GUE minors, maximal Brownian functionals and longest increasing subsequences

TL;DR

This paper establishes a connection between GUE matrix minors, Brownian motion functionals, and the asymptotic shape of Young diagrams from random words, providing new insights into longest increasing subsequences.

Contribution

It introduces equalities in law linking GUE minors spectra with Brownian functionals and derives the limiting shape of Young diagrams from these spectral properties.

Findings

Spectral equalities between GUE minors and Brownian functionals

Limiting shape of RSK Young diagrams from spectral data

Distribution of the longest increasing subsequence length

Abstract

We present equalities in law between the spectra of the minors of a GUE matrix and some maximal functionals of independent Brownian motions. In turn, these results allow to recover the limiting shape (properly centered and scaled) of the RSK Young diagrams associated with a random word as a function of the spectra of these minors. Since the length of the top row of the diagrams is the length of the longest increasing subsequence of the random word, the corresponding limiting law also follows.