Fluctuations of Interlacing Sequences

Abstract

In a series of works published in the 1990-s, Kerov put forth various applications of the circle of ideas centred at the Markov moment problem to the limiting shape of random continual diagrams arising in representation theory and spectral theory. We demonstrate on several examples that his approach is also adequate to study the fluctuations about the limiting shape.In the random matrix setting, we compare two continual diagrams: one is constructed from the eigenvalues of the matrix and the critical points of its characteristic polynomial, whereas the second one is constructed from the eigenvalues of the matrix and those of its principal submatrix. The fluctuations of the latter diagram were recently studied by Erdős and Schröder; we discuss the fluctuations of the former, and compare the two limiting processes.For Plancherel random partitions, the Markov correspondence establishes the equivalence between Kerov's central limit theorem for the Young diagram and the Ivanov-Olshanski central limit theorem for the transition measure. We outline a combinatorial proof of the latter, and compare the limiting process with the ones of random matrices.

Interlacing adjacent levels of $$\beta $$–Jacobi corners processes

“…Independently of the present article, the Central Limit Theorem for S f (k) in the context of Wigner matrices was also considered recently by Erdős-Schröder [ES17] and by Sodin [Sod16]. Since these authors consider only the asymptotics for a single k, the link to the Gaussian Free Field is less visible there, although we believe that it should be also present (at least in the case when the Wigner matrices have Gaussian entries, i.e.…”

“…for GOE, GUE, GSE). Despite the connections, our setup is quite different from [ES17], [Sod16]. In particular, these papers rely on matrix models and independence of matrix elements; we do not know how to extend such an approach to our settings of β-Jacobi corners process.…”

A random walk approach to linear statistics in random tournament ensembles

“…In particular, we are able to obtain these results using combinatorial methods, closely related to previous calculations for showing distributional convergence, but without explicit rates (see e.g. [28,49]). Moreover, this approach only requires estimates involving third order moments to show distributional convergence.…”

Spectral statistics of the uni-modular ensemble

“…For the UME (and more generally for Wigner matrices) the convergence of F n (M) to independent Gaussians (see Theorem 1 below) is well known (see e.g. [6] for instance) and can be obtained by showing the convergence of all moments via combinatorial methods. However the combinatorial approach does not easily lend itself to obtaining rates of convergence.…”

“…Examples of the two types of subgraphs. Type I with a = 1, p + = (2, 3, 4) and q + =(5,6,7,8,9,10) and type II subgraph with a = 1, b = 4, q + = (2, 3), p + =(5,6) …”