CLT for Circular beta-Ensembles at High Temperature
aa r X i v : . [ m a t h . P R ] O c t CLT for Circular beta-Ensembles at High Temperature
Adrien Hardy ∗ Gaultier Lambert † October 25, 2019
Abstract
We consider the macroscopic large N limit of the Circular beta-Ensemble at high tem-perature, and its weighted version as well, in the regime where the inverse temperaturescales as β/N for some parameter β >
0. More precisely, in the limit N → ∞ , the equilib-rium measure of this particle system is described as the unique minimizer of a functionalwhich interpolates between the relative entropy ( β = 0) and the weighted logarithmicenergy ( β = ∞ ). The purpose of this work is to show that the fluctuation of the empiricalmeasure around the equilibrium measure converges towards a Gaussian field whose covari-ance structure interpolates between the Lebesgue L ( β = 0) and the Sobolev H / ( β = ∞ ) norms. We furthermore obtain a rate of convergence for the fluctuations in the W metric. Our proof uses the normal approximation result of Lambert, Ledoux, and Webb[2017], the Coulomb transport inequality of Chafa¨ı, Hardy, and Ma¨ıda [2018], and a spec-tral analysis for the operator associated with the limiting covariance structure. Contents β ∈ [0 , + ∞ ] ∗ Universit´e de Lille, CNRS, UMR 8524, Inria - Laboratoire Paul Painlev´e, F-59000 Lille, France.Email: [email protected] † University of Zurich, Winterthurerstrasse 190, 8057 Z¨urich, Switzerland.Email: [email protected] Introduction and statement of the results
Let T := [ − π, π ] ≃ R / π Z be the one-dimensional torus that we equip with the metric( x, y )
7→ | e i x − e i y | = | x − y ) | . Given an inverse temperature parameter β >
0, the Circular-beta-ensemble is a celebrated particle system from random matrix theory of N particles on T with distribution 1 Z N Y i
For any β > V : T → R ,we consider N random interacting particles on T with joint probability distributiond P N ( x , . . . , x N ) := 1 Z N Y i 0) of the Green function G of the two-dimensional torus T × T , that is ∆ G = − π ( δ − T × T in the distributional sense, see e.g. [Borodin and Serfaty, 2013]. Thus P N describes agas of N unit charges, interacting according to the laws of electrostatic on the two-dimensionaltorus but constrained to stay on T ≃ T × { } ⊂ T × T , in presence of an external potential V ,at temperature N/ (2 β ). As we shall see below, in this temperature regime, one of the mainreasons to study the statistical properties of such a Coulomb gas for large N is that there isa subtle competition between the energy and entropy of the gas which results in non-trivialglobal fluctuations. This fact is somewhat surprising knowing that for any β ≥ 0, the localfluctuations of the Coulomb gas (6) are described by a Poisson point process with intensity µ Vβ – this follows from adapting the argument from Nakano and Trinh [2019] from R to T . Macroscopic behavior. First, we discuss the large N limit of the empirical measure µ N , see(1), when the x i ’s are distributed according to P N . If µ lies in the space M ( T ) of probabilitymeasures on T , define its logarithmic energy by E ( µ ) := Z Z log (cid:12)(cid:12)(cid:12) sin (cid:0) x − y (cid:1)(cid:12)(cid:12)(cid:12) − µ (d x ) µ (d y ) ∈ [0 , + ∞ ] (9)= Z Z log 1 | e i x − e i y | µ (d x ) µ (d y ) + log 2 . (10)Moreover, given any µ, ν ∈ M ( T ), the relative entropy of µ with respect to ν is given by K ( µ | ν ) := Z log (cid:18) d µ d ν (cid:19) d µ ∈ [0 , + ∞ ] (11)when µ is absolutely continuous with respect to ν ; set K ( µ | ν ) := + ∞ otherwise. The functionalof interest here is F Vβ : M ( T ) → [0 , + ∞ ] defined by F Vβ ( µ ) := β E ( µ ) + K ( µ | µ V ) . (12)4ote that when F Vβ ( µ ) is finite, then µ is absolutely continuous and, if µ (d x ) = µ ( x )d x , thenwe can alternately write F Vβ ( µ ) = β E ( µ ) + Z V d µ + Z log µ d µ + log(2 π ) . In particular, when µ has a density and R log µ d µ < ∞ , we see that F V ∞ ( µ ) := lim β →∞ β F βVβ ( µ ) = E ( µ ) + Z V d µ (13)is the celebrated weighted logarithmic energy from potential theory [Saff and Totik, 1997]. Thenext result can be extracted from the literature. Theorem 1.1. Let β ≥ and assume V : T → R is continuous. (a) The functional F Vβ : M ( T ) → [0 , + ∞ ] has compact level sets { F β ≤ α } , α ∈ R , and isstrictly convex. In particular it has a unique minimizer µ Vβ on M ( T ) . (b) The sequence ( µ N ) satisfies a large deviation principle in M ( T ) equipped with its weaktopology at speed βN with rate function µ F β ( µ ) − F β ( µ Vβ ) . In particular, µ N a . s . −−−−→ N →∞ µ Vβ in the probability space N N ( T N , B ( T ) ⊗ N , P N ) . When β = 0, this is Sanov’s theorem for i.i.d random variables and elementary propertiesof the relative entropy, see e.g. [Dembo and Zeitouni, 2010]. Moreover, the unique minimizerof F V is given by (7) and hence the notation is consistent. In the case where β > 0, statement(a) is classical (see e.g. the proof of Proposition 2.1 below) and (b) can be found in [Berman,2018, Garc´ıa-Zelada, 2018]. In fact, statement (a) of the theorem is also true for weakerregularity assumptions on V and also when β = ∞ . Moreover, if one considers back thefixed temperature setting by taking the particle system (6) after the scaling β N β and V N V , then statement (b) holds true at the same speed with rate function F V ∞ − F ∞ ( µ V ∞ ),see [Hiai and Petz, 2000, Anderson, Guionnet, and Zeitouni, 2010].We will derive several properties for µ Vβ in Section 2 but let us already mention that, dueto the rotational invariance, the equilibrium measure µ β for V = 0 is the uniform probabilitymeasure d x π on T for every β ∈ [0 , ∞ ]. For a general potential V , we shall see that µ Vβ has abounded density that is larger than a positive constant and is essentially as smooth as V is. Macroscopic fluctuations. Our main result is a central limit theorem (CLT) for the randomsigned measure ν N := √ N ( µ N − µ Vβ ) (14)tested against sufficiently smooth functions, with an explicit upper bound on the rate of con-vergence in the Wasserstein W metric; the latter is defined for random variables X, Y takingvalues in R d by W ( X, Y ) := inf Z ∈ Π( X,Y ) r E h k Z − Z k i Z = ( Z , Z ) with Z = X and Z = Y. To state the result, let us also write µ Vβ for the density of the equilibrium measure, so thatd µ Vβ ( x ) = µ Vβ ( x )d x , and introduce the operator L defined by − L φ = φ ′′ + 2 πβH ( µ Vβ φ ′ ) + (log µ Vβ ) ′ φ ′ (15)which acts formally on the space L ( T ) of real-valued square integrable functions on T equippedwith the scalar product h f, g i L := Z T f ( x ) g ( x ) d x π . Here H stands for the Hilbert transform defined on L ( T ) by Hψ ( x ) := − p . v . Z T ψ ( t )tan (cid:0) x − t (cid:1) d t π (16)where p . v . is the Cauchy principal value, that is the limit as ε → | e i x − e i t | > ε . Note that when β = 0 the operator L corresponds tothe Sturm-Liouville operator L φ = − φ ′′ + V ′ φ ′ . As we shall see from Proposition 4.3 below,for any β > L is well-defined and positive on the Sobolev-type space H := (cid:26) ψ ∈ L ( T ) : ψ ′ ∈ L ( T ) , Z ψ d µ Vβ = 0 (cid:27) , (17)which is an Hilbert space once equipped with the inner-product h φ, ψ i H := Z φ ′ ψ ′ d µ Vβ , (18)and moreover that its inverse L − is trace-class on H .The central result of this work is that ν N converges, in the sense of finite dimensionaldistributions, to a Gaussian process on H with covariance operator L − . Theorem 1.2 (CLT) . Let β > and V ∈ C , ( T ) . Assume ψ ∈ C γ +1 ( T ) for some integer γ ≥ and that R ψ d µ Vβ = 0 . Then we have ν N ( ψ ) = 1 √ N N X i =1 ψ ( x i ) law −−−−→ N →∞ N (cid:0) , σ Vβ ( ψ ) (cid:1) (19) where the variance is given by σ Vβ ( ψ ) := h ψ, L − ψ i H = Z ψ ′ ( L − ψ ) ′ d µ Vβ . (20) Moreover, there exists C = C ( β, V, ψ ) > such that W (cid:16) ν N ( ψ ) , N (cid:0) , σ Vβ ( ψ ) (cid:1)(cid:17) ≤ C s log NN γ − γ +1 . 