Fluctuations of interlacing sequences

Sasha Sodin

Published 2016-10-09Version 1

In a series of works published in the 1990-s, Kerov put forth various applications of the Markov correspondence 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 and the critical points of the characteristic polynomial, whereas the second one is constructed from the eigenvalues and the eigenvalues of a principal submatrix. The fluctuations of the latter 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, Markov's 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.