Solvable models in the KPZ class: approach through periodic and free boundary Schur measures

TL;DR

This paper develops a comprehensive theory for solvable stochastic models in the KPZ class by connecting $q$-Whittaker measures with boundary Schur measures, leading to explicit formulas and phase transition results.

Contribution

It introduces new determinantal and pfaffian formulas for KPZ models, including ASEP and Log Gamma polymer, and establishes the Baik-Rains phase transition in half-space KPZ.

Findings

Determinantal formulas for ASEP current distribution

Fredholm pfaffian formulas for Log Gamma polymer in half space

Baik-Rains phase transition for KPZ height function

Abstract

We explore probabilistic consequences of correspondences between $q$-Whittaker measures and periodic and free boundary Schur measures established by the authors in the recent paper [arXiv:2106.11922]. The result is a comprehensive theory of solvability of stochastic models in the KPZ class where exact formulas descend from mapping to explicit determinantal and pfaffian point processes. We discover new variants of known results as determinantal formulas for the current distribution of the ASEP on the line and new results such as Fredholm pfaffian formulas for the distribution of the point-to-point partition function of the Log Gamma polymer model in half space. In the latter case, scaling limits and asymptotic analysis allow to establish Baik-Rains phase transition for height function of the KPZ equation on the half line at the origin.