Fluctuations of random tilings and discrete Beta-ensembles Alice - PowerPoint PPT Presentation

Fluctuations of random tilings and discrete Beta-ensembles Alice Guionnet CNRS (´ ENS Lyon) Advances in Mathematics and Theoretical Physics, Roma, 2017 Joint work with A. Borodin, G. Borot, V. Gorin, J.Huang

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

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

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.

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.

Fluctuations of the surface Conjecture. (Kenyon- Okounkov) The recen- tered 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 Tili` ere-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.

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).

Trapezoids gluings

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

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

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 N 1 P Θ , w � ( ℓ i − ℓ j ) 2Θ[ h ( i ) , h ( j )] � ( ℓ ) = w ( ℓ i ) N Z Θ , w N i < j i =1 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.

Discrete β -ensembles ( β = 2 θ ) For configurations ℓ such that ℓ h i +1 − ℓ h i − θ h , h ∈ N , it is given by: N ( ℓ ) = 1 I θ h , h ′ ( ℓ h ′ P θ, w � � � j , ℓ h w ( ℓ h i ) i ) , Z N 1 ≤ h ≤ h ′ ≤ H 1 ≤ i ≤ Nh 1 ≤ j ≤ N ′ h , i < j where I θ ( ℓ ′ , ℓ ) = Γ( ℓ ′ − ℓ + 1)Γ( ℓ ′ − ℓ + θ ) Γ( ℓ ′ − ℓ )Γ( ℓ ′ − ℓ + 1 − θ ) Note that I θ ( ℓ ′ , ℓ ) ≃ | ℓ ′ − ℓ | 2 θ with = if θ = 1 , 1 / 2.

Discrete β -ensembles ( β = 2 θ ) For configurations ℓ such that ℓ h i +1 − ℓ h i − θ h , h ∈ N , it is given by: N ( ℓ ) = 1 I θ h , h ′ ( ℓ h ′ P θ, w � � � j , ℓ h w ( ℓ h i ) i ) , Z N 1 ≤ h ≤ h ′ ≤ H 1 ≤ i ≤ Nh 1 ≤ j ≤ N ′ h , i < j where I θ ( ℓ ′ , ℓ ) = Γ( ℓ ′ − ℓ + 1)Γ( ℓ ′ − ℓ + θ ) Γ( ℓ ′ − ℓ )Γ( ℓ ′ − ℓ + 1 − θ ) Note that I θ ( ℓ ′ , ℓ ) ≃ | ℓ ′ − ℓ | 2 θ with = if θ = 1 , 1 / 2. ◮ We can study the convergence, global fluctuations of the empirical measures N h N = 1 � µ h ˆ δ ℓ h i / N , 1 ≤ h ≤ H N h i =1 and fluctuations of the extreme particles.

Discrete β -ensembles ( β = 2 θ ) For configurations ℓ such that ℓ h i +1 − ℓ h i − θ h , h ∈ N , it is given by: N ( ℓ ) = 1 I θ h , h ′ ( ℓ h ′ P θ, w � � � j , ℓ h w ( ℓ h i ) i ) , Z N 1 ≤ h ≤ h ′ ≤ H 1 ≤ i ≤ Nh 1 ≤ j ≤ N ′ h , i < j where I θ ( ℓ ′ , ℓ ) = Γ( ℓ ′ − ℓ + 1)Γ( ℓ ′ − ℓ + θ ) Γ( ℓ ′ − ℓ )Γ( ℓ ′ − ℓ + 1 − θ ) Note that I θ ( ℓ ′ , ℓ ) ≃ | ℓ ′ − ℓ | 2 θ with = if θ = 1 , 1 / 2. ◮ We can study the convergence, global fluctuations of the empirical measures N h N = 1 � µ h ˆ δ ℓ h i / N , 1 ≤ h ≤ H N h i =1 and fluctuations of the extreme particles. ◮ Bufetov-Gorin 17’: the fluctuations of the surface of the whole tiling follows from the fluctuations on the gluing axis.

