ζ t ◦ φ t ( x i ) (cid:17) + N X i =1 log φ ′ t ( x i ) ! d ~X N = ˆ exp − β F N (Φ t ( ~X N ) , ˜ µ t ) + 2 N N X i =1 ζ V ( x i ) ! + N X i =1 log φ ′ t ( x i ) ! d ~X N , (5.6)since ζ V = ˜ ζ t ◦ φ t by definition. Using Lemma 5.2 we may write(5.7) K N,β (˜ µ t , ˜ ζ t ) K N,β ( µ V , ζ V ) = 1 K N,β ( µ V , ζ V ) ˆ R N exp − β F N ( ~X N , µ ) + 2 N N X i =1 ζ ( x i ) ! + (cid:18) − β (cid:19) N X i =1 log φ ′ t ( x i ) + β t A + Error ( ~X N ) ! d ~X N = E P (0) N,β exp (cid:18) − β (cid:19) N X i =1 log φ ′ t ( x i ) + β t A + Error ( ~X N ) !! , where the Error term is bounded as in (5.4). We may finally write N X i =1 log φ ′ t ( x i ) = N ˆ R log φ ′ t dµ V + Error ( ~X N )with an Error 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 V ˜ µ t ◦ φ t and thus(5.8) ˆ R log φ ′ t dµ V = ˆ R log µ V dµ V − ˆ R log µ t ◦ φ t dµ V = Ent( µ V ) − Ent( ˜ µ t ) . This yields (5.3). (cid:3)
Smallness of the anisotropy term.Proposition 5.4.
For any s such that | s | βN ≤ t max , we have (5.9) log E P VN,β (cid:18) exp (cid:18) − sN A (cid:19)(cid:19) = o N (1) . Proof.
Applying Cauchy-Schwarz to (5.3) we may write(5.10) E P VN,β (cid:20) exp (cid:18) β t A (cid:19)(cid:21) ≤ E P VN,β (cid:20) exp (cid:18) β t A + Error + Error (cid:19)(cid:21) E P VN,β [exp( − Error − Error )] ≤ K N,β (˜ µ t , ˜ ζ t ) K N,β ( µ V , ζ V ) exp (cid:18)(cid:18) − β (cid:19) N (Ent(˜ µ t ) − Ent( µ V )) (cid:19) E P VN,β (exp( − Error )) E P VN,β (exp( − Error )) . Inserting (2.7) and (5.4)–(5.5) into (5.10) we obtain that for t small enough,(5.11) log E P VN,β (cid:18) exp (cid:18) β t A (cid:19)(cid:19) ≤ C ( t N k ψ k C + t √ N k ψ k C ) + N δ N , for some sequence { δ N } N with lim N →∞ δ N = 0. Applying this to t = 4 ε/β with ε small(possibly depending on N ) and using H¨older’s inequality, we deducelog E P VN,β (cid:18) exp (cid:18) − sN A (cid:19)(cid:19) ≤ | s | N ε log E P VN,β (exp( ε A )) ≤ C | s | ε k ψ k C + C | s |√ N k ψ k C + C | s | ε δ N . In particular, choosing ε = √ δ N , we get (5.9). (cid:3) Conclusion: proof of Theorem 1.
Proof of Theorem 1.
Combining (4.2) and (4.24) for t = − sβN (where s is independent of N )and (5.9), together with the Cauchy-Schwarz inequality, we find(5.12) log E P VN,β [exp( s Fluct N ( ξ ))] = log Const + o N (1)+ O s + s √ N (cid:16) k ψ k C ( U ) + k ψ k C ( U ) (cid:17) + s N (cid:16) k ψ k C + k ψ k L ∞ k ξ k C (cid:17)! with(5.13) log Const = − s β ˆ ξ ′ ψdµ V − sβ (cid:18) − β (cid:19) ˆ ψ ′ dµ V Letting N → ∞ , we obtain,(5.14) lim N →∞ log E P VN,β [exp( s Fluct N ( ξ ))] = − s β ˆ ξ ′ ψdµ V − sβ (cid:18) − β (cid:19) ˆ ψ ′ dµ V 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 thatof a Gaussian of mean m ξ and variance v ξ , which implies convergence in law and proves themain theorem. (cid:3) LT FOR FLUCTUATIONS OF β -ENSEMBLES WITH GENERAL POTENTIAL 19 Appendix A. The one-cut regular case
In the one-cut noncritical case, every regular enough function is in the range of the operatorΞ, so that the map ψ can always be built. This allows to bootstrap the approach used forproving Theorem 1. In this appendix, we expand on how we can proceed in this simpler settingwithout refering to the result of [LS15] but assuming more regularity of ξ , and retrieve thefindings of [BG13b] (but without assuming analyticity), as well as a rate of convergence forthe Laplace transform of the fluctuations.A.1. The bootstrap argument.