6f course the theorem still holds for a general ψ ∈ C γ +1 ( T ) after replacing ψ by ψ − R ψ d µ Vβ in the left hand side of (19) and in the limiting variance (20) . When V = 0, we can obtainan explicit formula for the limiting variance. Lemma 1.3. When V = 0 , we have σ β ( ψ ) = 2 ∞ X k =1 11 + β/k | ˆ ψ k | . This identity follows from the fact that, using the invariance by rotation, it is easy todiagonalize the operator L – see the identity (73) below. Indeed, in this setting we have − L φ = φ ′′ + βH ( φ ′ ) and the eigenfunctions are given by the Fourier basis φ j ( x ) = e i jx since L φ j = ( j + β | j | ) φ j for every j ∈ Z .Recalling (2)–(3) and (4)–(5), observe that σ β ( ψ ) → k ψ k L as β → βσ β ( ψ ) →k ψ k H / as β → ∞ ; the factor 2 disappears due to the change of scale we made for temperature.In this sense σ β ( ψ ) interpolates between the Lebesgue L and the Sobolev H / (semi-)norm.In Section 8, we establish that for a general potential, we also have σ Vβ ( ψ ) → k ψ k L ( µ V ) as β → µ Vβ so that βσ Vβ ( ψ ) → k ψ k H / as well as βσ βVβ ( ψ ) → k ψ k H / as β → ∞ (seeProposition 8.3). This establishes that the Gaussian process which appears in Theorem 1.2interpolates from a white noise (Poisson statistics) to a H / noise (random matrix statistics).This also shows that the fluctuations become universal, in the sense that they do not dependon V , only when β = ∞ . Remark 1.1. Let us observe that the rate of convergence in Theorem 1.2 does not dependon the smoothness of V , but it improves with the regularity of the test function. Moreover, if ψ ∈ C ∞ ( T ), we have W (cid:16) ν N ( ψ ) , N (cid:0) , σ Vβ ( ψ ) (cid:1)(cid:17) ≤ C r log NN . We expect this rate to be optimal, maybe up to the factor √ log N .The proof of Theorem 1.2 is deferred to Section 4 and relies on a normal approximationtechnique introduced in [Lambert, Ledoux, and Webb, 2017], which is inspired from Stein’smethod; see Theorem 4.5 below. In [Lambert et al., 2017] this method has been used to inves-tigate the rate of convergence of the fluctuations for beta-Ensembles on R at fixed temperature.There is a substantial technical difference in the analysis which arises in the high temperatureregime due to the fact that the operator L has an extra Sturm-Liouville component. In par-ticular, the spectral properties of L are quite different and this yields changes in the rate ofconvergence as well as in the limiting variance.Stein’s method has also been used previously in the context of random matrix theory toinvestigate the rate of convergence for linear statistics of random matrices from the classi-cal compact groups [Fulman, 2012, D¨obler and Stolz, 2011, 2014] and for the Circular beta-Ensemble at fixed temperature [Webb, 2016]. There are also results from Chatterjee [2009]on linear statistics of Wigner matrices which are valid under strong assumptions on the law Note the operator L − is only defined on the Hilbert space H , see (17). 7f the entries and from Johnson [2015] on the eigenvalues of random regular graphs. For acomprehensive introduction to Stein’s method which includes several applications, we refer tothe survey [Ross, 2011].On the road to establish the CLT, we prove the following concentration inequality whichmay be of independent interest: let W ( µ, ν ) be the Wasserstein-Kantorovich distance of order 1between µ, ν ∈ M ( T ), defined byW ( µ, ν ) := inf π ∈ Π( µ,ν ) Z Z | e i x − e i y | π (d x, d y ) = sup k f k Lip ≤ Z f d( µ − ν ) (21)where Π( µ, ν ) is the set of probability measures on T × T with respective marginals µ and ν ;the second identity is known as the Kantorovich-Rubinstein dual representation for W , wherethe supremum is taken over Lipschitz functions T → R with Lipschitz constant at most one. Theorem 1.4 (Concentration) . Let β > and assume V : T → R has a weak derivative V ′ in L ( T ) . Then, there exists C = C ( µ Vβ ) > such that, for every N ≥ and r > , P N (cid:16) W ( µ N , µ Vβ ) > r (cid:17) ≤ e − β ( π Nr − N − C ) . We have an explicit expression for the constant C in terms of µ Vβ in (43). In particular,when V = 0, this upper bound holds with C = 2 log 2 + 3 / π − ≃ . 2, which does notdepend on β .In particular, this yields together with Borel-Cantelli lemma that W ( µ N , µ Vβ ) → β > V = 0, that W ( µ N , d x π ) → β may depend on N as long as β ≫ N − . For lower order temperature scales this should still be true but one needs to proveit differently; note also there is an interesting change of behavior for the partition function ofthe Gaussian-beta-ensemble around β ∼ N − pointed out in [Pakzad, 2018, Lemma 1.3].The proof of the theorem follows the same strategy than the one of [Chafa¨ı, Hardy, and Ma¨ıda,2018] and rely on their Coulomb transport inequality. Differences however arise due to thepresence of the relative entropy in F Vβ . In particular, one needs to study the regularity of thepotential of the equilibrium measure. Organisation of the paper. In sections 2 we obtain preliminary results on the equilibriummeasure µ Vβ and its logarithmic potential. Section 3 is devoted to the proof of Theorem 1.4. Insection 4, we provide the core of the proof of Theorem 1.2. In section 5, we obtain concentrationestimates for error terms by means of Theorem 1.4. In section 6, we investigates the spectralproperties of the operator L ; in particular we show that L − is trace-class. In section 7, westudy the regularity of the eigenfunctions of the operator L so as to complete the proof of themain theorem. Finally, in Section 8, we investigate the behavior of the variance σ Vβ as β → β → ∞ (random matrix regime). Notations, basic properties and conventions. From now, β > η is a measure on T , we will denote by η ( x ) its density with respect to the Lebesgue measured x when it exists. If S ⊂ T is a Borel set, we denote by | S | its Lebesgue measure.Recall that T is equipped with the metric ( x, y ) 7→ | e i x − e i y | and denote for any k ∈ N := { , , , . . . } and 0 < α ≤ C k,α ( T ) the space of k -times differentiable functions on T whose8 -th derivative is α -H¨older continuous, or Lipschitz continuous when α = 1. When 0 < α < C α instead of C ,α , since there is not ambiguity, and put k ψ k C α := sup x,y ∈ T x = y | ψ ( x ) − ψ ( y ) || e i x − e i y | α , k ψ k Lip := sup x,y ∈ T x = y | ψ ( x ) − ψ ( y ) || e i x − e i y | . Note that, for any 0 < α < 1, we have k ψ k C α ( T ) ≤ k ψ k Lip .We sometimes use as well the chordal metric d T ( x, y ) := inf k ∈ Z | x − y + 2 kπ | (22)instead of the reference metric since they are equivalent: π d T ( x, y ) ≤ | e i x − e i y | ≤ d T ( x, y ) forany x, y ∈ R . Moreover, since Rademacher’s theorem states that the Lipschitz constant forthe metric d T reads k f ′ k L ∞ , we have k f ′ k L ∞ ≤ k f k Lip ≤ π k f ′ k L ∞ . (23)Recall that ˆ ψ k = R T ψ ( x ) e − i kx d x π denotes the Fourier coefficient of ψ ∈ L ( T ). Let L ( T ) = { ψ ∈ L ( T ) : ˆ ψ = 0 } and H m ( T ) be the Sobolev subspace of L ( T ) of functions havingtheir m -th