Uniform bounds for exponential moment of maximum of a Dyck paths

TL;DR

This paper establishes uniform bounds for the exponential moments of the maximum height of uniformly chosen Dyck paths, which supports high-moment estimates in random matrix theory.

Contribution

It proves the convergence and uniform boundedness of exponential moments of Dyck path maxima normalized by sqrt(n), a novel result linking combinatorics and random matrix analysis.

Findings

Exponential moments of Dyck path maxima converge as n approaches infinity.

These moments are uniformly bounded in n, independent of path length.

Supports high-moment estimates in random matrix theory.

Abstract

Let D be a Dyck path chosen uniformly from the set of Dyck paths with 2n steps. We prove that the sequence of the exponential moments of the maximum of D normalized by the square root of n converges in the limit of infinite n, and therefore is bounded uniformly in n. This result justifies corresponding assumption used to prove certain estimates of high moments of large random matrices.