Discrete β -ensembles ( β = 2 θ ): law of large numbers Assume w ( x ) ≃ e − NV ( x / N ) , ( θ h , h ′ ) h , h ′ ≥ 0, θ h , h > 0. ◮ Fixed heigths : N h / N �→ ε h , Then N h 1 � i / N → µ h lim δ ℓ h a . s , ε N h N →∞ i =1 ◮ Random heigths : � h N h = N , Then N h / N → ε ∗ h and N h 1 � i / N → µ h lim δ ℓ h a . s , ε ∗ N h N →∞ i =1 Indeed, � Nh − N 2 E ( 1 i =1 δ ℓ h i / N , 1 ≤ h ≤ H ) P θ, w N ( ℓ ) ≃ e Nh where E has a unique minimizer.

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

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

Global fluctuations: Random Heights Assume � N i = N . Then [WIP Borot-Gorin-G ] ◮ N i N → ε ∗ i ◮ The heights are equivalent to discrete Gaussian ‘: N ( N h − E [ N h ] = x ) ≃ 1 Z e − 1 2 σ ( x ) 2 P θ, w

Global fluctuations: Random Heights Assume � N i = N . Then [WIP Borot-Gorin-G ] ◮ N i N → ε ∗ i ◮ The heights are equivalent to discrete Gaussian ‘: N ( N h − E [ N h ] = x ) ≃ 1 Z e − 1 2 σ ( x ) 2 P θ, w ◮ N h � � ( f h ( ℓ h i / N ) − E [ f h ( ℓ h ( N k − E [ N k ]) ∂ ε k µ h i / N )]) − ε ( f h ) | ε = ε ∗ i =1 k converges towards a centered Gaussian variable.

Discrete β -ensembles, edge fluctuations [Huang-G 17’] Under the previous assump- tions, the boundary fluctuates like a Tracy-Widom distribu- tion. � δ ℓ 1 i / N converges towards µ 1 with liquid region [ a , b ], 1 If N 1 µ 1 (( −∞ , a )) = 0, then for all t real � � N →∞ P θ, w N 2 / 3 ( ℓ 1 lim 1 / N − a ) ≥ t = f 2 θ 11 ( t ) N with f 2 θ the 2 θ -Tracy-Widom distribution appearing in the continuous β -ensembles. Corollary: Fluctuations of the first rows of Young diagrams under Jack deformation of Plancherel measure.

Continuous β -ensembles The distribution of continuous β -ensembles is given by 1 | λ i − λ j | β e − β N � N dP β, V � i =1 V ( λ i ) � ( λ ) = d λ i . N Z β, V ˜ i < j N 4 x 2 , and β = 1 (resp. β = 2 , 4), P β, V ◮ When V ( x ) = 1 is the N distribution of the eigenvalues of a symmetric (resp. Hermitian, resp. symplectic) N × N matrix with centered Gaussian entries with covariance 1 / N , the GOE (resp. GUE, resp. GSE) .

Continuous β -ensembles The distribution of continuous β -ensembles is given by 1 | λ i − λ j | β e − β N � N dP β, V � i =1 V ( λ i ) � ( λ ) = d λ i . N Z β, V ˜ i < j N 4 x 2 , and β = 1 (resp. β = 2 , 4), P β, V ◮ When V ( x ) = 1 is the N distribution of the eigenvalues of a symmetric (resp. Hermitian, resp. symplectic) N × N matrix with centered Gaussian entries with covariance 1 / N , the GOE (resp. GUE, resp. GSE) . ◮ when w ( x ) ≃ e − β NV ( x / N ) , dP 2 θ, V Z θ, w P θ, w Z 2 θ, V N ( ℓ ) ≃ ˜ N ( ℓ/ N ) N N d λ

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 = 2 θ = 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 [Erd¨ os-Yau et al, Tao-Vu]. An important step is to prove rigidity (particles are very close to their deterministic limit), see [Bourgade-Erd¨ os-Yau]

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