DLR equations for the Sineβ process and applications

Transcription

1 DLR equations for the Sineβ process and applications Thomas Leble (Courant Institute - NYU) Columbia University - 09/28/2018 Joint work with D. Dereudre, A. Hardy, M. Maı da (Universite Lille 1)

2 Log-gases N point particles X N = (x 1,..., x N ) in R Logarithmic pairwise interaction log x y Confining field/potential V (x) (continuous + growth at infinity, e.g. V (x) = x 2 ). Energy in the state X N H N ( X N ) := 1 log x i x j + 2 i j N NV (x i ) i=1

3 Gibbs measure Canonical Gibbs measure at (inverse) temperature β dp N,β ( X N ) := 1 ( exp βh N ( Z ) X N ) dx N N,β dx N = Lebesgue on R N, with Z N,β (the partition function) ˆ ( Z N,β := exp βh N ( ) X N ) dx N. R N Questions Asymptotic behavior of the system (N )? Fluctuations? Dependency on β?

4 Motivation I - Random Matrix Theory (RMT) Classical Gaussian Hermitian ensembles Large (N N) matrix with Gaussian coefficients in R, C, H (quaternions) Coefficients independent up to symmetry (symmetric, Hermitian, self-dual). Observation (Dyson, Ginibre) Joint law of eigenvalues explicitly computable (thanks to Gaussian distribution). Coincides with the canonical Gibbs measure of a log-gas. dp RMT (λ 1,..., λ N ) = 1 Z N,β exp ( βh N (λ 1,..., λ N )) dλ 1... dλ N β = 1, 2, 4 (GOE, GUE, GSE), V quadratic. For all β, existence of a tridiagonal matrix model whose eigenvalues are distributed as P N,β (Dimitriu-Edelman).

5 Motivation II - Statistical physics Statistical physics Model with singular, long-range interactions in R. Real-life implementations: as square of wave-function for certain quantum many-body systems (Calogero-Sutherland model). d = 1 before d = 2.

6 Global behavior Empirical measure Encodes the global/macroscopic behavior µ N := 1 N N δ xi. i=1 lim µ N = µ eq,v equilibrium measure N V quadratic Wigner s semicircle law dµ(x) = 1 4 x 2π 2 dx.

7

8 What is known about microscopic behavior? For β > 0, existence of a limit point process (Valkó-Virág, also Killip-Stoiciu) named Sine β point process (for V quadratic). Starting point: tridiagonal model of Dumitriu-Edelman. The description of the infinite process involves a coupled one-parameter family of stochastic differential equations driven by a two-dimensional standard Brownian motion. Now: spectrum of a random differential operator (Valkó-Virág). Universality of the microscopic behavior with respect to V Bourgade-Erdös-Yau-Lin, Bekerman-Figalli-Guionnet

9 Properties of Sine β Some properties of the limit are known: A CLT for the number of points in [0, R] Kritchevski-Valkó-Virág with standard deviation log R. Maximal deviation of the counting process Holcomb-Paquette, overcrowding Holcomb-Valkó, large deviations for the number of points Holcomb-Valkó, large gaps Valkó-Virág Behavior as β + : crystallization, and β 0: Poisson Allez-Dumaz.

10 Physical description of the limit process? Finite N: dp N,β ( X N ) := 1 ( exp βh N ( Z ) X N ) dx N N,β X N = N i=1 δ xi N i=1 δ Nxi rescaling C = i Z The energy H N ( X N ) H(C) The volume d X N dp(c) (Poisson) Perhaps δ pi infinite configuration dsine β (C) := 1 Z exp ( βh(c)) dp(c)??? Answer: no.

11 DLR equations Dobrushin-Lanford-Ruelle (DLR) formalism: condition on the exterior. Λ γ Λ c η γ Λ c

12 DLR for Sine β Theorem (Dereudre - Hardy - L. - Maïda) Given a compact Λ R, and given an exterior configuration γ, the law of the configuration η in Λ knowing γ Λ c is given by a Gibbs measure with density Sine β (η γ Λ c ) exp ( β (H(η) + M(η, γ Λ c ))) db(η), H(η) is the logarithmic interaction of η with itself, and M(η, γ Λ c ) corresponds to the effect of the exterior configuration in Λ. db(η) Bernoulli point process with a fixed number of points. Already known for integrable case (β = 2) Bufetov, Kuijlaars- Miña-Díaz.