first distributional derivatives in L ( T ). We will also use at several instances thecontinuous embedding H m +1 ( T ) ⊂ C m, / ( T ) for m ∈ N , sometimes known as the Sobolev-H¨older embedding theorem.Finally, we uses the letter C for a positive constant which may varies from line to line, andwhich may depend only on β > V unless stated otherwise. Acknowledgments. The authors wish to thank Benjamin Schlein and Sylvia Serfaty forinteresting discussions, and Severin Schraven for pointing out the reference [Brown et al., 2013].A. H. is supported by ANR JCJC grant BoB (ANR-16-CE23- 0003) and Labex CEMPI (ANR-11-LABX-0007-01). G.L. is supported by the grant SNSF Ambizione S-71114-05-01. In this section we study the minimizer µ Vβ of F β , see (12), and collect useful properties forlater. Given µ ∈ M ( T ), its logarithmic potential U µ : T → [0 , + ∞ ] is defined by U µ ( x ) := Z log (cid:12)(cid:12)(cid:12) sin (cid:0) x − y (cid:1)(cid:12)(cid:12)(cid:12) − µ (d y ) . Proposition 2.1. If V : T → R is a measurable and bounded function, then for any β ≥ , (a) F Vβ has a unique minimizer µ Vβ on M ( T ) . (b) µ Vβ is absolutely continuous and there exists a < δ < such that δ ≤ µ Vβ ( x )2 π ≤ δ − a.e.In particular, there exists < ℓ < such that ℓ ≤ U µ Vβ ≤ ℓ − on T . There exists a constant C Vβ ∈ R such that βU µ Vβ ( x ) + V ( x ) + log µ Vβ ( x ) = C Vβ a.e. (24)Part (c) of the proposition is usually referred as the Euler-Lagrange equation. Remark 2.1. If V = 0, then µ Vβ is the uniform measure d x π because of the rotational invariance.One can also check it satisfies (24) since, for any x ∈ T , U d x π ( x ) = Z T log (cid:12)(cid:12)(cid:12)(cid:12) − e i( x − y ) (cid:12)(cid:12)(cid:12)(cid:12) − d y π = log 2 . (25)Thus, the Euler-Lagrange constant reads C β = 2 β log 2 − log(2 π ). Remark 2.2. Part (a) of the proposition follows from well known results. Although part (b)and (c) seem to be part of the folklore, we were not able to locate (b) and (c) proven in fulldetails in the literature; the little subtlety is to take care of the sets where the density of µ Vβ may a priori vanish or be arbitrary close to zero due to the term log µ Vβ . Proof of Proposition 2.1. It is known that both mappings µ 7→ E ( µ ) and µ 7→ K ( µ | µ V )have compact level sets on M ( T ) and are strictly convex there, see [Saff and Totik, 1997,Dembo and Zeitouni, 2010], from which (a) directly follows. Moreover, since F β ( d x π ) < ∞ wehave E ( µ Vβ ) < ∞ and K ( µ Vβ | µ V ) < ∞ , and in particular µ Vβ is absolutely continuous.Let µ Vβ : T → R be any measurable function such that µ Vβ (d x ) = µ Vβ ( x )d x . We first claimthat the Borel set A := { x ∈ T : µ Vβ ( x ) = 0 } has null Lebesgue measure. Indeed, otherwisewe could define η := | A | − A ( x )d x ∈ M ( T ) and obtain, for any 0 < ε < F Vβ ((1 − ε ) µ Vβ + εη ) = F Vβ ( µ Vβ ) + ε (cid:18)Z (2 βU µ Vβ + V )d( η − µ ) + Z log η d η − Z log µ Vβ d µ Vβ (cid:19) + ε log ε + (1 − ε ) log(1 − ε ) + ε β E ( µ Vβ − η ) . This yields in turn F Vβ ((1 − ε ) µ Vβ + εη ) = F Vβ ( µ Vβ ) + ε ( C + log ε ) + O ( ε )when ε → C ∈ R and, since ε ( C + log ε ) + O ( ε ) is negative for every ε > µ Vβ is the unique minimizer. Thus | A | = 0 . We next prove a weak form of (c). Let φ : T → R be a measurable and bounded functionsatisfying R φ d µ Vβ = 0. Then, for any real | ε | ≤ k φ k − ∞ , we have (1 + εφ ) µ Vβ ∈ M ( T ) and F Vβ ((1 + εφ ) µ Vβ ) = F Vβ ( µ Vβ ) + ε Z (cid:0) βU µ Vβ + V + log µ Vβ (cid:1) φ d µ Vβ + ε E ( φ µ Vβ ) + Z (1 + εφ ) log(1 + εφ ) d µ Vβ . By definition of µ Vβ , the mapping ε F Vβ ((1 + εφ ) µ Vβ ) has a unique minimum at ε = 0 and,since R (1 + εφ ) log(1 + ε φ ) d µ Vβ = O ( ε ), we obtain Z (cid:0) βU µ Vβ + V + log µ Vβ (cid:1) φ d µ Vβ = 010or any such φ ’s. If η ∈ M ( T ) has a bounded density ψ with respect to µ Vβ , then by taking φ := ψ − Z (cid:0) βU µ Vβ + V + log µ Vβ (cid:1) d η = C Vβ := Z (cid:0) βU µ Vβ + V + log µ Vβ (cid:1) d µ Vβ . (26)Now, if one assumes A := { x ∈ T : 2 βU µ Vβ ( x ) + V ( x ) + log µ Vβ ( x ) > C Vβ } has µ Vβ -positivemeasure, then by taking η (d x ) := µ Vβ ( A ) − A ( x ) µ Vβ (d x ) in (26) we reach a contradiction.Since the same holds after replacing > by < we obtain2 βU µ Vβ + V + log µ Vβ = C Vβ , µ Vβ -a.e. (27)We are now equipped to prove (b) and (c). Using that U µ Vβ ≥ T , we obtain from (27)that µ Vβ ( x ) ≤ c e − V ( x ) µ Vβ -a.e for some c > 0, and thus the same holds true (Lebesgue)-a.e. Inparticular, since V is bounded by assumption, there exists C > µ Vβ ( x ) ≤ C π fora.e. x ∈ T . This yields in turn with (25) that U µ Vβ ( x ) ≤ CU d x π ( x ) = C log 2 on T . Next, let A κ := { x ∈ T : µ Vβ ( x ) ≤ κ } for any 0 < κ < 1. If µ Vβ ( A κ ) > 0, then by taking the measure η (d x ) := ( µ Vβ ( A κ )) − A κ ( x ) µ Vβ (d x ) in (26) we obtain C Vβ ≤ βC log 2 + k V k L ∞ + log κ and thus µ Vβ ( A κ ) = 0 for every κ > | A | = 0,this means that | A κ | = 0 for every κ > x log | sin( x ) | − is non-negative and integrable on T , the second claimsfollows as well.Finally, this yields that the equation (27) holds a.e. and thus (c) is proven. Corollary 2.2. For any µ ∈ M ( T ) satisfying E ( µ ) < ∞ , we have F Vβ ( µ ) − F Vβ ( µ Vβ ) = β E ( µ − µ Vβ ) + K ( µ | µ Vβ ) . Proof. One can assume µ has a density which satisfies R log µ d µ < ∞ since the identity isotherwise trivial. Similarly, one can assume E ( µ ) < ∞ so that E ( µ − µ Vβ ) makes sense (and isnon-negative), see [Saff and Totik, 1997, Lemma 1.8]. By integrating (24) against µ this yields C Vβ = 2 β Z U µ Vβ d µ + Z V d µ + Z log µ Vβ d µ. (28)In particular, we obtain by taking µ = µ Vβ and subtracting the resulting identity to (28), Z V d( µ − µ Vβ ) = 2 β E ( µ Vβ ) − β Z U µ Vβ d µ − Z log µ Vβ d µ + Z log µ Vβ d µ Vβ . The latter identity plugged into F Vβ ( µ ) − F Vβ ( µ Vβ ) yields the corollary.We also describe the behavior as β → β → ∞ of the equilibrium measure.11 emma 2.3. If V : T → R is measurable and bounded, then we have the weak convergences lim β → µ Vβ = µ V and lim β →∞ µ Vβ = µ ∞ = d x π . If we further assume V is lower semicontinuous and that µ V ∞ has a density which satisfies R log µ V ∞ d µ V ∞ < ∞ , then we have the weak convergence lim β →∞ µ βVβ = µ V ∞ . Note that V is lower semicontinous and does not take the value + ∞ ensures that F V ∞ islower semicontinuous and has a unique minimizer µ V ∞ on M ( T ), see [Saff and Totik, 1997]. Proof. First, since E is positive, µ Vβ minimizes F β , K ( µ V | µ V ) = 0 and E ( µ V ) < ∞ , we have K ( µ Vβ | µ V ) ≤ F β ( µ Vβ ) ≤ F β ( µ V ) = β E ( µ V ) −−−→ β → . Since µ 7→ K ( µ | µ V ) has for unique minimizer µ V and is lower semicontinuous on M ( T ), whichis weakly compact, this implies the weak convergence µ Vβ → µ V as β → µ ∞ = d x π is the unique minimizer of E on M ( T ). Since β E ( µ Vβ ) + K ( µ Vβ | µ V ) = F Vβ ( µ Vβ ) ≤ F Vβ ( d x π ) ≤ β E ( µ Vβ ) + K ( d x π | µ V )we obtain that K ( µ Vβ | µ V ) ≤ K ( d x π | µ V ) = R T V d x π < ∞ for every β > 0, and moreoverlim sup β →∞ E ( µ Vβ ) = lim sup β →∞ β F Vβ ( µ Vβ ) ≤ lim sup β →∞ β F Vβ ( d x π ) = E ( d x π ) . (29)Since E is lower semicontinuous on M ( T ), this similarly yields the weak convergence µ Vβ → d x π as β → ∞ .Finally, by observing that F βVβ ( µ ) = βF V ∞ ( µ ) + R log(2 πµ ) d µ provided µ ∈ M ( T ) has adensity, we have βF V ∞ ( µ βVβ ) + Z log(2 πµ βVβ ) d µ βVβ = F βVβ ( µ βVβ ) ≤ F βVβ ( µ V ∞ )= βF V ∞ ( µ V ∞ ) + Z log(2 