Fluctuations of random tilings and discrete Beta-ensembles

Transcription

1 Fluctuations of random tilings and discrete Beta-ensembles Alice Guionnet CRS (E S Lyon) Workshop in geometric functional analysis, MSRI, nov Joint work with A. Borodin, G. Borot, V. Gorin, J.Huang

2 Consider an hexagon with a hole and take a tiling at random. How does it look?

3 Petrov s picture. When the mesh of the tiling goes to zero, one can see a frozen region and a liquid region. Limits, fluctuations?

4 Tiling of the hexagon Kenyon s picture. Cohn, Larsen,Propp 98 : When tiling an hexagon, the shape of the tiling converges almost surely as the mesh goes to zero.

5 General domains Kenyon-Okounkov s picture. Cohn-Kenyon-Propp 00 and Kenyon- Okounkov 07 : The shape of the tiling (e.g the height function) converges almost surely for a large class of domains.

6 Fluctuations of the surface Conjecture.(Kenyon- Okounkov) The recentered height function converges to the Gaussian Free Field in the liquid region in general domains. (Kenyon-06 ) A class of domains with no frozen regions (Borodin Ferrari-08 ) Some infinite domains with frozen regions (Boutillier-de Tilière-09, Dubedat-11 ) On the torus (Petrov-12, Bufetov-G.-16 ) A class of simply-connected polygons (Berestycki-Laslier-Ray-16+) Flat domains, some manifolds Borodin-Gorin-G.- 16, and Bufetov-Gorin.-17 Polygons with holes trapezoid gluings.

7 Local fluctuations of the boundary of the liquid region Ferrari-Spohn 02, Baik-Kriecherbauer-McLaughlin-Miller 03 : appropriately rescaled, a generic point in the boundary of the liquid region converges to the Tracy-Widom distribution in the random tiling of the hexagon, the distribution of the fluctuations of the largest eigenvalue of the GUE. G-Huang. -17 : This extends to polygonal domains obtained by trapezoid gluings (on the gluing axis).

8 Trapezoids gluings

9 Trapezoids gluings We can glue arbitrary many trapezoids, where we may cut triangles or lines, Always along a single vertical axis

10 What is good about trapezoids? Fact: The total number of tilings of trapezoid with fixed along the border horizontal lozenges l > > l 1 is proportional to i<j l j l i j i Indeed Tilings = Gelfand Tsetlin patterns, enumerated through combinatorics of Schur polynomials or characters of unitary groups

11 Distribution of horizontal tiles H = 4 cuts The distribution of horizontal lozenges {l h i } along the axis of gluing has the form: l h i+1 l i + 1 P Θ,w (l) = 1 Z Θ,w (l i l j ) 2Θ[h(i),h(j)] w(l i ) i<j h(i) number of the cut. Θ symmetric H H matrix of 1 s, 1/2 s, and 0 s with 1 s on the diagonal. i=1

12 Discrete β-ensembles (β = 2θ) For configurations l such that l h i+1 lh i θ h,h, l h i [a h, b h ], b h 1 < a h < b h < a h+1, it is given by: P θ,w (l) = 1 h Z θ,w I θh,h (l j, l h i ) w h (l h i ), 1 h h H 1 i h 1 j h,i<j where I θ (l, l) = Γ(l l + 1)Γ(l l + θ) Γ(l l)γ(l l + 1 θ) ote that I θ (l, l) l l 2θ with = if θ = 1, 1/2.

13 Discrete β-ensembles (β = 2θ) For configurations l such that l h i+1 lh i θ h,h, l h i [a h, b h ], b h 1 < a h < b h < a h+1, it is given by: P θ,w (l) = 1 h Z θ,w I θh,h (l j, l h i ) w h (l h i ), 1 h h H 1 i h 1 j h,i<j where I θ (l, l) = Γ(l l + 1)Γ(l l + θ) Γ(l l)γ(l l + 1 θ) ote that I θ (l, l) l l 2θ with = if θ = 1, 1/2. We can study the convergence, global fluctuations of the empirical measures ˆµ h = 1 h δ l h h i /, 1 h H i=1 and fluctuations of the extreme particles of the liquid region.

14 Discrete β-ensembles (β = 2θ) For configurations l such that l h i+1 lh i θ h,h, l h i [a h, b h ], b h 1 < a h < b h < a h+1, it is given by: P θ,w (l) = 1 h Z θ,w I θh,h (l j, l h i ) w h (l h i ), 1 h h H 1 i h 1 j h,i<j where I θ (l, l) = Γ(l l + 1)Γ(l l + θ) Γ(l l)γ(l l + 1 θ) ote that I θ (l, l) l l 2θ with = if θ = 1, 1/2. We can study the convergence, global fluctuations of the empirical measures ˆµ h = 1 h δ l h h i /, 1 h H i=1 and fluctuations of the extreme particles of the liquid region. Bufetov-Gorin 17 : the fluctuations of the surface of the whole tiling follows from the fluctuations on the gluing axis.

