Fluctuations of interlacing sequences

TL;DR

This paper explores the fluctuations of interlacing sequences in random matrix theory and representation theory, comparing different diagrams and their limiting behaviors, extending Kerov's approach to study shape fluctuations.

Contribution

It demonstrates the applicability of Kerov's circle of ideas to analyze fluctuations around the limiting shape in various contexts, including random matrices and partitions.

Findings

Compared fluctuations of two types of continual diagrams in random matrices.

Established connections between Kerov's limit theorem and Ivanov--Olshanski theorem.

Provided a combinatorial proof linking representation theory and random matrix results.

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 Erd\H{o}s and Schr\"oder; 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.