πµ V ∞ ) d µ V ∞ Thus, if R log µ V ∞ d µ V ∞ < ∞ , after dividing by β > β → + ∞ , thisimplies lim sup β →∞ F V ∞ ( µ βVβ ) ≤ F V ∞ ( µ V ∞ )and the weak convergence µ βVβ → µ V ∞ as β → ∞ is obtained as well.12ext, we study the regularity of the equilibrium measure and its potential. Recall theHilbert transform H acting on the Hilbert space L ( T ) is defined in (16). We can also define Hµ for µ ∈ M ( T ) as soon as it has a density µ ( x ). Note that H acts in a simple fashion onthe Fourier basis: H (1) = 0 and, if k ∈ N \ { } , H (e i kx ) = ie i kx π Z T − e i k ( t − x ) − e i( t − x ) (1 + e i( t − x ) )d t = ie i kx . By taking the complex conjugate, this implies that for every k ∈ Z , H (e i kx ) = i sgn( k )e i kx (30)where we set sgn(0) := 0. This yields that H : L ( T ) → L ( T ) is a well-defined boundedoperator with adjoint H ∗ = − H . Moreover, when restricted to L ( T ), this turns H into anisometry which satisfies H − = − H . We will also use that this implies that for any f ∈ H ( T ), Hf also belong to the Sobolev space H ( T ) and that ( Hf ) ′ = H ( f ′ ). In the sequel, we willuse these properties of the Hilbert transform at several instances. Lemma 2.4. If V is measurable and bounded, then U µ Vβ ∈ H ( T ) and ( U µ Vβ ) ′ = πHµ Vβ .Proof. For any ϑ ∈ C ( T ) , by using the definition of the Cauchy principle value and doing anintegration by part we obtain, for every x ∈ T , U ϑ ′ ( x ) = Z T log (cid:12)(cid:12)(cid:12)(cid:12) sin (cid:18) x − t (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) − ϑ ′ ( t )d t = − p . v . Z T 12 tan (cid:0) x − t (cid:1) ϑ ( t )d t = πHϑ ( x ) . Next, using Fubini theorem and that H is a bounded operator on L ( T ) satisfying H ∗ = − H ,we obtain h ϑ ′ , U µ Vβ i L = h U ϑ ′ , µ Vβ i L = h πHϑ, µ Vβ i L = −h ϑ, πHµ Vβ i L . This shows that U µ Vβ has a distributional derivative given by πHµ Vβ . Moreover, since thedensity µ Vβ ( x ) belongs to L ( T ) by Proposition 2.1 (b), so does Hµ Vβ and thus ( U µ Vβ ) ′ ∈ L ( T ). Proposition 2.5. If V ∈ H ( T ) then U µ Vβ ∈ C , / ( T ) . Moreover, if V ∈ C m, ( T ) for some m ≥ , then µ Vβ ∈ C m, ( T ) . Note that V ∈ H ( T ) implies that V is continuous and this ensures the existence of µ Vβ . Proof. By differentiating the Euler-Lagrange equation (24) we obtain the distributional identity( µ Vβ ) ′ = − µ Vβ (2 βπHµ Vβ + V ′ ) . (31)Since Hµ Vβ ∈ L ( T ) and k µ Vβ k L ∞ < ∞ according to Proposition 2.1 (b), (31) yields that µ Vβ ∈ H ( T ) as soon as V ′ ∈ L ( T ). This also shows that ( Hµ Vβ ) ′ = H ( µ Vβ ) ′ ∈ L ( T ) and thus13 µ Vβ ∈ H ( T ) ⊂ C / ( T ). In particular, the first claim follows by Lemma 2.4. Moreover, if wefurther assume that V ∈ C , ( T ), then k ( µ Vβ ) ′ k L ∞ < ∞ by (31) and the second statement isproven for m = 0.Next, we differentiate (31) in order to obtain( µ Vβ ) ′′ = − ( µ Vβ ) ′ (2 βπHµ Vβ + V ′ ) − µ Vβ (2 βπH ( µ Vβ ) ′ + V ′′ ) . (32)If we assume V ∈ C , ( T ), then in particular it is C , and we have already shown that k ( µ Vβ ) ′ k L ∞ < ∞ . Together with (32) this provides ( µ Vβ ) ′′ ∈ L ( T ), thus ( H ( µ Vβ ) ′ ) ′ = H ( µ Vβ ) ′′ ∈ L ( T ), and this yields in turn H ( µ Vβ ) ′ ∈ C / ( T ). Using (32) again, we obtain k ( µ Vβ ) ′′ k L ∞ < ∞ and the claim holds for m = 1.The case m ≥ We now turn to the proof of Theorem 1.4. The proof follows the same strategy than the one in[Chafa¨ı, Hardy, and Ma¨ıda, 2018] and is based on combining a Coulomb transport inequalitytogether with an energy estimate after an appropriate regularization of the empirical measure.The regularization we use here is rather similar to [Ma¨ıda and Maurel-Segala, 2014] and thetechnical input with this respect here is the following lemma. Lemma 3.1. Given any configuration of distinct points x , . . . , x N ∈ T , there exists a config-uration y , . . . , y N ∈ T satisfying: min j = k | e i y j − e i y k | ≥ N , N X j =1 | e i x j − e i y j | ≤ , and X j = k log 1 | e i x j − e i x k | ≥ X j = k log 1 | e i y j − e i y k | − N. Proof. Given any ordered configuration x < . . . < x N in T , there exists at least one index j such that x j +1 − x j ≥ π/N . Thus, by permutation and translation, one can assume withoutloss of generality that − π + 2 N ≤ x < . . . < x N < π − N . Consider the increasing bijection x ∈ T ˜ x := tan( x/ ∈ R ∪ {±∞} which satisfies | e i x − e i y | = 2 | ˜ x − ˜ y |√ x p y . (33)We set ˜ y := ˜ x and ˜ y j +1 := ˜ y j + max(˜ x j +1 − ˜ x j , N − ) and then let y < . . . < y N ∈ T bethe configuration obtained by taking the image of the ˜ y j ’s by the inverse bijection. Since byconstruction ˜ y j − ˜ x j ≤ ( j − N − we have N X j =1 | e i x j − e i y j | ≤ N − N X j =1 ( j − ≤ . x j ’s we have max j | ˜ x j | ≤ | tan( π − N ) | ≤ N, which yieldsmax j | ˜ y j | ≤ N, and we thus obtain, for every j = k , | e i y j − e i y k | ≥ N (1 + 4 N ) ≥ N . Finally, we have X j = k log 2 | e i x j − e i x k | = X j = k log 1 | ˜ x j − ˜ x k | + ( N − N X j =1 log(1 + ˜ x j ) ≥ X j = k log 1 | ˜ y j − ˜ y k | + ( N − N X j =1 log(1 + ˜ x j )= X j = k log 2 | e i y j − e i y k | + ( N − N X j =1 log (cid:16) x j y j (cid:17) ≥ X j = k log 2 | e i y j − e i y k | − ( N − N X j =1 log (cid:16) x j + N − ( j − x j (cid:17) . Using that, for any 0 < c < x ∈ R log (cid:18) x + c ) x (cid:19) = log (cid:18) c √ c + 4 − c (cid:19) ≤ c, we obtain ( N − N X j =1 log x j + N − ( j − x j ! ≤ N which completes the proof of the lemma. Proof of Theorem 1.4. Recalling (6), if we set for convenience g ( x ) := log (cid:12)(cid:12)(cid:12) sin (cid:16) x (cid:17)(cid:12)(cid:12)(cid:12) − (34)then we can writed P N ( x , . . . , x N ) = 1 Z ′ N exp − βN X j = k g ( x j − x k ) − N X j =1 V ( x j ) N Y j =1 d x j π (35)for some new normalization constant Z ′ N > Step 1: Lower bound on the partition function. By writing Z ′ N = Z exp n − βN X j = k g ( x j − x k ) − N X j =1 (cid:0) V ( x j ) + log(2 πµ Vβ ( x j )) (cid:1)o N Y j =1 µ Vβ (d x j )15nd using Jensen’s inequality, we obtainlog Z ′ N ≥ Z − βN X j = k g ( x j − x k ) − N X j =1 (cid:0) V ( x j ) + log(2 πµ Vβ ( x j )) (cid:1) N Y j =1 µ Vβ (d x j )= − β ( N − E ( µ Vβ ) − N Z ( V + log(2 πµ Vβ )) d µ Vβ = − N F β ( µ Vβ ) + β E ( µ Vβ ) . (36) Step 2: Regularization and energy estimates. Given any configuration x , . . . , x N ∈ T ofdistinct points, let y , . . . , y N ∈ T be as in Lemma 3.1 and set e µ N := 1 N n X i =1 δ y i ∗ λ N − , λ ε := [0 ,ε ] ( x ) d xε . Since g ′ ( x ) = − (2 tan( x/ − , a Taylor-Lagrange expansion yields for any | u | ≤ | x | / | g ( x + u ) − g ( x ) | ≤ | u | | x/ | ≤ | u | sin | x/ | . (37)Since Lemma 3.1 yields sin( | y j − y k | / ≥ N − / 10 and sin( | y k − y j − u | / ≥ N − / − | u | when j = k , we obtain from Lemma 3.1 again and (37) that, for N ≥ X j = k g ( x j − x k ) ≥ X j = k g ( y j − y k ) − N ≥ X j = k Z g ( y j − y k + u ) λ N − (d u ) − N ≥ X j = k Z Z g ( y j − y k + u − v ) λ N − (d u ) λ N − (d v ) − N = N E ( e µ N ) − N E ( λ N − ) − N. (38)Next, by using that 2 | sin( a ) | ≥ | a | when | a | ≤ 1, we obtain the upper bound E ( λ ε ) ≤ Z log 1 | x − y | λ ε (d x ) λ ε (d y ) + log 2 = − log ε + 3 / . (39)If we set c := E ( µ Vβ ) + 16 + 3 / Z ′ N exp − βN X j = k g ( x j − x k ) − n X j =1 V ( x j ) ≤ e β (5 log N + c ) e − N (cid:0) β E ( e µ N ) − F β ( µ Vβ ) (cid:1) − P Nj =1 V ( x j ) = e β (5 log N + c ) e − N (cid:0) F β ( e µ N ) − F β ( µ Vβ ) −K ( e µ N | µ Vβ ) (cid:1) + N R Q d( e µ N − µ N ) N Y j =1 πµ Vβ ( x j )16here