We will consider the whole family P ( t ) N,β of probabilitymeasures d P ( t ) N,β ( ~X N ) = 1 K N,β (˜ µ t , ˜ ζ t ) exp − β F N ( ~X N , ˜ µ t ) + 2 N N X i =1 ˜ ζ t ( x i ) !! d ~X N , where K N,β (˜ µ t , ˜ ζ t ) is as in (2.4). We will also emphasize the t dependence by writingfluct ( t ) N := X i δ x i − N ˜ µ t and using similarly the notation Fluct ( t ) and A ( t ) .Let us first explain the main computational point for the bootstrap argument. Differenti-ating (4.2) with respect to t and using (4.5), we obtain − βN E P (0) N,β [Fluct (0) N ( ξ )] = E P (0) N,β " − β A (0) [ ~X N , ψ ] + (cid:18) − β (cid:19) ddt | t =0 N X i =1 log φ ′ t ( x i ) . Note that here all the error terms in (4.2) have disappeared because they were in factor of t .Also this is true as well for all t ∈ [ − t max , t max ], i.e.(A.1) E P ( t ) N,β [Fluct ( t ) N ( ξ )] = − βN E P ( t ) N,β " − β A ( t ) [ ~X N , ψ ] + (cid:18) − β (cid:19) ddt N X i =1 log φ ′ t ( x i ) . We may in addition write that(A.2) ddt N X i =1 log φ ′ t ( x i ) = N ˆ ddt log φ ′ t d ˜ µ t + Fluct ( t ) N (cid:18) ddt log φ ′ t (cid:19) so that(A.3) E P ( t ) N,β [Fluct ( t ) N ( ξ )] = − β (cid:18) − β (cid:19) ˆ ddt log φ ′ t d ˜ µ t − βN E P ( t ) N,β (cid:20) − β A ( t ) [ ~X N , ψ ] + (cid:18) − β (cid:19) Fluct ( t ) N (cid:18) ddt log φ ′ t (cid:19)(cid:21) . This provides a functional equation which gives the expectation of the fluctuation in terms ofa constant term plus a lower order expectation of another fluctuation and the A term (whichitself 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 , since ˜ µ 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 ofexponential moments given in Corollary 4.5 into the control of a weak norm of Fluct ( t ) N . Herewe use the Sobolev spaces H α ( R ). Lemma A.1.
Under the same assumptions, for α ≥ we have (A.4) (cid:12)(cid:12)(cid:12)(cid:12) E P ( t ) N,β h k fluct ( t ) N k H − α i(cid:12)(cid:12)(cid:12)(cid:12) ≤ C, 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(A.5) k u k H − α ( R ) ≤ C ˆ r α − k u ∗ Φ( r, · ) k L ( R ) dr where Φ( r, · ) is the standard heat kernel, i.e. Φ( r, x ) = √ πr e − | x | r . It follows that(A.6) E P ( t ) N,β h k fluct ( t ) N k H − α ( R ) k i ≤ C ˆ r α − E P ( t ) N,β h k fluct ( t ) N ∗ Φ( r, · ) k L ( R ) i dr. On the other hand we may easily check that, letting ξ x,r := Φ( r, x − · ), we have(A.7) E P ( t ) N,β h k fluct ( t ) N ∗ Φ( r, · ) k L ( R ) i = ˆ E P ( t ) N,β (cid:20)(cid:16)
Fluct ( t ) N ( ξ x,r ) (cid:17) (cid:21) dx. Applying the result of Corollary 4.5 to ξ x,r gives us a control on the second moment ofFluct ( t ) N [ ξ x,r ] of the form E P ( t ) N,β h (Fluct ( t ) N ( ξ x,r )) i ≤ C (cid:16) k ξ x,r k C + k ξ x,r k C (cid:17) . Inserting into (A.6) and (A.7), we are led to E P ( t ) N,β h k fluct ( t ) N k H − α ( R ) i ≤ C ˆ ˆ r α − C (cid:16) k ξ x,r k C + k ξ x,r k C (cid:17) dx dr. Since U is bounded, the right-hand side can be bounded by C ´ r α − (1 + r − / ) dr , whichconverges if α > / (cid:3) A.3.
Proof of Theorem 2.
