Localization in log-gamma polymers with boundaries

Abstract

Consider the directed polymer in one space dimension in log-gamma environment with boundary conditions, introduced by Seppäläinen [35]. In the equilibrium case, we prove that the end point of the polymer converges in law as the length increases, to a density proportional to the exponent of a zero-mean random walk. This holds without space normalization, and the mass concentrates in a neighborhood of the minimum of this random walk. We have analogous results out of equilibrium as well as for the middle point of the polymer with both ends fixed. The existence and the identification of the limit relies on the analysis of a random walk seen from its infimum.

Free energy of directed polymers in random environment in $1+1$-dimension at high temperature

“…There are a lot of progressions for Z d -lattice model in three decades [9,11,15,16,12,17,22,5]. Recently, the KPZ universality class conjecture for d = 1 case has been focused and was confirmed for a certain environment [30,19,14]. The recent progressions are reviewed in [13].…”

The endpoint distribution of directed polymers

“…In view of the result (1.19) of Comets and Nguyen [19], it seems plausible that the single copy condition holds for the log-gamma polymer in 1+1 dimensions. Unfortunately, we have been unable to determine whether or not the single copy condition holds in general.…”

“…Hence where the final inequality is a consequence of (A. 19). Considering the second case, we have Next we analyze the second difference, D 2 , from (A.27).…”

“…Regarding spatial fluctuations, Comets and Nguyen [19] found an explicit limiting endpoint distribution in the equilibrium case of the log-gamma model. More precisely, they showed that the endpoint distribution, when re-centered around the most likely site, converges in law to a certain random distribution on Z.…”

“…of size O(1)). In the notation of this paper, if x n = arg max ρ n (ω n = ·), then the result of [19] says lim K→∞ lim inf n→∞ ρ n ( ω n − x n 1 ≤ K) = 1 a.s. (1.19) This provides an affirmative case of the so-called "favorite region" conjecture, which speculates that in strong disorder, directed polymers on the lattice localize their endpoint to a region of fixed diameter. We emphasize once more that this is a statement about quenched endpoint distributions.…”