Q := V + log µ Vβ = C Vβ − βU µ Vβ by (24). Using Corollary 2.2, we deduce from (35) thatfor any r > P N (cid:0) E ( e µ N − µ Vβ ) > r (cid:1) ≤ e − β ( Nr − N − c ) Z e N R Q d( e µ N − µ N ) ( µ Vβ ) ⊗ N (d x ) . (40)Finally, since by assumption V ∈ H ( T ), Proposition 2.5 yields U µ Vβ is Lipschitz and, usingagain Lemma 3.1, we have (cid:12)(cid:12)(cid:12)(cid:12) N Z Q d( e µ N − µ N ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ β N X j =1 Z (cid:12)(cid:12) U µ Vβ ( y j + u ) − U µ Vβ ( x j ) (cid:12)(cid:12) λ N − (d u ) ≤ β k U µ Vβ k Lip N X j =1 | e i x j − e i y j | + 2 N Z | sin( u/ | λ N − (d u ) ≤ β k U µ Vβ k Lip . Together with (40), we have finally obtained the energy estimate P N (cid:0) E ( e µ N − µ Vβ ) > r (cid:1) ≤ e − β ( Nr − N − ˜ C ) , (41)where ˜ C := E ( µ Vβ ) + 3 k U µ Vβ k Lip + 16 + 3 / Step 4: The Coulomb transport inequality and conclusion. Lemma 3.1 yields,W ( µ N , e µ N ) ≤ N N X j =1 Z | e i x j − e i( y j + u ) | λ N − (d u ) ≤ N + 2 Z | sin( u/ | λ N − (d u ) ≤ N . (42)Since both e µ N and µ Vβ have finite logarithmic energy, it follows from [Chafa¨ı et al., 2018,Theorem 1.1] and the discussion below that, for every ε > ( e µ N , µ Vβ ) ≤ π E ( e µ N − µ Vβ ) . Moreover, using that12 W ( µ N , µ Vβ ) ≤ W ( e µ N , µ Vβ ) + W ( µ N , e µ N ) ≤ W ( e µ N , µ Vβ ) + 4 N , we obtain for any r > P N (cid:16) W ( µ N , µ Vβ ) > r (cid:17) ≤ P N (cid:16) W ( e µ N , µ Vβ ) > r − N (cid:17) ≤ P N (cid:18) E ( e µ N − µ Vβ ) > C T (cid:16) r − N (cid:17)(cid:19) ≤ e − β ( π Nr − N − C ) where the constant is given by C := E ( µ Vβ ) + 3 k U µ Vβ k Lip + 16 + 32 + log 2 + 1 π (43)and the proof of the theorem is complete. 17 Main steps for the proof of Theorem 1.2 In this section, we explain the main strategy to prove Theorem 1.2. It is based on the multi-dimensional Gaussian approximation result from [Lambert, Ledoux, and Webb, 2017] com-bined with the previous concentration inequality and a study of the spectral properties of theoperator L .Consider the differential operator given by L := ∆ − ∇ (cid:0) βN H ( x ) + P Nj =1 V ( x j ) (cid:1) · ∇ = N X j =1 ∂ x j + 2 βN X i = j ∂ x j (cid:16) x j − x i (cid:17) − N X j =1 V ′ ( x j ) ∂ x j , which satisfies the integration by part identity R f ( − L g ) d P N = R ∇ f · ∇ g d P N for any smoothfunctions f, g : T N → R .Recalling that ν N = √ N (ˆ µ N − µ Vβ ), we first show that ν N ( φ ), seen as a mapping T N → R ,is an approximate eigenfunction for L as long as φ is a (strong) eigenfunction of the differentialoperator L defined in (15). More precisely, we have the approximate commutation relation: Lemma 4.1. For any φ ∈ C ( T ) we have L ν N ( φ ) = − ν N ( L φ ) + β √ N ζ N ( φ ) (44) where we introduced ζ N ( φ ) := Z Z φ ′ ( x ) − φ ′ ( y )2 tan( x − y ) ν N (d x ) ν N (d y ) − Z φ ′′ d µ N . (45) Proof. If we set Φ( x ) := P Nj =1 φ ( x j ) then we have L Φ( x ) = N X j =1 φ ′′ ( x j ) + βN N X i = j φ ′ ( x j )tan( x j − x i ) − N X j =1 φ ′ ( x j ) V ′ ( x j )= (cid:18) − βN (cid:19) N X j =1 φ ′′ ( x j ) + βN N X i,j =1 φ ′ ( x j ) − φ ′ ( x i )2 tan( x j − x i ) − N X j =1 φ ′ ( x j ) V ′ ( x j ) . (46)Next, it is convenient to introduce the operator Ξ defined byΞ ψ ( x ) := Z ψ ( x ) − ψ ( t )2 tan( x − t ) µ Vβ (d t ) , (47)which is a weighted version of the Hilbert transform H defined in (16). Indeed, we can write1 N N X i,j =1 φ ′ ( x j ) − φ ′ ( x i )2 tan( x j − x i )= 2 √ N Z Ξ( φ ′ ) d ν N + N Z Ξ( φ ′ ) d µ Vβ + Z Z φ ′ ( x ) − φ ′ ( y )2 tan( x − y ) ν N (d x ) ν N (d y )18nd this yields together with (46) and (45), L Φ = N Z ( φ ′′ + β Ξ( φ ′ ) − φ ′ V ′ ) d µ Vβ + √ N Z ( φ ′′ + 2 β Ξ( φ ′ ) − φ ′ V ′ ) d ν N + βζ N ( φ ) . (48)By (15), observe that the variational equation (31) yields φ ′′ + 2 β Ξ( φ ′ ) − V ′ φ ′ = − L φ (49)where we used that, by (47), Ξ ψ = π (cid:0) H ( ψµ Vβ ) − ψH ( µ Vβ ) (cid:1) . (50)Moreover, we obtain by using that H ∗ = − H , Z Ξ ψ d µ Vβ = − π Z H ( µ Vβ ) ψ d µ Vβ + π h H ( µ Vβ ψ ) , µ Vβ i L = − π Z H ( µ Vβ ) ψ d µ Vβ . By integrating (31) against φ ′ d x , this yields together with an integration by parts: Z ( φ ′′ + β Ξ( φ ′ ) − φ ′ V ′ ) d µ Vβ = 0 . (51)By combining (48)–(51), we have finally shown that L Φ = −√ Nν N ( L φ ) + βζ N ( φ )and the result follows by linearity of L since Φ = √ Nν N ( φ ) + N R φ d µ Vβ .It turns out the random variables ζ N ( φ ) are of smaller order of magnitude than the fluctu-ations provided φ is smooth enough. More precisely, we have the following estimates. Lemma 4.2. There exists a constant C = C ( β, V ) > such that, for any N ≥ , for any φ ∈ C , ( T ) , we have E h(cid:12)(cid:12) ζ N ( φ ) (cid:12)(cid:12) i ≤ C k φ ′′′ k (log N ) . (52) Moreover, for any Lipschitz function g : T → R E "(cid:12)(cid:12)(cid:12)(cid:12)Z g d ν N (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k g k log N. (53)The proof of this lemma is based on Theorem 1.4 and is postponed to Section 5.Another important input is the existence of an eigenbasis of H for the operator L thatbehaves like an eigenbasis of a Sturm-Liouville operator. Note that by (17), H is a separableHilbert space and it follows from Proposition 2.1(b) that the associated norm satisfies δ k ψ ′ k L ≤ k ψ k H ≤ δ − k ψ ′ k L , a fact we will use at several instances below. 19 roposition 4.3. Assume that V ∈ C m, ( T ) for some m ≥ . Then there exists a family ( φ j ) ∞ j =1 of functions φ j : T → R such that: (a) L φ j = κ j φ j where ( κ j ) ∞ j =1 is an increasing sequence of positive numbers. (b) ( φ j ) ∞ j =1 is an orthonormal basis of the Hilbert space H . (c) There exists α > such that κ j ∼ αj as j → ∞ . for every j ≥ . (d) φ j ∈ C m ( T ) and there exist constants C k such that for every k ∈ { , . . . , m } , k φ ( k ) j k Lip ≤ C k κ k +12 j . The proof of Proposition 4.3 is postponed to the sections 6 and 7. Proposition 4.4. Assume that the external potential V ∈ C , ( T ) . There exists a constant C = C ( β, V ) > such that, if we set F := (cid:0) √ κ ν N ( φ ) , . . . , √ κ d ν N ( φ d ) (cid:1) , then we have for every N ≥ and d ≥ , W (cid:0) F, N (0 , I d ) (cid:1) ≤ C √ N β log N vuut d X j =1 κ j + vuut log N d X j =1 κ j d X j =1 κ j . Here N (0 , I d ) stands for a real standard Gaussian random vector in R d . This propositionis a consequence of the previous concentration estimates together with the following generalnormal approximation given by [Lambert et al., 2017, Proposition 2.1]; for F = ( F , . . . , F d ) ∈C ( T N , R d ) we set L F := ( L F , . . . , L F d ) and denote by k · k R d the Euclidean norm of R d . Theorem 4.5. For any given F = ( F , . . . , F d ) ∈ C ( T N , R d ) , let Γ := h ∇ F i · ∇ F j i di,j =1 and see both F and Γ as random variables defined on the probability space ( T N , B ( T ) ⊗ N , P N ) .Given any d × d diagonal matrix K with positive diagonal entries, we have W (cid:0) F, N (0 , I d ) (cid:1) ≤ r E h k F + K − L F (cid:13)(cid:13) R d i + r E h(cid:13)(cid:13) I d − K − Γ (cid:13)(cid:13) R d × d i . Proof of Proposition 4.4. By Proposition 4.3 (a) and Lemma 4.1, we have for every j ≥ L F j = − κ j F j + β r κ j N ζ N ( φ j ) . As a consequence, taking K := diag( κ , . . . , κ d ), we obtain E h k F + K − L F (cid:13)(cid:13) R d i = β N d X j =1 κ − j E h(cid:12)(cid:12) ζ N ( φ j ) (cid:12)(cid:12) i E h k F + K − L F (cid:13)(cid:13) R d i ≤ Cβ (log N ) N d X j =1 κ j . (54)Next, since for any i, j ∈ N ,Γ ij = ∇ F i · ∇ F j = √ κ i κ j Z φ ′ i φ ′ j d