Techniques from harmonic analysis and asymptotic results in - PowerPoint PPT Presentation

Techniques from harmonic analysis and asymptotic results in probability theory Pierre-Loïc Méliot 2018, July 3rd University Paris-Sud (Orsay)

Objective: present some mathematical tools combinatorics, arithmetics, random matrix theory, etc. ), when their size n goes to infinity; lated quantities (moments, cumulants). Red line: a concrete problem on certain random variables stemming from the representation theory of symmetric groups. 0. Fine asymptotics of the Plancherel measures 1. Random objects chosen in a group or in its dual 2. Mod-Gaussian convergence and the method of cumulants 3. Perspectives 1 ▶ which allow us to study various random objets (stemming from ▶ which rely on several versions of the Fourier transform, or on re-

Plancherel measures 6 longest increasing subword, 7 4 2 1 8 3 5 2 We denote P ( n ) the set of integer partitions of n (Young diagrams). The Plancherel measure on P ( n ) is the probability measure P n [ λ ] = (dim λ ) 2 , n ! where dim λ is the number of standard tableaux with shape λ . λ = ( 4 , 3 , 1 ) = ; T = . The measure P n ▶ plays an essential role in the solution of Ulam’s problem of the ▶ has its fluctuations related to the spectra of random matrices.

1 2 0 3 − 2 − 1 A random partition λ under the Plancherel measure P n = 400 .

Random character values and Kerov’s central limit theorem The random characters 4 Each partition λ ∈ P ( n ) corresponds to an irreducible representation ( V λ , ρ λ ) of S ( n ) : V λ complex vector space with dimension dim λ, ρ λ : S ( n ) → GL ( V λ ) . X k ( λ ) = tr ( ρ λ ( c k )) tr ( ρ λ ( id )) , c k k -cycle are important random variables for studying the fluctuations of P n . Kerov’s central limit theorem (1993) ensures that for any k ≥ 2, n k / 2 X k ( λ ) ⇀ n → + ∞ N R ( 0 , k ) . λ ∼ P n

If X is a real-valued random variable, its cumulants are the coefficients A possible proof of Kerov’s CLT relies on Śniady’s estimate (2006) viation principles, etc. obtain Berry–Esseen estimates, concentration inequalities, large de- Idea: with a better control, one can make the CLT more precise and 2 5 κ ( r ) ( X ) of ∞ ∑ κ ( r ) ( X ) log( E [ e zX ]) = z r . r ! r = 1 � � ( ) � � � κ ( r ) ( n k / 2 X k ( λ )) n 1 − r � = O k , r . Conjecture: (2011) there exist constants C = C k such that � � � � � κ ( r ) ( n k / 2 X k ( λ )) � ≤ ( Cr ) r n 1 − r 2 . ∀ ( n , r ) ,

Random objects chosen in a group or in its dual

Groups and observables We are interested in two kinds of random objects: necting random elements g ). G , which encode combinatorial prob- lems (examples: Plancherel measure on partitions; systems of interacting particles). goes to infinity. 6 G = a group (finite, or compact, or reductive) ; � G = set { ( V λ , ρ λ ) } of the irreducible representations of G . 1. random variables g ∈ G , or objects constructed from such vari- ables (examples: g = g t random walk on G ; Γ random graph con- 2. random representations λ ∈ � Goal: study g or λ when the size of the group or of the random object

One obtains relevant information by considering the real- or matrix- valued random variables lem of the choice of a basis; crystal theory). functions), which satisfy orthogonality formulas. partitions, or subset of a lattice). 7 Ingredients: and by computing their moments (Poincaré; Diaconis; Kerov–Vershik). ρ λ ( g ) ch λ ( g ) = tr ( ρ λ ( g )) ; 1. The set � G (or � G K for random variables in G / K ) is explicit (set of 2. Idem for the dimensions d λ and the characters (or the spherical 3. The matrices ρ λ ( g ) are much more complicated to describe (prob- Example: for Plancherel measures, E n [ X k ( λ )] = E n [ ch λ ( c k ) ] = 1 ( k = 1 ) . d λ

Cutoff phenomenon for Brownian motions X in particular the L 1 -distance (total variation). dx p dx 8 The speed of convergence is given by the L p -distances know that Laplace–Beltrami operator, starting from a fixed base point x 0 ), we X = compact Lie group ( SU ( n ) , SO ( n ) , etc. ) or compact symmetric quotient ( S n , Gr ( n , d , k ) , etc. ) . If ( x t ) t ∈ R + is the Brownian motion on X (diffusion associated to the ( µ t = law of x t ) ⇀ t → + ∞ Haar . � � (∫ ) 1 / p � � d µ t ( x ) � � − 1 d p ( µ t , Haar ) = , � �

The same cutoff phenomenon occurs at the same time for the dis- Theorem (M., 2013) Sketch of proof: 2. Before the cutoff, one can find discriminating functions which be- 9 In each class, there exists an explicit positive constant α such that ∀ ε > 0 , d TV ( µ t , Haar ) ≥ 1 − C if t = α ( 1 − ε ) log n , n c ε d TV ( µ t , Haar ) ≤ C if t = α ( 1 + ε ) log n . n c ε tances d p , p ∈ ( 1 , + ∞ ) . 1. After the cutoff, one can compute d 2 ( µ t , Haar ) ≥ d TV ( µ t , Haar ) . have differently under µ t and under µ ∞ = Haar .

