Crossing random walks and stretched polymers at weak disorder

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 $d\ge 3$ 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.

Published in at http://dx.doi.org/10.1214/10-AOP625 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• Coincidence of Lyapunov exponents for random walks in weak random potentials
• Finite connections for supercritical Bernoulli bond percolation in 2D

Cited by in corpus (10)

• Stretched Polymers in Random Environment
• Localization Transition for Polymers in Poissonian Medium
• Crossing velocities for an annealed random walk in a random potential
• Lyapunov exponents, shape theorems and large deviations for the random walk in random potential
• Asymptotics of even-even correlations in the Ising model
• Lyapunov exponents of random walks in small random potential: the lower bound
• Superdiffusivity for Brownian Motion in a Poissonian Potential with Long Range Correlation I: Lower Bound on the Volume Exponent
• An Almost-Sure CLT for Stretched Polymers
• Crossing speeds of random walks among "sparse" or "spiky" Bernoulli potentials on integers
• Ornstein-Zernike behavior for Ising models with infinite-range interactions