µ N , and using that the φ j ’s are orthonormal, we obtain E h(cid:13)(cid:13) I d − K − Γ (cid:13)(cid:13) R d × d i = E h(cid:13)(cid:13) I d − K − / Γ K − / (cid:13)(cid:13) R d × d i = 1 N d X i,j =1 E "(cid:12)(cid:12)(cid:12)(cid:12)Z φ ′ i φ ′ j d ν N (cid:12)(cid:12)(cid:12)(cid:12) . Proposition 4.3 (d) moreover yield, for any i, j ∈ N , k φ ′ i φ ′ j k Lip ≤ k φ ′ i k Lip k φ j k Lip + k φ i k Lip k φ ′ j k Lip ≤ C (cid:0) κ i √ κ j + √ κ i κ j (cid:1) and it thus follows from Lemma 4.2 that E h(cid:13)(cid:13) I d − K − Γ (cid:13)(cid:13) R d × d i ≤ C log NN d X j =1 κ j d X j =1 κ j . (55)The proposition follows by combining estimates (54) and (55) together with Theorem 4.5.We are finally in position to prove Theorem 1.2 by decomposing a general test functioninto the eigenbasis ( φ j ) ∞ j =1 and by using Proposition 4.4. Proof of Theorem 1.2. Assume that V ∈ C , ( T ) and let ψ ∈ C γ +1 ( T ) for some integer γ ≥ R ψ d µ Vβ = 0. Thus ψ ∈ H and we have byProposition 4.3 (b), ψ H = ∞ X j =1 h ψ, φ j i H φ j (56)Moreover, since ψ lies in the domain of L γ and using that L is symmetric, we have (cid:12)(cid:12) h ψ, φ j i H (cid:12)(cid:12) = 1 κ γj (cid:12)(cid:12) h L γ ψ, φ j i H (cid:12)(cid:12) ≤ κ γj k L γ ψ k H . (57)In particular, by Proposition 4.3 (c), the series (56) converges uniformly on T .Next, given any d ∈ N , let us consider the truncation of ψ , ψ [ d ] := d X j =1 h ψ, φ j i H φ j . γ > k ψ − ψ [ d ] k Lip ≤ C ∞ X j = d +1 (cid:12)(cid:12) h ψ, φ j i H (cid:12)(cid:12) √ κ j ≤ C k L γ ψ k H ∞ X j = d +1 κ − γ +1 / j ≤ C k L γ ψ k H d − γ − . Thus, by definition of the W metric and Lemma 4.2, this yieldsW (cid:16) ν N ( ψ ) , ν N ( ψ [ d ] ) (cid:17) ≤ s E (cid:20)(cid:12)(cid:12)(cid:12) Z ( ψ − ψ [ d ] ) d ν N (cid:12)(cid:12)(cid:12) (cid:21) ≤ C k L γ ψ k H √ log Nd γ − . (58)Next, if we set η m := P mj =1 κ − j (cid:12)(cid:12) h ψ, φ j i H (cid:12)(cid:12) for m ∈ N ∪ {∞} , then we obtain from (57) andProposition 4.3 (c) thatW (cid:0) N (0 , η d ) , N (0 , η ∞ ) (cid:1) = ( √ η ∞ − √ η d ) ≤ η ∞ − η d = ∞ X j = d +1 κ − j (cid:12)(cid:12) h ψ, φ j i H (cid:12)(cid:12) ≤ C k L γ ψ k H d − γ − . (59)Last, Proposition 4.4 and Proposition 4.3 (c) yieldsW (cid:16) ν N ( ψ [ d ] ) , N (0 , η d ) (cid:17) = W d X j =1 h ψ, φ j i H √ κ j ν N ( φ j ) , N (0 , η d ) ≤ √ η ∞ W (cid:0) F, N (0 , I d ) (cid:1) ≤ r Cη ∞ N (cid:16) β log N d / + p log N d (cid:17) . (60)Finally, by combining the estimates (58)–(60) and taking for d the integer part of N / γ +1 ,we obtain W (cid:0) ν N ( ψ ) , N (0 , η ∞ ) (cid:1) ≤ C ψ s log NN γ − γ +1 where C ψ > η ∞ , k L γ ψ k H , β and V only. It remains to check that η ∞ equals tothe variance σ Vβ ( ψ ) given in (20); this is proven in Proposition 6.3 below. The proof of thetheorem is therefore complete. If we use the Kantorovich-Rubinstein dual representation of W and take r := R p log N/N in Theorem 1.4, then under the same assumptions and using the same notation as in that22heorem we obtain the following estimate: there exists C = C ( µ Vβ , β ) > κ = κ ( β ) > R ≥ N ≥ P N sup k f k Lip ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z f d ν N (cid:12)(cid:12)(cid:12)(cid:12) > p R log N ! ≤ CN − κR . (61)We also need the next estimate. Lemma 5.1. There exists κ = κ ( β ) > and a constant R > such that, for any function ψ ∈ C , ( T ) , one has for every R ≥ R and N ≥ , P N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Z ψ ( x ) − ψ ( y )2 tan( x − y ) ν N (d x ) ν N (d y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > R k ψ ′′ k Lip log N ! ≤ CN − κ ′ R . Proof. The strategy is to prove that the random functionΨ N ( x ) := Z ψ ( x ) − ψ ( y )2 tan( x − y ) ν N (d y )has Lipschitz constant controlled by k ψ ′′ k Lip √ log N with high probability and then to use (61).Since ψ ∈ C , ( T ), we verify that for any x ∈ R ,Ψ ′ N ( x ) = − Z ψ ( x ) − ψ ( y ) − ψ ′ ( x ) sin( x − y )4 sin ( x − y ) ν N (d y ) . (62)We now provide an upper bound on the Lipschitz constant of the integrand of Ψ ′ N which isuniform in x . Indeed, we havedd y ψ ( x ) − ψ ( y ) − ψ ′ ( x ) sin( x − y )4 sin ( x − y ) ! = (cid:0) ψ ( x ) − ψ ( y ) (cid:1) cos( x − y ) − (cid:0) ψ ′ ( x ) + ψ ′ ( y ) (cid:1) sin( x − y )4 sin ( x − y ) . (63)Let et us recall that we introduced d T ( x, y ) in (22). Two Taylor-Lagrange expansions yield,for any x, y ∈ T , ψ ( x ) − ψ ( y ) = d T ( x, y )2 (cid:0) ψ ′ ( x ) + ψ ′ ( y ) (cid:1) − d T ( x, y ) (cid:0) ψ ′′ ( u ) − ψ ′′ ( v ) (cid:1) for some u, v ∈ T , so that ψ ( x ) − ψ ( y ) = d T ( x, y )2 (cid:0) ψ ′ ( x ) + ψ ′ ( y ) (cid:1) + O (cid:0) d T ( x, y ) (cid:1) k ψ ′′ k Lip . Together with (63), this implies that there exists a constant C > x,y ∈ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dd y ψ ( x ) − ψ ( y ) − ψ ′ ( x ) sin( x − y )4 sin ( x − y ) !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:0) k ψ ′′ k Lip + k ψ ′ k L ∞ (cid:1) . Since the mean value theorem yields that k ψ ′ k L ∞ ≤ π k ψ ′′ k L ∞ ≤ π k ψ ′′ k Lip , (64)23e deduce from (61) that there exist constants κ ′ ≥ κ and R > R ≥ R and N ≥ P N (cid:16) k Ψ N k Lip > k ψ ′′ k Lip p R log N (cid:17) ≤ CN − κ ′ R . Therefore, by (61) again, we obtain P N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Z ψ ( x ) − ψ ( y )2 tan( x − y ) ν N (d x ) ν N (d y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > k ψ ′′ k Lip R log N ! ≤ P N (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) Z Ψ N d ν N (cid:12)(cid:12)(cid:12)(cid:12) > k ψ ′′ k Lip R log N, k Ψ N k Lip ≤ k ψ ′′ k Lip p R log N (cid:19) + CN − κ ′ R ≤ CN − κ ′ R . which completes the proof of the lemma. Proof of Lemma 4.2. Using (61) and that, for any real random variable X and α > E [ X ] = α Z ∞ P ( | X | ≥ √ αR ) d R, (65)we obtain for any N ≥ 10 and any Lipschitz function g : T → R that E "(cid:12)(cid:12)(cid:12)(cid:12)Z g d ν N (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18) κ log N (cid:19) k g k log N (66)and the second statement of the lemma is obtained.Next, according to (45) and since µ N is a probability measure, we have (cid:12)(cid:12) ζ N ( φ ) (cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) Z Z φ ′ ( x ) − φ ′ ( y )2 tan( x − y ) ν N (d x ) ν N (d y ) (cid:12)(cid:12)(cid:12)(cid:12) + k φ ′′ k L ∞ . Using the inequality k φ ′′ k L ∞ ≤ π k φ ′′′ k Lip obtained as in (64), we deduce from Lemma 5.1that for all R ≥ R and N ≥ P (cid:16)(cid:12)(cid:12) ζ N ( φ ) (cid:12)(cid:12) ≥ R k φ ′′′ k Lip log N (cid:17) ≤ CN − κ ′ R . Thus, combined with (65) this yields E h(cid:12)(cid:12) ζ N ( φ ) (cid:12)(cid:12) i ≤ k φ ′′′ k (log N ) (cid:18) R + 2 C ( κ ′ log N ) (cid:19) (67)and the proof of the lemma is complete. 