Spatial tightness at the edge of Gibbsian line ensembles

TL;DR

This paper proves the tightness of the spatial process at the edge of Gibbsian line ensembles under certain conditions, with applications to the log-gamma polymer and implications for KPZ universality.

Contribution

It introduces a continuous grand monotone coupling for Gibbsian line ensembles, enabling the proof of edge tightness and Brownian absolute continuity.

Findings

Edge of Gibbsian line ensembles is tight under specified conditions.

Application to log-gamma polymer shows KPZ $2/3$ exponent behavior.

Subsequential limits are Brownian absolutely continuous.

Abstract

Consider a sequence of Gibbsian line ensembles, whose lowest labeled curves (i.e., the edge) have tight one-point marginals. Then, given certain technical assumptions on the nature of the Gibbs property and underlying random walk measure, we prove that the entire spatial process of the edge is tight. We then apply this black-box theory to the log-gamma polymer Gibbsian line ensemble which we construct. The edge of this line ensemble is the transversal free energy process for the polymer, and our theorem implies tightness with the ubiquitous KPZ class $2/3$ exponent, as well as Brownian absolute continuity of all the subsequential limits. A key technical innovation which fuels our general result is the construction of a continuous grand monotone coupling of Gibbsian line ensembles with respect to their boundary data (entrance and exit values, and bounding curves). {\em Continuous} means that the Gibbs measure varies continuously with respect to varying the boundary data, {\em grand} means that all uncountably many boundary data measures are coupled to the same probability space, and {\em monotone} means that raising the values of the boundary data likewise raises the associated measure. This result applies to a general class of Gibbsian line ensembles where the underlying random walk measure is discrete time, continuous valued and log-convex, and the interaction Hamiltonian is nearest neighbor and convex.