Symmetrized models of last passage percolation and non-intersecting lattice paths

TL;DR

This paper develops a theory connecting symmetrized last passage percolation models with averages over classical groups, using non-intersecting lattice paths and random matrix techniques.

Contribution

It introduces a unified framework for analyzing symmetrized percolation models via non-intersecting lattice paths and random matrix integration methods.

Findings

Expressed last passage times in terms of classical group averages

Derived probabilities for additional symmetrizations

Connected percolation models with random matrix theory

Abstract

It has been shown that the last passage time in certain symmetrized models of directed percolation can be written in terms of averages over random matrices from the classical groups $U(l)$, $Sp(2l)$ and $O(l)$. We present a theory of such results based on non-intersecting lattice paths, and integration techniques familiar from the theory of random matrices. Detailed derivations of probabilities relating to two further symmetrizations are also given.