First, by (5.6) and in view of Lemma 5.2, we may write(A.8) ddt | t =0 log K N,β (˜ µ t , ˜ ζ t ) = E P (0) N,β " − β A (0) [ ~X N , ψ ] + (cid:18) − β (cid:19) ddt | t =0 N X i =1 log φ ′ t ( x i ) . Similarly, we have for all t (A.9) ddt log K N,β (˜ µ t , ˜ ζ t ) = E P ( t ) N,β " − β A ( t ) [ ~X N , ψ ] + (cid:18) − β (cid:19) ddt N X i =1 log φ ′ t ( x i ) . Indeed, ˜ µ t has the same regularity as µ V . LT FOR FLUCTUATIONS OF β -ENSEMBLES WITH GENERAL POTENTIAL 21 For any test function φ ( x, y ) we may write ¨ φ ( x, y ) d fluct ( t ) N ( x ) d fluct ( t ) N ( y ) ≤ k φ k C α ( U × U ) k fluct ( t ) N k H − α ( R ) and so by the result of Lemma A.1, we find(A.10) (cid:12)(cid:12)(cid:12)(cid:12) E P ( t ) N,β (cid:18) ¨ φ ( x, y ) d fluct ( t ) N ( x ) d fluct ( t ) N ( y ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C k φ k C α ( U × U ) . We may return to (4.26) and, using (A.10), write that(A.11) (cid:12)(cid:12)(cid:12)(cid:12) E P ( t ) N,β h A ( t ) [ ~X N , ψ ] i(cid:12)(cid:12)(cid:12)(cid:12) ≤ C k ψ k C α +1 ( U ) . On the other hand, by differentiating (4.30) applied with ξ = ddt log φ ′ t , we have(A.12) (cid:12)(cid:12)(cid:12)(cid:12) E P ( t ) N,β (cid:20) ˆ ddt log φ ′ t d fluct ( t ) N (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:16) k ψ k C ( U ) + k ψ k C ( U ) (cid:17) Inserting (5.8) and (A.11) and (A.12), (A.2) into (A.9), and integrating between 0 and t = − s/N β , we obtain(A.13)log K N,β (˜ µ t , ˜ ζ t ) K N,β ( µ V , ζ V ) = (cid:18) − β (cid:19) N (Ent( ˜ µ t ) − Ent( µ )) + sN OC (cid:16) k ψ k C ( U ) + k ψ k C ( U ) (cid:17) . Comparing (A.13) with (5.3), we obtainlog E P (0) N,β (cid:18) exp (cid:18) β t A (0) + Error ( ~X N ) + Error ( ~X N ) (cid:19)(cid:19) = sN O (cid:16) k ψ k C α +1 ( U ) + k ψ k C ( U ) + k ψ k C ( U ) (cid:17) Using the bounds of (5.4)-(5.5) and the Cauchy-Schwarz inequality, we deduce thatlog E P (0) N,β (cid:20) exp (cid:18) β t A (0) (cid:19)(cid:21) = sN O (cid:16) k ψ k C α +1 ( U ) + k ψ k C ( U ) + k ψ k C ( U ) + s k ψ k C + √ N k ψ k C (cid:17) . This can be inserted in place of (5.9) into (4.2) yields (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) log E P VN,β [exp( s Fluct N ( ξ ))] + (cid:18) − β (cid:19) sβ ˆ ψ ′ dµ V + s β ˆ ξ ′ ψdµ V (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = sN O (cid:16) k ψ k C α +1 ( U ) + k ψ k C ( U ) + k ψ k C ( U ) + s k ψ k C + √ N k ψ k C (cid:17) . 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 ourassumptions with n = 0 , m = 0. In this section, we explain how to iterate the proceduredescribed above to obtain a relative expansion of the partition function, namely an expansionof log Z WN,β − log Z VN,β to any order of 1 /N . Up to applying an affine transformation to one ofthe 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 rearrange-ment) φ from µ V to µ W is C ∞ on Σ and can be extended as a C ∞ function with compactsupport on R . We let ψ := φ − Id, which is smooth, and for t ∈ [0 ,
1] the map φ t := Id + tψ isa C ∞ -diffeomorphism, by the properties of optimal transport. We let ˜ µ t := φ t µ V as before. We can integrate (A.9) to obtainlog K N,β ( µ W , ζ W ) K N,β ( µ V , ζ V )= ˆ E P ( t ) N,β (cid:20) − β A ( t ) [ ~X N , ψ ] + (cid:18) − β (cid:19) N ˆ ddt log φ ′ t d ˜ µ t + (cid:18) − β (cid:19) ˆ ddt log φ ′ t d fluct ( t ) N (cid:21) dt = N (cid:18) − β (cid:19) (Ent( µ W ) − Ent( µ V ))+ ˆ E P ( t ) N,β (cid:20) − β A ( t ) [ ~X N , ψ ] + (cid:18) − β (cid:19) Fluct N (cid:20) ˆ ddt log φ ′ t d fluct ( t ) N (cid:21)(cid:21) dt. The integral on the right-hand side is of order 1, and we claim that the terms in the integralcan 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 ) N,β h Fluct ( t ) N ( ddt log φ ′ t ) i which can be computed up to an error O (1 /N ) by the result of Theorem 2. The term E P ( t ) N,β h − β A ( t ) [ ~X N , ψ ] i can on the other handbe deduced from the knowledge of the covariance structure of the