24 Spectral theory: Proof of Proposition 4.3 (a)–(c) In this section, we always assume V ∈ C , ( T ). In particular it follows from Proposition 2.5that (log µ Vβ ) ′ is Lipschitz continuous. Recalling (15), we write L = A + 2 πβ W (68)where we introduced the operators on L ( T ), A φ := − φ ′′ − (log µ Vβ ) ′ φ ′ = − ( φ ′ µ Vβ ) ′ µ Vβ , W φ := − H ( φ ′ µ Vβ ) . (69)Note that A is a Sturm-Liouville operator in the sense that it reads − A = dd x (cid:18) p ( x ) dd x (cid:19) + q ( x )with p := log µ Vβ and q := 0; we refer to [Marchenko, 2011, Brown et al., 2013] for generalreferences on Sturm-Liouville equations.We first check that L is a positive operator on H , as a consequence of the next lemma. Lemma 6.1. The operators A and W are both positive on H .Proof. We have for any function φ ∈ H , h A φ, φ i H = − Z T (cid:18) ϕ ′ µ Vβ (cid:19) ′ ϕ d x = Z T | ϕ ′ | d xµ Vβ ≥ ϕ := φ ′ µ Vβ . Moreover, if one decomposes ϕ in the Fourier basis, then we have h W φ, φ i H = − Z T H ( ϕ ) ′ ϕ d x = X k ∈ Z | k || ˆ ϕ k | = k ϕ k H / ≥ A (with periodic boundary condi-tions) are well known, see for instance [Brown et al., 2013, Chapter 2 and 3], from which onecan obtain the basic properties: Lemma 6.2. There exists a orthonormal basis ( ϕ j ) ∞ j =1 of H consisting of (weak) eigenfunctionsof A associated with positive eigenvalues. Moreover, if A ϕ j = λ j ϕ j with < λ ≤ λ ≤ · · · , then there exists α > such that, as j → ∞ , λ j ∼ αj . roof. Since for any smooth function φ : T → R we have h A φ, φ i L ( µ Vβ ) := Z A φ φ d µ Vβ = Z | φ ′ | d µ Vβ ≥ , (71)we see that A is a positive Sturm-Liouville operator on L ( µ Vβ ) whose domain is H ( µ Vβ ) := { φ ∈ L ( µ Vβ ) : φ ′ ∈ L ( µ Vβ ) } = H ( T ), where we used Proposition 2.1 (b) for this equality. Itthen follows from the general properties of the Sturm-Liouville operators that there exists anorthonormal basis of L ( µ Vβ ) consisting of eigenfunctions ( e ϕ j ) + ∞ j =0 ⊂ H ( T ) of A associated tonon-negative increasing eigenvalues ( λ j ) ∞ j =0 . Moreover, by Weyl’s law (see e.g. [Brown et al.,2013, Theorem 3.3.2] in our setting), there exists α > λ j ∼ αj as j → ∞ .The smallest eigenvalue λ = 0 comes with the eigenfunction e ϕ = 1 which is orthogonalto H in L ( T ), see (17). Since the e ϕ j ’s are orthonormal in L ( µ Vβ ), we have for any j ≥ Z e ϕ j d µ Vβ = h e ϕ j , e ϕ i L ( µ Vβ ) = 0 (72)and thus ( e ϕ j ) ∞ j =1 ⊂ H . Moreover, since we have for any i, j ∈ N , h e ϕ i , e ϕ j i H = h e ϕ ′ i , e ϕ ′ j i L ( µ Vβ ) = h e ϕ i , A e ϕ j i L ( µ Vβ ) = λ j δ ij , it follows that λ > e ϕ would be a non-zero constant function and this wouldcontradict (72)). Finally, if we set ϕ j := e ϕ j / p λ j , then the family ( ϕ j ) ∞ j =1 is an orthonormalbasis of H that satisfies the requirements of the lemma. Proposition 6.3. Proposition 4.3 (a)–(c) hold true. More precisely, there exists a orthonormalbasis ( φ j ) ∞ j =1 of H such that L φ j = κ j φ j with < κ ≤ κ ≤ · · · and we have κ j ∼ αj as j → ∞ for the same α > than in Lemma 6.2. In particular, L − is a well defined traceclass operator on H and, for any ψ ∈ H , we have h ψ, L − ψ i H = ∞ X j =1 κ j (cid:12)(cid:12) h ψ, φ j i H (cid:12)(cid:12) . (73) Proof. We use here basic results from operator theory, see e.g. [Kato, 1995]. Lemma 6.2yields that A is a positive self-adjoint operator on H and that A − is trace-class. Since W is non-negative and self-adjoint on H , it follows that L − = ( A + 2 πβ W ) − is a positiveself-adjoint compact operator on H . The spectral theorem for self-adjoint compact operatorsthen yields the existence of an orthonormal family ( φ j ) ∞ j =1 in H and an increasing sequence ofpositive numbers ( κ j ) ∞ j =1 such that L − = P j κ − j φ j ⊗ φ j . In particular L φ j = κ j φ j weaklyfor every j ≥ 1. Moreover, since L − is positive, the family ( φ j ) is necessarily a completeorthonormal family in H : part (a) and (b) are thus proven.26riting h· , ·i instead of h· , ·i H for simplicity, the min-max theorem (see e.g. [Reed and Simon,1978, Theorem XIII.2]) yields, for any j ≥ κ j = max S j − min ψ ∈ S ⊥ j − k ψ k =1 h L ψ, ψ i , where the maximum is taken over all subspace S j − ⊂ H of dimension j − 1. By taking˜ S j := span( ϕ , . . . , ϕ j ) where ( ϕ j ) ∞ j =1 is as in Lemma 6.2, this provides κ j ≥ min ψ ∈ ˜ S ⊥ j − k ψ k =1 h L ψ, ψ i≥ min ψ ∈ ˜ S ⊥ j − k ψ k =1 h A ψ, ψ i = λ j (74)where we also used that W ≥ j ≥ κ j = min S j max ψ ∈ S j k ψ k =1 h L ψ, ψ i≤ max ψ ∈ ˜ S j k ψ k =1 h L ψ, ψ i≤ λ j + 2 πβ max ψ ∈ ˜ S j k ψ k =1 h W ψ, ψ i . (75)Next, by using (70), the Cauchy-Schwarz inequality, that H is an isometry of L ( T ) such that H ( ψ ) ′ = H ( ψ ′ ) for every ψ ∈ H ( T ), the second equality in (69) and Proposition 2.1 (b), weobtain for any ℓ ≥ h W ϕ ℓ , ϕ ℓ i H ≤ π k ( ϕ ′ ℓ µ Vβ ) ′ k L k ϕ ′ ℓ µ Vβ k L = 2 π k µ Vβ A ϕ ℓ k L k ϕ ′ ℓ µ Vβ k L ≤ πδ − k A ϕ ℓ k L ( µ Vβ ) k ϕ ℓ k H = 2 πδ − λ / ℓ . For the last step, we used that by definition, k ϕ ℓ k H = 1 and k ϕ ℓ k L ( µ Vβ ) = λ − / ℓ (see the endof the proof of Lemma 6.2). Together with (75), this yields κ j ≤ λ j + 4 π βδ − λ / j . Finally, combined with (74) and Lemma 6.2, the proof of the proposition is complete. We start with the following lemma. 27 emma 7.1. Suppose that V ∈ C , ( T ) . There exists C = C ( β, V ) > such that, if φ ∈ H satisfies k φ k H = 1 and L φ = κ φ weakly for some κ > , then φ ′′ ∈ L ( T ) and k φ k L ≤ C κ − / .Proof. First, since Z φ d µ Vβ = 0 and φ is continuous, there exists ξ ∈ T such that φ ( ξ ) = 0.Thus, by the Cauchy-Schwarz inequality, k φ k L ∞ ≤ sup x ∈ T (cid:12)(cid:12)(cid:12)(cid:12)Z [ ξ,x ] φ ′ d θ (cid:12)(cid:12)(cid:12)(cid:12) ≤ π k φ ′ k L . (76)By Proposition 2.1(b), this yields in turn k φ k L ∞ ≤ πδ − / k φ k H = 2 π/ √ δ. (77)Since µ Vβ ∈ C , ( T ) according to Proposition 2.5 and using that the Hilbert transform H preserves the L ( T ) norm, we see the functions H ( µ Vβ φ ′ ) and (log µ Vβ ) ′ φ ′ are in L ( T ).Together with the definition (15) of L , this implies that − φ ′′ = κ φ + 2 πβH ( µ Vβ φ ′ ) + (log µ Vβ ) ′ φ ′ (78)belongs to L ( T ). Recalling (69), an integration by parts shows that |h φ, A φ i L | ≤ δ − Z φ A φ d µ Vβ = δ − k φ k H = δ − . Moreover, by (69), using Cauchy-Schwarz inequality and (77), we have |h φ, W φ i L | ≤ k φ k L k φ ′ µ Vβ k L ≤ δ − / k φ k L ∞ k φ k H ≤ πδ − . Put together, by (68), this yields k φ k L = κ − h φ, L φ i L ≤ κ − δ − (cid:0) π β (cid:1) which completes the proof.We finally turn to the proof of the last statement of Proposition 4.3 and thus complete theproof of Theorem 1.2. Proof of Proposition 4.3 (d). Assume V ∈ C m, ( T ) for some m ≥ 1. In particular, Proposi-tion 2.5 yields µ Vβ ∈ C m, ( T ) and, thanks to Proposition 2.1(b), we also have log µ Vβ ∈ C m, ( T )and thus k (log µ Vβ ) ( m +1) k L ∞ < ∞ .Starting from (78) and using that k H · k L ≤ k · k L , that k φ j k H = 1 and Lemma 7.1, wesee there exists C = C ( β, V ) > j ≥ k φ ′′ j k L ≤ κ j k φ j k L + 2 πβ k φ ′ j µ Vβ k L + k (log µ Vβ ) ′ φ ′ j k L ≤ κ j k φ j k L + (cid:0) πβδ − / + δ − / k (log µ Vβ ) ′ k L ∞ (cid:1) k φ j k H ≤ C √ κ j . (79)28ombined with (76) and (23), this yields that Proposition 4.3 (d) holds true when k = 0.Next, we use that for any ψ ∈ H ( T ), we have by (76) k Hψ k L ∞ ≤ π k ( Hψ ) ′ k L = 2 π k ψ ′ k L . Thus, since µ Vβ ∈ C , ( T ) and φ ′′ j ∈ L ( T ), according to (78), we have for every j ≥ k H ( µ Vβ φ ′ j ) k L ∞ ≤ π k ( µ Vβ φ ′ j ) ′ k L ≤ π (cid:0) k ( µ Vβ ) ′ k L ∞ k φ ′ j k L + δ − k φ ′′ j k L (cid:1) ≤ C √ κ j (80)for some C = C ( β, V ) > 0; note that we used again that k φ ′ j k L ≤ δ − / k φ j k H . By using thisestimate in (78) together with (77) and using the proposition for k = 0, we obtain k φ ′′ j k L ∞ ≤ C κ j . This proves the proposition when k = 1. Note that, in particular, φ ′′ j ∈ L ∞ ( T ).Assume now that m ≥ k = 2. Observe that, since φ ′′ j ∈ L ( T ),the right hand side of equation (78) has a weak derivative in L and we obtain, for any j ≥ − φ ′′′ j = κ j φ ′ j + 2 πβH ( µ Vβ φ ′ j ) ′ + (log µ Vβ ) ′′ φ ′ j + (log µ Vβ ) ′ φ ′′ j . (81)Together with (79) and the upper bounds used to prove it, this yields k φ ′′′ j k L ≤ C κ j and in particular φ ′′′ j ∈ L ( T ). Similarly as in (80), this implies in turn that k H ( µ Vβ φ ′ j ) ′ k L ∞ ≤ C κ j . By using this estimate combined together with the proposition for k = 0 and k = 1, we obtainfrom (81) that k φ ′′′ j k L ∞ ≤ C κ / j and the proof of the proposition is complete when k = 2.The setting where k ≥ k − β ∈ [0 , + ∞ ] In this final section, we study the limits of σ Vβ ( ψ ) as β → β → ∞ . We provide sufficientconditions on V so that the variance interpolates between the L and the H / (semi-)norms,as it is the case when V = 0, see Lemma 1.3. 