Random geometric graphs of X ; 1 in two distinct regimes: 10 ( X , d ) = compact symmetric space endowed with the geodesic distance . The geometric graph with level L > 0 and size N ∈ N on X is the graph Γ X geom ( N , L ) obtained: ▶ by taking N independent points x 1 , . . . , x N under the Haar measure ▶ by connecting x i to x j if d ( x i , x j ) ≤ L . We are interested in the spectra of the adjacency matrix of Γ X geom ( N , L ) , 1. Gaussian regime: L is fixed and N → + ∞ . 2. Poisson regime: L = ( ℓ N ) dim X with ℓ > 0, and N → + ∞ .

11 Random geometric graph on the sphere S 2 , with N = 100 points and level L = π 8 (stereographic projection).

Gaussian regime If L is fixed and N goes to infinity, there exist a.s. limits L 1 with N G, Theorem (M., 2017) G the dominant weights. We 12 We assume that X = G is a compact Lie group, and we denote d the rank of G , W its Weyl group and λ ∈ � denote the spectrum of Γ geom ( N , L ) e − 1 ( N , L ) ≤ e − 2 ( N , L ) ≤ · · · ≤ 0 ≤ · · · ≤ e 2 ( N , L ) ≤ e 1 ( N , L ) ≤ e 0 ( N , L ) . e i ( N , L ) e i ( L ) = lim N →∞ for any i ∈ Z . This limiting spectrum is the spectrum of a compact operator on L 2 ( G ) , and it consists in ( d λ ) 2 values c λ for each λ ∈ � ( ) d ∑ ε ( w ) � c λ = √ 2 ( L ∥ λ + ρ − w ( ρ ) ∥ ) . d λ vol ( t / t Z ) 2 π J d w ∈ W

C 0 13 ⊞ ⊟ ⊟ ⊞ ⊞ λ ⊟ ω 2 ω 1 A limiting eigenvalue c λ for each dominant weight, given by an alter- nate sum of values of Bessel functions ( G = SU ( 3 ) ).

Poisson regime Theorem (M., 2018) N N implies the convergence in probability One has a Benjamini–Schramm local convergence 14 N 1 ( ℓ ) In the Poisson regime, L = L N = dim X is chosen so that each vertex of Γ X geom ( N , L N ) has O ( 1 ) neighbors. Γ X geom ( N , L N ) → Γ ∞ , where Γ ∞ is the geometric graph with level 1 obtained from a Poisson point process on R dim X with intensity ℓ vol ( X ) , rooted at the point 0 . This ∑ µ N = 1 δ e i ( N , L ) ⇀ N → + ∞ µ ∞ ; i = 1 the limiting measure µ ∞ is determined by its moments.

15 volving certain graphs (circuits and reduced circuits), and one can give functionals in the Kashiwara–Lusztig crystal theory. This conjecture is Example: G , and to an interpretation of these 5 dx 3 2 k 3 C ∫ R x r µ ∞ ( dx ) have a combinatorial expansion in- The moments M r = an explicit formula if r ≤ 7. M 5 = e + 5 e + 5 e = I 5 ( ℓ ′ ) 4 + 5 I 3 I 2 ( ℓ ′ ) 3 + 5 I 3 ( ℓ ′ ) 2 ( ) ∫ ( ∂ Φ − � J R Ω )( x ) with ℓ ′ = ℓ vol ( t / t Z ) and I k = ( 2 π ) d / 2 ( δ ( x )) k − 2 . A general formula for M r is related to a conjecture on certain func- tionals of the representations λ ∈ � true for tori, which should allow us to find the support of µ ∞ .

Mod-Gaussian convergence and the method of cumulants

16 The previous problems have been solved by computing the moments of observables of the random objects. To get more precise informa- tion, one can look at the fine asymptotics of the Fourier or Laplace transform of the observables. General framework: deviation estimates, local limit theorems, concentration inequalities. One obtains: an extended CLT, speed of convergence estimates, large with Mod- φ convergence ∫ e zx ϕ ( dx ) = e η ( z ) ; ϕ infinitely divisible law with R ( X n ) n ∈ N sequence of real random variables; ( t n ) n ∈ N sequence of parameters growing to + ∞ E [ e zX n ] lim e t n η ( z ) = ψ ( z ) locally uniformly in z ∈ C . n → + ∞

The method of cumulants N n D n N n D n A 3 3 1 17 such that In the Gaussian case ( η ( z ) = z 2 2 ), consider ( S n ) n ∈ N which satisfies the hypotheses of the method of cumulants with parameters ( A , D n , N n ) : (MC1) There exist A > 0 and two sequences N n → + ∞ and D n = o ( N n ) � � � � � κ ( r ) ( S n ) � ≤ N n ( 2 D n ) r − 1 r r − 2 A r . ∀ r ≥ 1 , (MC2) There exist σ 2 ≥ 0 and L such that = σ 2 ( ( )) κ ( 2 ) ( S n ) 1 + o ( D n / N n ) ; κ ( 3 ) ( S n ) N n ( D n ) 2 = L ( 1 + o ( 1 )) . 1. If σ 2 > 0, then S n − E [ S n ] √ var ( S n ) = Y n ⇀ N R ( 0 , 1 ) , and more precisely, ( ) √ d Kol ( Y n , N R ( 0 , 1 )) = O . σ 3