Non-existence of bi-infinite polymer Gibbs measures

TL;DR

This paper proves that in the inverse-gamma directed polymer model on a square lattice, nontrivial bi-infinite Gibbs measures almost surely do not exist, highlighting the model's unique path behavior in typical environments.

Contribution

It establishes the non-existence of nontrivial bi-infinite Gibbs measures in the inverse-gamma polymer model, using novel coupling and comparison techniques.

Findings

Bi-infinite Gibbs measures are almost surely absent in typical environments.

Paths with endpoints diverging in opposite directions tend to escape, indicating non-existence.

The proof leverages Busemann functions and planar comparison arguments.

Abstract

We show that nontrivial bi-infinite polymer Gibbs measures do not exist in typical environments in the inverse-gamma (or log-gamma) directed polymer model on the planar square lattice. The precise technical result is that, except for measures supported on straight-line paths, such Gibbs measures do not exist in almost every environment when the weights are independent and identically distributed inverse-gamma random variables. The proof proceeds by showing that when two endpoints of a point-to-point polymer distribution are taken to infinity in opposite directions but not parallel to lattice directions, the midpoint of the polymer path escapes. The proof is based on couplings, planar comparison arguments, and a recently discovered joint distribution of Busemann functions.