Tightness for interlacing geometric random walk bridges

TL;DR

This paper studies line ensembles formed by interlaced geometric random walk bridges, proving tightness and Brownian Gibbs property, and demonstrating convergence of certain Schur processes to Airy wanderer ensembles.

Contribution

It establishes tightness and the Brownian Gibbs property for interlaced geometric random walk bridges, and proves convergence of spiked Schur processes to Airy wanderer line ensembles.

Findings

Line ensembles are tight under one-point tightness.

Any subsequential limit satisfies the Brownian Gibbs property.

Sequences of spiked Schur processes converge to Airy wanderer ensembles.

Abstract

We investigate a class of line ensembles whose local structure is described by independent geometric random walk bridges, which have been conditioned to interlace with each other. The latter arise naturally in the context Schur processes, including their versions in a half-space and a finite interval with free or periodic boundary conditions. We show that under one-point tightness of the curves, these line ensembles are tight and any subsequential limit satisfies the Brownian Gibbs property. As an application of our tightness results, we show that sequences of spiked Schur processes, that were recently considered in arXiv:2408.08445, converge uniformly over compact sets to the Airy wanderer line ensembles.