The Annals of Probability

Crossing random walks and stretched polymers at weak disorder

Dmitry Ioffe and Yvan Velenik

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We consider a model of a polymer in ℤd+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 → ∞, when d ≥ 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.

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Ann. Probab., Volume 40, Number 2 (2012), 714-742.

First available in Project Euclid: 26 March 2012

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Polymer central limit theorem diffusivity Ornstein–Zernike theory quenched random environment


Ioffe, Dmitry; Velenik, Yvan. Crossing random walks and stretched polymers at weak disorder. Ann. Probab. 40 (2012), no. 2, 714--742. doi:10.1214/10-AOP625.

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