Path counting and random matrix theory

TL;DR

This paper explores combinatorial identities related to Dyck and Motzkin paths, connecting them to properties of random matrix ensembles and open problems in combinatorial proofs.

Contribution

It introduces new combinatorial identities linked to random matrix theory and provides bijective proofs, with some identities remaining open for combinatorial interpretation.

Findings

Established three identities involving Dyck and Motzkin paths

Connected combinatorial identities to properties of $eta$-Hermite and $eta$-Laguerre ensembles

Presented two new identities with open problems for combinatorial proofs

Abstract

We establish three identities involving Dyck paths and alternating Motzkin paths, whose proofs are based on variants of the same bijection. We interpret these identities in terms of closed random walks on the halfline. We explain how these identities arise from combinatorial interpretations of certain properties of the $\beta$-Hermite and $\beta$-Laguerre ensembles of random matrix theory. We conclude by presenting two other identities obtained in the same way, for which finding combinatorial proofs is an open problem.