Kerov's interlacing sequences and random matrices

TL;DR

This paper investigates the asymptotic behavior of Kerov's interlacing functions derived from random matrices, revealing their convergence to well-known limit shapes like the Vershik-Kerov-Logan-Shepp curve and the Marchenko-Pastur law.

Contribution

It provides explicit scaling limits of Kerov's functions for Wigner and Wishart matrices, connecting them to classical limit spectral laws.

Findings

Wigner matrices' Kerov functions converge to the Vershik-Kerov-Logan-Shepp curve.

Wishart matrices' Kerov functions relate to the Marchenko-Pastur law.

Explicit descriptions of the scaling limits are provided.

Abstract

To a $N \times N$ real symmetric matrix Kerov assigns a piecewise linear function whose local minima are the eigenvalues of this matrix and whose local maxima are the eigenvalues of its $(N-1) \times (N-1)$ submatrix. We study the scaling limit of Kerov's piecewise linear functions for Wigner and Wishart matrices. For Wigner matrices the scaling limit is given by the Verhik-Kerov-Logan-Shepp curve which is known from asymptotic representation theory. For Wishart matrices the scaling limit is also explicitly found, and we explain its relation to the Marchenko-Pastur limit spectral law.