Last passage percolation in lower triangular domain

TL;DR

This paper investigates last passage percolation in a lower triangular domain, revealing connections to Schur measures and demonstrating that asymptotic distributions follow GOE and GSE Tracy-Widom laws, including a new generalization.

Contribution

It establishes the asymptotic behavior of LPP in a lower triangular domain, linking it to Tracy-Widom distributions and generalizing the GSE case.

Findings

Asymptotics governed by GOE Tracy-Widom distribution

Asymptotics governed by GSE Tracy-Widom distribution

Generalization of GSE Tracy-Widom distribution for truncated case

Abstract

Last passage percolation (LPP) in an $n\times n$ lower triangular domain has nice connections with various generalizations of Schur measures. LPP along an anti-diagonal, from $(1,n)$ to $(n,1)$, gives a distribution of a highest column of a random composition with respect to a Demazure measure (a non-symmetric analog of a Schur measure). LPP along a main diagonal, from $(1,1)$ to $(n,n)$, is distributed as a marginal of a Pfaffian Schur process. In the first case we show that the asymptotics for the constant specialization is governed by the GOE Tracy-Widom distribution, in the second case - by the GSE Tracy-Widom distribution. In the latter case we were also able to study the truncated lower triangular case, obtaining an interesting generalization of the GSE Tracy-Widom distribution.