15 Discrete β-ensembles (β = 2θ): law of large numbers Assume w h (x) e V h(x/), V h C 0, (θ h,h ) h,h 0, θ h,h > 0. Fixed heigths : h / ε h, Then lim 1 h h δ l h i / µh ε a.s, i=1 Random heigths : θ = ΛΛ T with Λ fixed. Then h / ε h and 1 h δ l h i / µh ε a.s, Indeed lim h i=1 P θ,w (l) E( 1 h e 2 h i=1 δ l h i /, h/,1 h H) where E has a unique minimizer.

16 Assumption on the equilibrium measures towards fluctuations ote that for all h: We shall assume 0 dµh ε dx The liquid regions {0 < dµh ε dx θ 1 hh < θ 1 hh } are connected, The equilibrium measures are off critical: at the boundary of the liquid region they behave like a square root. w h (x) w h (x 1) = φ+,h (x) φ,h (x), φ±,h analytic, φ±,h = φ± h + 1 φ± 1,h + o( 1 ) Rmk: Off-criticality should be generically true.

17 Global fluctuations: fixed heights Assume h / ε h, Theorem (Borodin-Gorin-G 15 Borot-Gorin-G 17 ) Then for any analytic functions f h : ) (f h (l h i /) E[f h (l h i /)]) (0, Σ(f )). ( h i=1 h

18 Global fluctuations: Random Heights Assume Λ given. Then [WIP Borot-Gorin-G ] i ε i The heights are equivalent to discrete Gaussian : P θ,w ( h E[ h ] = x) 1 Z e 1 2σ (x)2

19 Global fluctuations: Random Heights Assume Λ given. Then [WIP Borot-Gorin-G ] i ε i The heights are equivalent to discrete Gaussian : P θ,w ( h E[ h ] = x) 1 Z e 1 2σ (x)2 (f (l h i /) E[f (l h i /)]) k i,h ( k E[ k ]) εk µ h ε(f ) ε=ε + G f where G f is a centered Gaussian variable, independent from the filling fractions.

20 Discrete β-ensembles, edge fluctuations [Huang-G 17 ] Under the previous assumptions, the boundary fluctuates like a Tracy-Widom distribution. 1 If δl 1 1 i / converges towards µ1 with liquid region [a, b], µ 1 ((, a)) = 0, then for all t real lim Pθ,w ( 2/3 (l 1 1/ a) t ) = f 2θ11 (t) with f 2θ the 2θ-Tracy-Widom distribution appearing in the continuous β-ensembles. Corollary: Fluctuations of the first rows of Young diagrams under Jack deformation of Plancherel measure.

21 Continuous β-ensembles The distribution of continuous β-ensembles is given by dp β,v (λ) = 1 λ 1 Z β,v i<j λ i λ j β e β i=1 V (λ i ) dλ i. When V (x) = 1 4 x 2, and β = 1 (resp. β = 2, 4), P β,v is the distribution of the eigenvalues of a symmetric (resp. Hermitian, resp. symplectic) matrix with centered Gaussian entries with covariance 1/, the GOE (resp. GUE, resp. GSE).

22 Continuous β-ensembles The distribution of continuous β-ensembles is given by dp β,v (λ) = 1 λ 1 Z β,v i<j λ i λ j β e β i=1 V (λ i ) dλ i. When V (x) = 1 4 x 2, and β = 1 (resp. β = 2, 4), P β,v is the distribution of the eigenvalues of a symmetric (resp. Hermitian, resp. symplectic) matrix with centered Gaussian entries with covariance 1/, the GOE (resp. GUE, resp. GSE). when w(x) e βv (x/), and H = 1, Z θ,w Pθ,w (l) Z 2θ,V dp 2θ,V dλ (l/)

23 Some techniques to deal with β-ensembles Integrable systems. In some cases, these distributions have particular symmetries allowing for special analysis, e.g when β = 2θ = 2 the density is the square of a determinant, allowing for orthogonal polynomials analysis [Mehta, Deift, Baik, Johansson etc] General case. -Dyson-Schwinger (or loop) Equations. Use equations for the correlators, e.g the moments of the empirical measures, and try to solve them asymptotically [Johansson, Shcherbina, Borot-G, etc] -Universality. Compare your law to one you can analyze [Erdös-Yau et al, Tao-Vu]. An important step is to prove rigidity (particles are very close to their deterministic limit), see [Bourgade-Erdös-Yau]

24 Continuous β-ensembles: Dyson-Schwinger equations(dse) Let ˆµ = 1 δ λi : ˆµ (f ) = 1 i=1 f (λ i ) If f is a smooth test function, then integration by parts yields the DSE: [ E ˆµ (V f ) 1 ] f (x) f (y) d ˆµ (x)d ˆµ (y) 2 x y = ( 1 β 1 2 )E[ˆµ (f 0)] i=1

25 Continuous β-ensembles: Dyson-Schwinger equations Linearizing this equation around its limit, we get where E[(ˆµ µ)(ξf )] = ( 1 β 1 2 )E[ˆµ (f )] + 1 f (x) f (y) 2 E[ d(ˆµ µ)(x)d(ˆµ µ)(y)] x y f (x) f (y) Ξf (x) = V (x)f (x) dµ(y). x y If we can show that the last term is negligible and Ξ is invertible (off-criticality), we can solve asymptotically this equation to get E[ˆµ µ](f ) ( 1 β 1 2 )E[ˆµ ((Ξ 1 f ) )] We can get an infinite system of DSE for all moments of ˆµ and get large expansion up to any order. This gives the CLT.

