Crossing random walks and stretched polymers at weak disorder

TL;DR

This paper studies a polymer model in a random environment, proving that under certain conditions, the quenched and annealed free energies coincide and the polymer exhibits diffusive behavior similar to the annealed case.

Contribution

The authors strengthen previous results by showing the quenched and annealed partition function ratio converges under weaker disorder assumptions, leading to diffusive scaling.

Findings

Quenched and annealed free energies coincide in the limit.

Partition function ratio converges under weaker assumptions.

Polymer exhibits diffusive behavior with the same diffusivity as the annealed model.

Abstract

We consider a model of a polymer in $\mathbb{Z}^{d+1}$, constrained to join 0 and a hyperplane at distance $N$. The polymer is subject to a quenched nonnegative random environment. Alternatively, the model describes crossing random walks in a random potential (see Zerner [Ann Appl. Probab. 8 (1998) 246--280] or Chapter 5 of Sznitman [Brownian Motion, Obstacles and Random Media (1998) Springer] for the original Brownian motion formulation). It was recently shown [Ann. Probab. 36 (2008) 1528--1583; Probab. Theory Related Fields 143 (2009) 615--642] that, in such a setting, the quenched and annealed free energies coincide in the limit $N\to\infty$, when $d\geq3$ and the temperature is sufficiently high. We first strengthen this result by proving that, under somewhat weaker assumptions on the distribution of disorder which, in particular, enable a small probability of traps, the ratio of quenched and annealed partition functions actually converges. We then conclude that, in this case, the polymer obeys a diffusive scaling, with the same diffusivity constant as the annealed model.