An identity in distribution between full-space and half-space log-gamma polymers

TL;DR

This paper establishes a distributional identity between two partition functions of the log-gamma polymer model and demonstrates a phase transition in free energy depending on boundary noise strength, confirming a long-standing physics prediction.

Contribution

It proves a novel distributional identity for log-gamma polymers and rigorously establishes a phase transition in free energy related to boundary noise.

Findings

Distributional identity between point-to-point and point-to-line partition functions

Existence of a phase transition in free energy depending on boundary noise

First rigorous proof of such a transition in a positive temperature model

Abstract

We prove an identity in distribution between two kinds of partition functions for the log-gamma directed polymer model: (1) the point-to-point partition function in a quadrant, (2) the point-to-line partition function in an octant. As an application, we prove that the point-to-line free energy of the log-gamma polymer in an octant obeys a phase transition depending on the strength of the noise along the boundary. This transition of (de)pinning by randomness was first predicted in physics by Kardar in 1985 and proved rigorously for zero temperature models by Baik and Rains in 2001. While it is expected to arise universally for models in the Kardar-Parisi-Zhang universality class, this is the first positive temperature model for which this transition can be rigorously established.