26 Concentration inequalities First concentration inequalities (Maida Maurel Segala) Let λ i = λ i 1 + max{λ i λ i 1, p }, λ 1 = λ 1 and µ = u 1 δ λ i, u uniform law on [0, p 1 ]. Then P β,v (D( µ, µ V ) t) e Cp ln β 2 t2 2 with D(µ, ν) 2 = = Indeed, one can show that 0 ln x y d(µ ν)(x)d(µ ν)(y) 1 t e itx 2 d(µ ν)(x) dt dp β,v dλ ecp ln β 2 2 D( µ,µ V ) 2 Dyson-Schwinger equations can be used to improve the concentration inequalities to get a concentration of order 1/.

27 Continuous β-ensembles: Fluctuations at the edge Dumitriu-Edelman 02 : Take V (x) = βx 2 /2. Then P β,βx2 /2 is the law of the eigenvalues of Y β 1 ξ ξ 1 Y β H β 2 ξ 2 0. = ξ 1 Y β where ξ i are iid (0, 1) and Y β i χ iβ independent.

28 Continuous β-ensembles: Fluctuations at the edge Dumitriu-Edelman 02 : Take V (x) = βx 2 /2. Then P β,βx2 /2 is the law of the eigenvalues of Y β 1 ξ ξ 1 Y β H β 2 ξ 2 0. = ξ 1 Y β where ξ i are iid (0, 1) and Y β i χ iβ independent. Ramirez-Rider-Viràg 06 : The largest eigenvalue fluctuates like Tracy-Widom β distribution. Bourgade-Erdòs-Yau 11, Shcherbina 13, Bekerman-Figalli-G 13 : Universality: This remains true for general potentials provided off-criticality holds.

29 Discrete β-ensembles: ekrasov s equation Recall l i+1 l i θ and set P θ,w (l) = 1 Γ(l j l i + 1)Γ(l j l i + θ) w(li, ) Z Γ(l j l i )Γ(l j l i + 1 θ) 1 i<j and assume there exists φ ± Then φ (ξ)e P θ,w is analytic. [ i=1 analytic so that w(x, ) w(x 1, ) = φ+ (x) φ (x). ( 1 θ ) ] + φ + ξ l (ξ)e P θ,w i [ i=1 ( ) ] θ 1 + ξ l i 1

30 Discrete β-ensembles: ekrasov s equation Recall l i+1 l i θ and set P θ,w (l) = 1 Γ(l j l i + 1)Γ(l j l i + θ) w(li, ) Z Γ(l j l i )Γ(l j l i + 1 θ) 1 i<j and assume there exists φ ± Then φ (ξ)e P θ,w is analytic. [ i=1 analytic so that w(x, ) w(x 1, ) = φ+ (x) φ (x). ( 1 θ ) ] + φ + ξ l (ξ)e P θ,w i [ i=1 ( ) ] θ 1 + ξ l i 1 Gives asymptotic equations for G (z) := 1 which can be analyzed as DSE. i=1 1 z l i

31 Consequences of ekrasov s equation One can estimate all moments of (G (z) G(z)) for z C\R, proving CLT,

32 Consequences of ekrasov s equation One can estimate all moments of (G (z) G(z)) for z C\R, proving CLT, One can estimate all moments of (G (z) G(z)) for Iz = 1 1 δ, δ > 0 proving rigidity : for any a > 0 P θ,w (sup l i γ i i where µ((, γ i )) = i/. a )2 ) e (log min{i/, 1 i/} 1/3

33 Consequences of ekrasov s equation One can estimate all moments of (G (z) G(z)) for z C\R, proving CLT, One can estimate all moments of (G (z) G(z)) for Iz = 1 1 δ, δ > 0 proving rigidity : for any a > 0 P θ,w (sup l i γ i i where µ((, γ i )) = i/. a )2 ) e (log min{i/, 1 i/} 1/3 One can compare the law of the extreme particles, at distance of order 1/3 1 (the mesh of the tiling) with the law of the extreme particles for the continuous model and deduce the 2θ- Tracy-Widom fluctuations.

34 Some open questions Critical case. In continuous setting, Bekerman, Leblé, Serfaty 17 derived CLT for functions in Im(Ξ). What about the discrete case? Can we get universality of local fluctuations? How are the fluctuations for functions which are not in Im(Ξ)? Universality for CLT. What if we have true Coulomb gas interaction in the discrete case? General domains? Local fluctuations in the bulk. When θ = 1 correlations functions in the bulk converge to discrete Sine process. What about other θ?

35 Many thanks to Alexei Borodin, Vadim Gorin and Leonid Petrov for the pictures. And thank you for your attention!