Integral formulas for two-layer Schur and Whittaker processes

TL;DR

This paper derives contour integral formulas for two-layer Schur and Whittaker processes, enabling explicit computation of multipoint distributions and applications like KPZ growth rate analysis.

Contribution

It introduces explicit contour integral formulas and Doob transformed Markov process representations for two-layer Schur and Whittaker processes, advancing understanding of their joint distributions.

Findings

Derived contour integral formulas for multipoint distributions.

Expressed processes as Doob transformed Markov processes.

Computed KPZ growth rate with arbitrary boundary parameters.

Abstract

Stationary measures of last passage percolation with geometric weights and the log-gamma polymer in a strip of the $\mathbb Z^2$ lattice are characterized in arXiv:2306.05983 using variants of Schur and Whittaker processes, called two-layer Gibbs measures. In this article, we prove contour integral formulas characterizing the multipoint joint distribution of two-layer Schur and Whittaker processes. We also express them as Doob transformed Markov processes with explicit transition kernels. As an example of application of our formulas, we compute the growth rate of the KPZ equation on $[0,L]$ with arbitrary boundary parameters.