Gelfand-Tsetlin polytopes and random contractions away from the limiting shapes

TL;DR

This paper investigates the extremal eigenvalues of random matrix compressions related to Gelfand-Tsetlin polytopes, providing explicit estimates and showing outlier events are exponentially unlikely, with applications in quantum information.

Contribution

It offers explicit uniform bounds for extremal eigenvalues of matrix compressions and demonstrates the rarity of outlier behavior using determinantal point processes.

Findings

Explicit uniform estimates for extremal eigenvalues

Outlier eigenvalues occur with exponentially small probability

Connections to quantum information applications

Abstract

In this paper, we consider a sequence of selfadjoint matrices $A_n$ having a limiting spectral distribution as $n\to \infty$, and we consider a sequence of full flags $\{0\le p_1^n\le\ldots\le p_i^n\le\ldots\le 1_n\}$ chosen at random according to the uniform measure on full flag manifolds. We are interested in the behaviour of the extremal eigenvalues of $p_i^nA_np_i^n$. This problem is known to be equivalent to the study of uniform probability measures on Gelfand-Tsetlin polytopes. Our main results consist in explicit uniform estimates for extremal eigenvalues, and the fact that an outlier behavior has an exponentially small probability. This problem is of intrinsic interest in random matrix theory, but it has also a strong motivation and some applications in quantum information, which we discuss. The proofs rely on a reinterpretation of the problem with the help of determinantal point processes and the techniques are based on steepest descent analysis.