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.
Highlights
1.1 Directed polymersThe directed polymer model is a stochastic model of a random path that interacts with a random environment
In its simplest formulation on an integer lattice Zd, positive random weights tYxuxPZd are assigned to the lattice vertices and the quenched probability of a finite lattice path π is declared to be proportional to the product ś xPπ
Gibbs measures to construct global solutions to a stochastic Burgers equation on the line, subject to random kick forcing at discrete time intervals
Summary
Let pYxqxPZ2 be an assignment of strictly positive real weights on the vertices of Z2. Define point-to-point polymer partition functions between vertices o ď p in Z2 by. The quenched polymer probability distribution on the set Xo,p is defined by. When the weights ω “ pYxq are random variables on some probability space pΩ, A, Pq, the averaged or annealed polymer distribution Po,p on Xo,p is defined by ż. It is convenient to use the unnormalized quenched polymer measure, which is the sum of path weights: Zo,ppAq “. The formulation above reveals that when the limit of the ratio Zxei,p{Zx,p exists for each fixed x as p tends to infinity, Q0,p converges weakly to a Markov chain. The limiting Markov chains are examples of rooted semi-infinite polymer Gibbs measures, which we discuss
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