29 onvention: In this section, we denote the Hilbert space H and the operators L , A , and W defined in the previous sections by H β , L β , A β , and W β respectively to emphasize on thedependence on the parameter β ≥ L − β . Lemma 8.1. Let V ∈ C , ( T ) . If f ∈ H β for some β > , then L − β f ∈ H ( T ) .Proof. If f ∈ H β , then by Proposition 4.3 we have the convergent expansion in H β , L − β f = ∞ X j =1 h f, φ j i H β κ j φ j . By differentiating this formula and using the estimate (79) this shows that, if V ∈ C , ( T ),there exists a constant C = C ( β, V ) > (cid:13)(cid:13)(cid:13)(cid:0) L − β f (cid:1) ′′ (cid:13)(cid:13)(cid:13) L ≤ ∞ X j =1 |h f, φ j i H β | κ j k φ ′′ j k L ≤ C ∞ X j =1 |h f, φ j i H β |√ κ j ≤ C k f k H β . Proposition 8.2. If V ∈ C , ( T ) then we have for every ψ ∈ H ( T ) , lim β → σ Vβ ( ψ ) = σ V ( ψ ) . Proof. Let ψ ∈ H ( T ) and set ψ β := ψ − R ψ d µ Vβ for any β ≥ 0. In particular ψ β is continuouson T and Lemma 2.3 yieldslim β → k ψ β k L ( µ Vβ ) = k ψ k L ( µ V ) = σ V ( ψ ) . (82)We also use the integration by part formula Z φ A β ψ d µ Vβ = Z φ ′ ψ ′ d µ Vβ (83)which holds for any φ ∈ H ( T ) and ψ ∈ H ( T ). Note that ψ β ∈ H β and, by Lemma 8.1, that L − β ψ β ∈ H . Since L β ≥ A β > H β , we obtain together with (83), σ Vβ ( ψ ) = D ψ β , L − β ψ β E H β = D ψ β , A β L − β ψ β E L ( µ Vβ ) ≤ k ψ β k L ( µ Vβ ) . (84)Combined with (82), this gives lim sup β → σ Vβ ( ψ ) ≤ σ V ( ψ ) . 30s for the lower bound, by using (83) again, that A β = L β − πβ W β and L β L − β φ = φ for every φ ∈ H ( T ), we have σ Vβ ( ψ ) = Z ψ β A β ( L − β ψ β ) d µ Vβ = k ψ β k L ( µ Vβ ) − πβ Z ψ β W β ( L − β ψ β ) d µ Vβ . (85)Since W β ( φ ) = − H ( φ ′ µ Vβ ) and the Hilbert transform satisfies H ∗ = − H on L ( T ), Z ψ β W β ( L − β ψ β ) d µ Vβ = Z H ( ψ β µ Vβ )( L − β ψ β ) ′ d µ Vβ . (86)Since µ Vβ is bounded by Proposition 2.1, we have ψ β µ Vβ ∈ L ( T ), and so does H ( ψ β µ Vβ )which moreover satisfies R T H ( ψ β µ Vβ ) d x = 0. As a consequence, H ( ψ β µ Vβ ) has a primitive ϑ β : T → R that we can pick so that R ϑ β d µ Vβ = 0. Thus, ϑ β ∈ H β and we obtain by usingthe Cauchy-Schwarz inequality (recalling that L − β > H β ) and (84), (cid:12)(cid:12)(cid:12)(cid:12)Z H ( ψ β µ Vβ )( L − β ψ β ) ′ d µ Vβ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)D ϑ β , L − β ψ β E H β (cid:12)(cid:12)(cid:12)(cid:12) ≤ rD ϑ β , L − β ϑ β E H β D ψ β , L − β ψ β E H β ≤ k ϑ β k L ( µ Vβ ) k ψ β k L ( µ Vβ ) . (87)To bound the term k ϑ β k L ( µ Vβ ) , first note that the variational constant C Vβ from (24) satisfies C Vβ = 2 F β ( µ Vβ ) − K ( µ Vβ | µ V ) ≤ F β ( µ V ) = 2 β E ( µ V ) , where we used that K ( µ Vβ | µ V ) ≥ 0, that µ Vβ is the minimizer of F β , and that K ( µ V | µ V ) = 0.Thus, since U µ Vβ ≥ 0, this yields together with Proposition 2.1(c) that µ Vβ ( x ) ≤ e β E ( µ V ) − V ( x ) on T . In particular, there exists C = C ( V ) > β ∈ [0 , k µ Vβ k L ∞ ≤ C /π . As a consequence, using (76), we obtain for β ∈ [0 , k ϑ β k L ( µ Vβ ) ≤ k ϑ β k L ∞ ≤ π k H ( ψ β µ Vβ ) k L = 2 π k ψ β µ Vβ k L ≤ C k ψ β k L ( µ Vβ ) . Combined with (85)–(87), this finally yieldslim inf β → σ Vβ ( ψ ) ≥ lim inf β → k ψ β k L ( µ Vβ ) (1 − πCβ ) = σ V ( ψ ) , where the last identity follows from (82). The proof of the proposition is thus complete.31 roposition 8.3. If V ∈ C , ( T ) and β min T µ Vβ → ∞ as β → ∞ , then for any ψ ∈ H ( T ) , lim β →∞ βσ Vβ ( ψ ) = k ψ k H / . (88) If we assume instead that β min T µ βVβ → ∞ as β → ∞ , then we also have lim β →∞ βσ βVβ ( ψ ) = k ψ k H / . (89) Remark 8.1. Let us comment on the assumptions of Proposition 8.3. First, the condition that ψ ∈ H ( T ) seems only technical and we expect the result still holds provided that ψ ∈ H ( T ).Next, we know from Proposition 2.1(b) that min T µ Vβ > T µ βVβ > β > 0. However, we expect that the later quantity decays to zero as β → ∞ . Indeed, onecan verify from the Euler-Lagrange equation that if V ∈ C , ( T ) is not constant, then theminimizer µ V ∞ of the functional (13) does not have full support on T . On the other-hand, ifthe potential V is fixed, then we already know from Lemma 2.3 that µ Vβ → d x π weakly. InLemma 8.4 below, we establish that, if this convergence holds in L p for p > c log β/β for c > β min T µ Vβ → + ∞ as β → + ∞ is satisfied.We are now ready to prove of Proposition 8.3. Proof. We start by proving (88). Recall that by definition, we have for every φ ∈ H ( T ), k φ k H / = h φ ′ , H ( φ ) i L = −h H ( φ ′ ) , φ i L . By Lemma 6.1, the operator W − β is well-defined on H β . Moreover, by (69) and since H − = − H on L ( T ), we have for every φ ∈ H β , (cid:0) W − β φ (cid:1) ′ µ Vβ = − Hφ + Z T (cid:0) W − β φ (cid:1) ′ d µ Vβ . Recall that ψ β = ψ − R ψ d µ Vβ for any β ≥ ψ β ∈ H β . Using further that L β ≥ πβ W β as operators on H β , we obtain for every β > βσ Vβ ( ψ ) = β D ψ β , L − β ψ β E H β (90) ≤ π D ψ β , W − β ψ β E H β = − (cid:10) ψ ′ β , H ( ψ β ) (cid:11) L = k ψ β k H / = k ψ k H / . (91)As for the lower bound, recalling that W β ( φ ) = − H ( φ ′ µ Vβ ) and writing 2 πβ W β = L β − A β ,32ince H is an isometry of L ( T ) and c ψ ′ = 0, we obtain βσ Vβ ( ψ ) = 2 πβ D ψ ′ , ( L − β ψ β ) ′ µ Vβ E L = 2 πβ D H ( ψ ′ ) , H (cid:0) ( L − β ψ β ) ′ µ Vβ (cid:1)E L = − πβ D H ( ψ ′ ) , W β ( L − β ψ β ) E L = − h H ( ψ ′ ) , ψ β i L + D H ( ψ ′ ) , A β ( L − β ψ β ) E L = k ψ k H / + D H ( ψ ′ ) , A β ( L − β ψ β ) E L . (92)Now, we set ϑ β := H ( ψ ′ ) µ Vβ . Since by assumption ψ ′′ ∈ L ( T ), k ( µ Vβ ) ′ k L ∞ < ∞ by Proposition 2.5 and H maps L ( T ) into L ( T ), it easily follows from Proposition 2.1(b) that ϑ β ∈ H β . Moreover, since L − β ( ψ β ) ∈ H ( T ) according to Lemma 8.1, we can use Remark 83 and the Cauchy-Schwarz inequality(recalling that L − β > H β ) to obtain D H ( ψ ′ ) , A β ( L − β ψ β ) E L = 12 π D ϑ β , A β ( L − β ψ β ) E L ( µ Vβ ) = − π D ϑ β , L − β ψ β E H β ≥ − π σ Vβ ( ψ ) σ Vβ ( ϑ β ) . (93)Next, using (84) and Proposition 2.1(b), we then have12 π σ Vβ ( ϑ β ) ≤ k ϑ β k L ( µ Vβ ) ≤ δ − / k H ( ψ ′ ) k L = δ − / k ψ k H for some δ = δ ( β ) > βδ → ∞ as β → ∞ . Combined with(92)–(93) this then yields βσ Vβ ( ψ ) ≥ k ψ k H / − δ − / k ψ k H σ Vβ ( ψ ) . By computing the roots of the polynomial function x x − ( βδ ) − / k ψ k H x + k ψ k H / thisprovides in turn,lim inf β → + ∞ p βσ Vβ ( ψ ) ≥ lim inf β → + ∞ s k ψ k H / + k ψ k H βδ − k ψ k H √ βδ = k ψ k H / and, together with the upper bound (90), the claim (88) is proven.Since the proof of (89) is identical to the one of (88) after replacing µ Vβ by µ βVβ everywherein the above arguments, the proposition is obtained.33 emma 8.4. Let V ∈ H ( T ) and p > . Suppose that for all β sufficiently large, k µ Vβ − d x π k L p ≤ κ p log ββ for a constant κ p > which is sufficiently small, then β inf T µ Vβ → + ∞ as β → + ∞ .Proof. Recall that U d x π = log 2. 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