fluctuations. Let F denotethe Fourier transform. In view of (4.26), using the identity ψ ( x ) − ψ ( y ) x − y = ˆ ψ ′ ( sx + (1 − s ) y ) ds and the Fourier inversion formula we may write(A.14) E P ( t ) N,β h A ( t ) [ ~X N , ψ ] i = E P ( t ) N,β " ¨ R × R ˆ ψ ′ ( sx + (1 − s ) y ) ds d fluct ( t ) N ( x ) d fluct ( t ) N ( y ) = ˆ ˆ λ F ( ψ )( λ ) E P ( t ) N,β h Fluct ( t ) N ( e isλ · )Fluct ( t ) N ( e i (1 − s ) λ · ) i ds dλ. On the other hand, let ϕ s,λ be the map associated to e isλ · by Lemma 3.3. Separating the realpart and the imaginary part we may use the results of the previous subsection to e isλ · andobtain E P ( t ) N,β h Fluct ( t ) N ( e isλ · ) i = (cid:18) − β (cid:19) ˆ ϕ ′ s,λ d ˜ µ t + O ( 1 N ) . By polarization of the expression for the variance (see (1.16)) and linearity E P ( t ) N,β h Fluct ( t ) N ( e isλ · )Fluct ( t ) N ( e i (1 − s ) λ · ) i = E P ( t ) N,β h Fluct ( t ) N ( e isλ · ) i E P ( t ) N,β h Fluct ( t ) N ( e i (1 − s ) λ · ) i + 2 β (cid:16) ¨ (cid:18) ϕ s,λ ( u ) − ϕ s,λ ( v ) u − v (cid:19) ϕ (1 − s ) ,λ ( u ) − ϕ (1 − s ) ,λ ( v ) u − v ! d ˜ µ t ( u ) d ˜ µ t ( v )+ ˆ V ′′ t ϕ s,λ ϕ (1 − s ) ,λ d ˜ µ t (cid:17) + O ( 1 N ) . Letting N → ∞ , we may then find the expansion up to O (1 /N ) of E P ( t ) N,β h − β A ( t ) [ ~X N , ψ ] i .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 expansionto arbitrary order of 1 /N as desired. LT FOR FLUCTUATIONS OF β -ENSEMBLES WITH GENERAL POTENTIAL 23 Appendix B. Auxiliary proofs
B.1.
Proof of Lemma 2.1.
Proof.
Denoting △ the diagonal in R × R we may write H VN ( ~X N ) = X i = j − log | x i − x j | + N N X i =1 V ( x i )= ¨ △ c − log | x − y | (cid:16) N X i =1 δ x i (cid:17) ( x ) (cid:16) N X i =1 δ x i (cid:17) ( y ) + N ˆ R V ( x ) (cid:16) N X i =1 δ x i (cid:17) ( x ) . Writing P Ni =1 δ x i as N µ V + fluct N we get(B.1) H VN ( ~X N ) = N ¨ △ c − log | x − y | dµ V ( x ) dµ V ( y ) + N ˆ R V dµ V + 2 N ¨ △ c − log | x − y | dµ V ( x ) d fluct N ( y ) + N ˆ R V d fluct N + ¨ △ c − log | x − y | d fluct N ( x ) d fluct N ( y ) . We now recall that ζ V was defined in (1.7), and that ζ V = 0 in Σ V . With the help of this wemay rewrite the medium line in the right-hand side of (B.1) as2 N ¨ △ c − log | x − y | dµ V ( x ) d fluct N ( y ) + N ˆ R V d fluct N = 2 N ˆ R (cid:18) − log | · | ∗ dµ V )( x ) + V (cid:19) d fluct N = 2 N ˆ R ( ζ V + c ) d fluct N = 2 N ˆ R ζ V d (cid:16) N X i =1 δ x i − N µ V (cid:17) = 2 N N X i =1 ζ V ( x i ) . 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 isabsolutely continuous with respect to the Lebesgue measure, we may include the diagonalback into the domain of integration. By that same argument, one may recognize in the firstline of the right-hand side of (B.1) the quantity N I V ( µ V ). (cid:3) 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 linein the plane. We view R as identified with R × { } ⊂ R = { ( x, y ) , x ∈ R , y ∈ R } . Let usdenote by δ R the uniform measure on R × { } , i.e. such that for any smooth ϕ ( x, y ) (with x ∈ R , y ∈ R ) we have ˆ R ϕδ R = ˆ R ϕ ( x, dx. Given ( x , . . . , x N ) in R N , we identify them with the points ( x , , . . . , ( x N ,
0) in R . Fora fixed ~X N and a given probability density µ we introduce the electric potential H µN by(B.2) H µN = ( − log | · | ) ∗ N X i =1 δ ( x i , − N µδ R ! . Next, we define versions of this potential which are truncated hence regular near the pointcharges. For that let δ ( η ) x denote the uniform measure of mass 1 on ∂B ( x, η ) (where B denotesan Euclidean ball in R ). We define H µN,η in R by(B.3) H µN,η = ( − log | · | ) ∗ N X i =1 δ ( η )( x i , − N µδ R ! . These potentials make sense as functions in R and are harmonic outside of the real axis.Moreover, H µN,η solves(B.4) − ∆ H µN,η = 2 π N X i =1 δ ( η )( x i , − N µδ R ! . Lemma B.1.
For any probability density µ , ~X N in R N and η in (0 , , we have (B.5) F N ( ~X N , µ ) ≥ π ˆ R |∇ H µN,η | + N log η − N k µ k L ∞ η. Proof.
