Pfaffian Schur processes and last passage percolation in a half-quadrant

TL;DR

This paper analyzes last passage percolation in a half-quadrant using Pfaffian Schur processes, revealing fluctuation distributions that depend on diagonal weights and establishing a universal crossover phenomenon in KPZ models.

Contribution

It introduces a novel analysis of half-space last passage percolation via Pfaffian Schur processes and uncovers a universal fluctuation crossover near critical diagonal weights.

Findings

Fluctuations are GSE, GOE, or Gaussian depending on diagonal weights.

Away from the diagonal, fluctuations follow GUE Tracy-Widom distribution.

Discovered a universal crossover between Tracy-Widom distributions in KPZ models.

Abstract

We study last passage percolation in a half-quadrant, which we analyze within the framework of Pfaffian Schur processes. For the model with exponential weights, we prove that the fluctuations of the last passage time to a point on the diagonal are either GSE Tracy-Widom distributed, GOE Tracy-Widom distributed, or Gaussian, depending on the size of weights along the diagonal. Away from the diagonal, the fluctuations of passage times follow the GUE Tracy-Widom distribution. We also obtain a two-dimensional crossover between the GUE, GOE and GSE distribution by studying the multipoint distribution of last passage times close to the diagonal when the size of the diagonal weights is simultaneously scaled close to the critical point. We expect that this crossover arises universally in KPZ growth models in half-space. Along the way, we introduce a method to deal with diverging correlation kernels of point processes where points collide in the scaling limit.