13 Step 1: Defining the objects H(η) := 1 2 M(η, γ Λ c ) := 2 x y x y log x y dη(x)dη(y) log x y dη(x)dγ Λ c (y) Problem: if potential ψ(y) := log x y dη(x), we have ψ(y) log y as y. Fix a reference measure η 0 in Λ, with η 0 = η, define: M(η, γ Λ c ) := 2 log x y d(η η 0 )(x)dγ Λ c (y), x y formally shifts the energy by a constant partition function. Now ˆ ψ(y) := log x y d(η dη 0 )(x) 1 as y. y

14 Average density of points = 1, and ˆ lim R [ R,R]\Λ 1 dy converges y Need to compare dγ Λ c (y) to dy, discrepancy estimates: Discr [0,R] (γ) = γ [0,R] R R typically In fact E[Discr 2 [0,R]] R. Note that CLT says Discr of order log R, but convergence in law... M(η, γ Λ c ) := 2 log x y d(η η 0 )(x)dγ Λ c (y) converges a.s.. x y

15 Step 2: A reference model The Circular Unitary Ensemble (CUE β ): a log-gas on the unit circle, or with periodic interaction ( ) log x y sin. 2πN Converges also to Sine β (Killip-Stoiciu + Nakano, or Valkó-Virág). DLR equations are easy for CUE β, finite N. Then use convergence as N (and the fact that the limit objects exist...)

16 Step 3: Rigidity Yields canonical DLR equations, with fixed exterior and number of points, because we need η 0 = η. In fact: rigidity in the sense of Ghosh-Peres Proposition (Rigidity of Sine β ) γ Λ c almost surely prescribes the number of points inside Λ. Was known for β = 2, and recently Chhaibi-Najnudel (β arbitrary). Otherwise the cost of creating a point at 0 would be finite!

17 0 x Correlation 0 { x? 0 0 Create 0 Move x logjxj

18 Application: fluctuations Finite-N one-cut case (e.g. V quadratic) ˆ Fluct N [ϕ] := ( N ) ϕ δ xi Nµ eq,v Gaussian. i=1 No normalisation! Fluctuations are O(1). Well-known since Johansson, Shcherbina, Borot-Guionnet, see also Lambert-Ledoux-Webb, L.-Serfaty Also holds for ϕ living on mesoscopic scales (Bekerman-Lodhia) Fluctuations of Sine β?

19 CLT for fluctuations of Sine β Let ϕ be a C 4, compactly supported function. Define ( x ) ϕ l (x) := ϕ, l and fluctuations ˆ Fluct[ϕ l ](C) := ϕ l (x)(dc dx) Theorem (L.) If C is distributed under Sine β, then, in law Fluct[ϕ l ](C) Gaussian as l centered, variance: H 1/2 norm on the real line 1 2βπ 2 R R ( ) φ(x) φ(y) 2 dxdy. x y

20 How does DLR help? For finite-n, classical trick, computing Laplace transform = perturbation of potential ( ( exp β 1 2 i j log x i x j + )) N i=1 NV (x i) ( ( R exp β 1 N 2 i j log x i x j + )) N i=1 NV (x i) dx, N so E PN,β [e t N ( ( R exp β 1 = N 2 R exp N i=1 ϕ(x i ) ] i j log x i x j + N i=1 N ( V tϕ βn ( ( β 1 2 i j log x i x j + N i=1 NV (x i) Then comparison of partition functions. ) )) (x i ) dx N )) dx N

21 Fix l, and λ l, and γ in R \ [ λ, λ]. By DLR E Sineβ (e tfluct γ Λ c ) = ( ( exp β H(η) + M(η, γ Λ c ) t β ϕl (dη dx))) db(η), exp ( β (H(η) + M(η, γλ c ))) db(η) Could see M(η, γ Λ c ) as the potential.. Or re-write energy using background dx. In any case t β ϕ l( change in the potential ) change in the background density

22 Background density goes from dx to dx + t β µ λ where ˆ log x y µ λ (y) = ϕ(x) + const, Tricomi s formula ˆ PV 1 x y µ λ(y) = ϕ (x) µ λ (y) 1 λ λ 2 y PV 2 t 2 dy. 2 t y

23 To compare partition functions with background dx and dx + t β µ λ Find a transportation map Φ t, with Φ t = Id outside [ λ, λ]. Do the change of variable η Φ t (η). Compare the energies H(η) + M(η, γ Λ c ) vs H(Φ t (η)) + M(Φ t (η), γ Λ c ) Should have Φ t = Id + tψ +... and Taylor expand. (Transport in Bekerman-L.-Serfaty, and L.-Serfaty (2d), present in various forms in Shcherbina, Bekerman-Figalli-Guionnet, Guionnet-Shlyakhtenko).

24 All errors are expressed in terms of dη dx. Need local laws. For finite N, ϕ macroscopic, controls are easy to get + bootstrap (one term in only O(1) at first glance). For finite N, ϕ mesoscopic (Bekerman-Lodhia) use the local laws of Bourgade-Erdös-Yau Here we have E Sineβ [Discr [0,R] ] R, which is enough. This follows from the LDP at microscopic scale of L.-Serfaty, because Sine β must have finite renormalized energy.

25 Perspectives Non smooth test functions? Correlations? Forrester Two-dimensional log-gases?

26 Thank you for your attention!