First we notice that ´ R |∇ H N,η | is a convergent integral and that(B.6) ˆ R |∇ H N,~η | = 2 π ¨ − log | x − y | d N X i =1 δ ( η ) x i − N µδ R ! ( x ) d N X i =1 δ ( η ) x i − N µδ R ! ( y ) . 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(B.7) ˆ B R |∇ H N,η | = ˆ ∂B R H N,η ∂H N ∂ν + 2 π ˆ B R H N,η N X i =1 δ ( η ) x i − N µδ R ! . In view of the decay of H N and ∇ H N , the boundary integral tends to 0 as R → ∞ , and sowe may write ˆ R |∇ H N,η | = 2 π ˆ R H N,η N X i =1 δ ( η ) x i − N µ ! and thus (B.6) holds. We may next write(B.8) ¨ − log | x − y | d N X i =1 δ ( η ) x i − N µδ R ! ( x ) d N X i =1 δ ( η ) x i − N µδ R ! ( y ) − ¨ △ c − log | x − y | d fluct N ( x ) d fluct N ( y )= − N X i =1 log η + X i = j ¨ − log | x − y | (cid:16) δ ( η ) x i δ ( η ) x j − δ x i δ x j (cid:17) +2 N N X i =1 ¨ − log | x − y | (cid:16) δ x i − δ ( η ) x i (cid:17) µ. We have used the fact that for any x i , ¨ − log | x − y | δ ( η ) x i ( x ) δ ( η ) x i ( y ) = − log η, 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 to ´ − log | x − y | δ x i outside of B ( x i , η ), and smaller otherwise. Since its Laplacian is − πδ ( η ) x i , LT FOR FLUCTUATIONS OF β -ENSEMBLES WITH GENERAL POTENTIAL 25 a negative measure, this is also a superharmonic function, so by the maximum principle, itsvalue 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 deducefrom these considerations, and reversing the roles of i and j , that for each i = j , − ˆ log | x − y | δ ( η ) x i δ ( η ) x j ≤ − ˆ log | x − y | δ x i δ ( η ) x j ≤ − ˆ log | x − y | δ x i δ x j . We may also obviously write ˆ − log | x − y | δ x i δ x j − ˆ − log | x − y | δ ( η ) x i δ ( η ) x j ≤ − log | x i − x j | | x i − x j |≤ η . We conclude that the second term in the right-hand side of (B.8) is nonpositive, equal to 0 ifall the balls are disjoint, and bounded below by P i = j log | x i − x j | | x i − x j |≤ η . Finally, by theabove 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) as2 N N X i =1 ˆ B ( x i ,η ) ( − log | x − x i | + log η )) dµδ R . But we have that(B.9) ˆ B (0 ,η ) ( − log | x | + log η ) δ R = η so if µ ∈ L ∞ , this last term is bounded by 2 k µ k L ∞ N η . Combining with all the above resultsyields the proof. (cid:3) Proof of Proposition 2.4.
We now apply Lemma B.1 for µ V with η = N . We obtain(B.10) 12 π ˆ R |∇ H µN,η | ≤ F N ( ~X N , µ V ) + N log N + C ( k µ V k L ∞ + 1) N. Let ξ be a Lipschitz, compactly supported test function in R , and let χ ( y ) be a smooth cutofffunction such that χ ( y ) = 1 for | y | ≤ χ ( y ) = 0 for | y | ≥ k χ ′ k L ∞ ≤
1. We then extend ξ in R by ˜ χ defined as ˜ χ ( x, y ) := ξ ( x ) χ ( y ) . It is easy to check that for any ( x, y ), |∇ ˜ χ ( x, y ) | ≤ | ξ ′ ( x ) | + | ξ ( x ) | , and ˜ χ 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 η = N )(B.11) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ fluct N − N X i =1 δ ( η ) x i − N µ V δ R !! ˜ χ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ N X i =1 ( δ x i − δ ( η ) x i ) ! ˜ χ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Iη k ξ ′ k L ∞ ≤ k ξ ′ k L ∞ , where we have bounded I by N in the last inequality. In view of (B.4), we also have(B.12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ N X i =1 δ ( η ) x i − N µ V δ R ! ˜ χ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 12 π (cid:12)(cid:12)(cid:12)(cid:12) ˆ R ∇ H µ V N,η · ∇ ( ˜ χ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ π ( k ξ ′ k L ( R ) + k ξ k L ( R ) ) k∇ H µ V N,η k L ( R ) . Combining (B.10), (B.11) and (B.12), we obtain(B.13) (cid:12)(cid:12)(cid:12)(cid:12) ˆ ξ fluct N (cid:12)(cid:12)(cid:12)(cid:12) ≤ k ξ ′ k L ∞ + (cid:16) k ξ ′ k L ( R ) + k ξ k L ( R ) (cid:17) (cid:16) F N ( ~X N , µ V ) + N log N + C ( k µ V k L ∞ + 1) N (cid:17) . (cid:3) B.3.
Proof of Lemma 3.1.
Proof.
