Let U N denote a Haar Unitary matrix of dimension N, and consider the fieldin probability. This provides a verification up to second order of a conjecture of Fyodorov, Hiary and Keating, improving on the recent first order verification of Arguin, Belius and Bourgade.
Consider d uniformly random permutation matrices on n labels. Consider the sum of these matrices along with their transposes. The total can be interpreted as the adjacency matrix of a random regular graph of degree 2d on n vertices. We consider limit theorems for various combinatorial and analytical properties of this graph (or the matrix) as n grows to infinity, either when d is kept fixed or grows slowly with n. In a suitable weak convergence framework, we prove that the (finite but growing in length) sequences of the number of short cycles and of cyclically nonbacktracking walks converge to distributional limits. We estimate the total variation distance from the limit using Stein's method. As an application of these results we derive limits of linear functionals of the eigenvalues of the adjacency matrix. A key step in this latter derivation is an extension of the Kahn-Szemerédi argument for estimating the second largest eigenvalue for all values of d and n.
We study the global fluctuations for linear statistics of the form n i=1 f (λ i ) as n → ∞, for C 1 functions f , and λ 1 , . . . , λn being the eigenvalues of a (general) β-Jacobi ensemble [18,29]. The fluctuation from the meanis given asymptotically by a Gaussian process. We compute the covariance matrix for the process and show that it is diagonalized by a shifted Chebyshev polynomial basis; in addition, we analyze the deviation from the predicted mean for polynomial test functions, and we obtain a law of large numbers.In the case that β ∈ {1, 2, 4} and n 1 , n 2 ∈ N, they admit full matrix models (as J = W 1/2where W 1 , W 2 are Wishart matrices, hence the "double Wishart" name. For an extensive study of the β = 1 case, as well as a clear exposition of how these models arose, we refer to [36]; the other cases (β = 2, 4) can be dealt with similarly.Recently, it was shown that in these "classical" cases a different kind of model can be constructed, starting from random projections, rather than random Wishart matrices; or, equivalently, that "chopping off" an appropriate corner of a unitary Haar matrix will yield a matrix whose singular values, squared, are distributed according to (1) (discovered in [14], rediscovered in [18]).The greatest generality is achieved by the tridiagonal model [18,29], which covers any β > 0, and removes the condition that n 1 , n 2 ∈ N. We give the model below (hereafter referred to as the Edelman-Sutton model, as it appears most clearly in their work [18]). Given the matrix B β defined as
We show that random walk on a stationary random graph with positive anchored expansion and exponential volume growth has positive speed. We also show that two families of random triangulations of the hyperbolic plane, the hyperbolic Poisson Voronoi tessellation and the hyperbolic Poisson Delaunay triangulation, have 1-skeletons with positive anchored expansion. As a consequence, we show that the simple random walks on these graphs have positive speed. We include a section of open problems and conjectures on the topics of stationary geometric random graphs and the hyperbolic Poisson Voronoi tessellation. arXiv:1409.4312v2 [math.PR]
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