Non-existence of bi-infinite polymer Gibbs measures

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. Contents1. Introduction Organization of the paper Notation and conventions 2. Polymer Gibbs measures 2.1. Directed polymers 2.2. Infinite Gibbs measures 2.3. Bi-infinite Gibbs measures in the inverse-gamma polymer 3. Stationary inverse-gamma polymer 4. Estimates for paths across a large square 5. Proof of the main theorem Appendix A. General properties of planar directed polymers A.1. Ratio weights and nested polymers A.2. Inequalities for point-to-point partition functions A.3. Ordering of path measures A.4. Polymers on the upper half-plane Appendix B. The inverse-gamma polymer B.1. Inverse-gamma weights B.2. Two jointly ratio-stationary polymer processes B.3. Wandering exponent Appendix C. Bound on the running maximum of a random walk References