Since µ V minimizes the logarithmic potential energy (1.6), for any bounded continuousfunction h , (4.6) holds. Of course, an identity like (4.6) extends to complex-valued functions,and applying it to h = z −· for some fixed z ∈ C \ Σ V leads to(B.14) G ( z ) − G ( z ) V ′ ( R ( z )) + L ( z ) = 0 , where G is the usual Stieltjes transform of µ V (B.15) G ( z ) = ˆ z − y dµ V ( y ) , and L is defined by(B.16) L ( z ) = ˆ V ′ ( R ( z )) − V ′ ( y ) z − y dµ V ( y ) . Solving (B.14) for G yields(B.17) G ( z ) = 12 (cid:18) V ′ ( R ( z )) − q V ′ ( R ( z )) − L ( z ) (cid:19) . As is well-known, since µ V is continuous on Σ V , the quantity − π I ( G ( x + iε )) convergestowards the density µ V ( x ) as ε → + , hence we have for x in Σ V (B.18) µ V ( x ) = S ( x ) σ ( x ) = − π ) ( V ′ ( x ) − L ( x )) . This proves that µ V has regularity C p − at any point where it does not vanish. Assumingthe form (1.11) for S , we also deduce that the function S has regularity at least C p − − k onΣ V .Applying (B.17) on R \ Σ, we obtain12 V ′ ( x ) − ˆ x − y dµ V ( y ) = 12 q V ′ ( x ) − L ( x ) , 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 smallenough such that ζ ′ does not vanish on U \ Σ V and on which we can write ζ ′ as in (3.1). (cid:3) LT FOR FLUCTUATIONS OF β -ENSEMBLES WITH GENERAL POTENTIAL 27 B.4.
Proof of Lemma 3.2.
Proof.
We first prove that the image of F is indeed contained in C ( U ).For ( t, ψ ) = (0 , F (0 ,
0) = ζ V + c and ζ V is in C ( R ) by the regularityassumptions on V . We may also write F ( t, ψ ) = F (0 , − ˆ log | φ ( · ) − φ ( y ) || · − y | dµ V ( y ) + 12 ( V t ◦ φ − V ◦ φ ) , and since k ψ k C ( U ) ≤ /
2, the second and third terms are also in C ( U ).Next, we compute the partial derivatives of F at a fixed point ( t , ψ ) ∈ [ − , × B . It iseasy to see that ∂ F ∂t (cid:12)(cid:12)(cid:12) ( t ,ψ ) = 12 ξ ◦ φ , and the map ( t , ψ ) ξ ◦ φ is indeed continuous.The Fr´echet derivative of F with respect to the second variable can be computed as follows F ( t , ψ + ψ ) = − ˆ log (cid:12)(cid:12)(cid:12)(cid:0) φ ( · ) − φ ( y ) (cid:1) + (cid:0) ψ ( · ) − ψ ( y ) (cid:1)(cid:12)(cid:12)(cid:12) dµ V ( y ) + 12 V t ◦ ( φ + ψ )= F ( t , ψ ) − ˆ log (cid:12)(cid:12)(cid:12) ψ ( · ) − ψ ( y ) φ ( · ) − φ ( y ) (cid:12)(cid:12)(cid:12) dµ V ( y ) + 12 (cid:0) V t ◦ ( φ + ψ ) − V t ◦ φ (cid:1) = F ( t , ψ ) − ˆ ψ ( · ) − ψ ( y ) φ ( · ) − φ ( y ) dµ V ( y ) + 12 ψ V ′ t ◦ φ + ε t ,ψ ( ψ ) , where ε t ,ψ ( ψ ) is given by ε t ,ψ ( ψ ) = − ˆ (cid:20) log (cid:12)(cid:12)(cid:12) ψ ( · ) − ψ ( y ) φ ( · ) − φ ( y ) (cid:12)(cid:12)(cid:12) − ψ ( · ) − ψ ( y ) φ ( · ) − φ ( y ) (cid:21) dµ V ( y )+ 12 (cid:0) V t ◦ ( φ + ψ ) − V t ◦ φ − ψ V ′ t ◦ φ (cid:1) . By differentiating twice inside the integral we get the bound k ε t ,ψ ( ψ ) k C ( U ) ≤ C ( t , ψ ) k ψ k C ( U ) , with a