GOE and ${\rm Airy}_{2\to 1}$ marginal distribution via symplectic Schur functions

TL;DR

This paper derives formulas for the Tracy-Widom GOE distribution and the ${\rm Airy}_{2\to 1}$ process marginal distribution as limits of last passage percolation models, using symplectic Schur functions.

Contribution

It provides new integral expressions for last passage times and connects them to symplectic Schur functions, deriving known distributions as scaling limits.

Findings

Fredholm determinant formula for Tracy-Widom GOE distribution

Distribution of ${\rm Airy}_{2\to 1}$ process as a limit of percolation models

New integral representations involving symplectic Schur functions

Abstract

We derive Sasamoto's Fredholm determinant formula for the Tracy-Widom GOE distribution, as well as the one-point marginal distribution of the ${\rm Airy}_{2\to1}$ process, originally derived by Borodin-Ferrari-Sasamoto, as scaling limits of point-to-line and point-to-half-line last passage percolation with exponentially distributed waiting times. The asymptotic analysis goes through new expressions for the last passage times in terms of integrals of (the continuous analog of) symplectic and classical Schur functions, obtained recently in [BZ19a].