constant depending on V . It implies that ∂ F ∂ψ (cid:12)(cid:12)(cid:12) ( t ,ψ ) [ ψ ] = − ˆ ψ ( · ) − ψ ( y ) φ ( · ) − φ ( y ) dµ V ( y ) + 12 ψ V ′ t ◦ φ , and we can check that this expression is also continuous in ( t , ψ ). In particular, we mayobserve that(B.19) ∂ F ∂ψ (cid:12)(cid:12)(cid:12) (0 , [ ψ ] = − Ξ V [ ψ ] . Finally, we prove the bound (3.3). For any fixed ( t, ψ ) ∈ [ − , × B , we write F ( t, ψ ) − F (0 ,
0) = ˆ d F ( st, sψ ) ds ds = ˆ (cid:16) t ∂ F ∂t (cid:12)(cid:12)(cid:12) ( st,sψ ) + ∂ F ∂ψ (cid:12)(cid:12)(cid:12) ( st,sψ ) [ ψ ] (cid:17) ds , we get(B.20) kF ( t, ψ ) − F (0 , − t ξ + Ξ V [ ψ ] k C ( U ) ≤ ˆ (cid:18) t k ξ ◦ φ s − ξ k C ( U ) + (cid:13)(cid:13)(cid:13)(cid:13) ∂ F ∂ψ (cid:12)(cid:12)(cid:12) ( st,sψ ) [ ψ ] − ∂ F ∂ψ (cid:12)(cid:12)(cid:12) (0 , [ ψ ] (cid:13)(cid:13)(cid:13)(cid:13) C ( U ) ! ds, with φ s = Id + sψ . It is straightforward to check that k ξ ◦ φ s − ξ k C ( U ) ≤ C k ξ k C ( U ) k ψ k C ( U ) . To control the second term inside the integral we write ∂ F ∂ψ (cid:12)(cid:12)(cid:12) ( st,sψ ) [ ψ ] − ∂ F ∂ψ (cid:12)(cid:12)(cid:12) (0 , [ ψ ]= − ˆ (cid:18) ψ ( · ) − ψ ( y ) φ s ( · ) − φ s ( y ) − ψ ( · ) − ψ ( y ) · − y (cid:19) dµ V ( y ) + 12 (cid:0) V ′ st ◦ φ s − V ′ (cid:1) ψ and we obtain (cid:13)(cid:13)(cid:13)(cid:13) ∂ F ∂ψ (cid:12)(cid:12)(cid:12) ( st,sψ ) [ ψ ] − ∂ F ∂ψ (cid:12)(cid:12)(cid:12) (0 , [ ψ ] (cid:13)(cid:13)(cid:13)(cid:13) C ( U ) ≤ ˆ (cid:13)(cid:13)(cid:13)(cid:13) ψ ( · ) − ψ ( y ) φ s ( · ) − φ s ( y ) − ψ ( · ) − ψ ( y ) · − y (cid:13)(cid:13)(cid:13)(cid:13) C ( U ) dµ V ( y )+ (cid:13)(cid:13)(cid:0) V ′ st ◦ φ s − V ′ (cid:1) ψ (cid:13)(cid:13) C ( U ) We now use that (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) ψ ( · ) − ψ ( y ) φ s ( · ) − φ s ( y ) − ψ ( · ) − ψ ( y ) · − y (cid:17)(cid:13)(cid:13)(cid:13)(cid:13) C ( U ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) ψ ( · ) − ψ ( y ) · − y (cid:19)(cid:18) · − yφ s ( · ) − φ s ( y ) − (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) C ( U ) ≤ C k ψ k C ( U ) (cid:13)(cid:13)(cid:13)(cid:13) · − yφ s ( · ) − φ s ( y ) − (cid:13)(cid:13)(cid:13)(cid:13) C ( U ) = Cs k ψ k C ( U ) (cid:13)(cid:13)(cid:13)(cid:13) ψ ( · ) − ψ ( y ) φ s ( · ) − φ s ( y ) (cid:13)(cid:13)(cid:13)(cid:13) C ( U ) ≤ C k ψ k C ( U ) (cid:13)(cid:13)(cid:13)(cid:13) · − yφ s ( · ) − φ s ( y ) (cid:13)(cid:13)(cid:13)(cid:13) C ( U ) ≤ C k ψ k C ( U ) . 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 / C ( U ) so its composition with thefunction x → / (1 + x ) is bounded in C ( U ). We conclude by checking that k (cid:0) V ′ st ◦ φ s − V ′ (cid:1) ψ k C ( U ) ≤ C (cid:16) k V k C ( U ) k ψ k C ( U ) + t k ψ k C ( U ) (cid:17) k ψ k C ( U ) . (cid:3) LT FOR FLUCTUATIONS OF β -ENSEMBLES WITH GENERAL POTENTIAL 29 B.5.
Proof of Lemma 3.3.
Proof.
First, we solve the equation Ξ V [ ψ ] = ξ + c ξ in ˚Σ V , where Ξ V is operator defined in(1.12). For x in ˚Σ V , we have the following equation(B.21) V ′ ( x )2 = P.V. ˆ x − y dµ V ( y ) . In particular, for x in ˚Σ V , it implies(B.22) Ξ V [ ψ ]( x ) := P.V. ˆ Σ V ψ ( y ) y − x µ V ( y ) dy, and we might thus try to solve(B.23) P.V. ˆ Σ V ψ ( y ) y − x µ V ( y ) dy = 12 ξ + c ξ . Equation (B.23) is a singular integral equation, we refer to [Mus92, Chap. 10-11-12] for adetailed treatment. In particular, it is known that if the conditions (1.14) are satisfied, thenthere exists a solution ψ to(B.24) P.V. ˆ Σ V ψ ( y ) y − x dy = 12 ξ + c ξ on ˚Σ V , which is explicitly given by the formula(B.25) ψ ( x ) = − σ ( x )2 π P.V. ˆ Σ V ξ ( y ) σ ( y )( y − x ) dy. Since we have, for x in ˚Σ V P.V. ˆ Σ V σ ( y )( y − x ) dy = 0 , we may re-write (B.25) as(B.26) ψ ( x ) = − σ ( x )2 π ˆ Σ V ξ ( y ) − ξ ( x ) σ ( y )( y − x ) dy on ˚Σ V , where the integral is now a definite Riemann integral. From (B.26) we deduce that the map ψ σ is of class C r − in ˚Σ V and extends readily to a C r − function on Σ V .For d = 0 , . . . , r − x ∈ Σ V , we compute that (cid:18) ψ σ (cid:19) ( d ) ( x ) = − d !2 π ˆ Σ V ξ ( y ) − R s i ,d +1 ξ ( y ) σ ( y )( y − s i ) d +1 dy. In particular, if conditions (1.15) hold, in view of Lemma 3.1 the map ψ ( x ) := ψ ( x ) S ( x ) σ ( x )extends to a function of class ( p − − k ) ∧ ( r − − k ), hence C on Σ V , and in view of (B.24)it satisfies Ξ V [ ψ ] = ξ + c ξ on Σ V .Now, we define ψ outside Σ V . By definition, for x outside Σ V , the equationΞ V [ ψ ]( x ) = 12 ξ ( x ) + c ξ can be written as ψ ( x ) ˆ x − y dµ V ( y ) − ˆ ψ ( y ) x − y dµ V ( y ) − ψ ( x ) V ′ ( x ) = 12 ξ ( x ) + c ξ , and thus the choice (3.5) ensures that Ξ V [ ψ ] = ξ + c ξ . Moreover, ψ is clearly of class C r ∧ ( p − on R \ Σ V . It remains to check that ψ has the desired regularity at the endpoints of Σ V . Letus consider ˜ ψ an extension of ψ in C l with l := ( p − − k ) ∧ ( r − − k ), which coincides with ψ on Σ V (given for instance by a Taylor expansion at the endpoints). As ψ and ˜ ψ are equalon the support we can rewrite (3.5) as ´ ψ ( y ) x − y dµ V ( y ) + ξ ( x )2 + c ξ ´ x − y dµ V ( y ) − V ′ ( x ) = − ´ ˜ ψ ( x ) − ˜ ψ ( y ) x − y dµ V ( y ) + ˜ ψ ( x ) ´ x − y dµ V ( y ) + ξ ( x )2 + c ξ ´ x − y dµ V ( y ) − V ′ ( x )= ˜ ψ ( x ) + ξ ( x )2 + c ξ − Ξ V [ ˜ ψ ]( x ) ´ x − y dµ V ( y ) − V ′ ( x ) . Since Ξ V [ ψ ] = ξ + c ξ on Σ V , the numerator on the right hand side of the last equation andits first l derivatives vanish at any endpoint α . From Lemma (3.1) we conclude that ψ is ofclass l − k = ( p − − k ) ∧ ( r − − k ) at α , hence C from (1.13). (cid:3) B.6.
Proof of Lemma 4.2.
Proof.
The first item is a consequence of the fact that µ V minimizes the logarithmic potentialenergy (1.6) and hence as is well-known ´ − log | · − y | dµ V ( y ) + V is constant on the supportof µ V . Differentiating this and integrating against hdµ V gives the result. For the secondrelation, by definition of ψ we have ξ c ξ = ˆ ψ ( x ) − ψ ( y ) x − y dµ V ( y ) − ψV ′ , and thus ξ ′ = 2 ˆ ψ ( y ) − ψ ( x ) − ψ ′ ( x )( y − x )( x − y ) dµ V ( y ) − ψ ′ V ′ − ψV ′′ . Integrating both sides against ψµ V yields ˆ ξ ′ ψdµ V = 2 ¨ ( ψ ( y ) − ψ ( x ) − ψ ′ ( x )( y − x )) ψ ( x )( x − y ) dµ V ( y ) dµ V ( x ) − ˆ ψψ ′ V ′ dµ V − ˆ V ′′ ψ dµ V . Using (4.6) for the second term we obtain ˆ ξ ′ ψdµ V = 2 ¨ ( ψ ( y ) − ψ ( x ) − ψ ′ ( x )( y − x )) ψ ( x )( y − x ) dµ V ( y ) dµ V ( x ) − ¨ ψψ ′ ( y ) − ψψ ′ ( x ) y − x dµ V ( x ) dµ V ( y ) − ˆ V ′′ ψ dµ V . We may then combine the first two terms in the right-hand side to obtain (4.7). (cid:3)
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Department of Mathematics, Massachusetts Institute of Technology, 77 Mas-sachusetts Ave, Cambridge, MA 02139-4307 USA.
E-mail address : [email protected] (T. Lebl´e) Courant Institute, New York University, 251 Mercer st, New York, NY 10012, USA.
E-mail address : [email protected] (S. Serfaty) Courant Institute, New York University, 251 Mercer st, New York, NY 10012, USA.
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