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March 2012 Crossing random walks and stretched polymers at weak disorder
Dmitry Ioffe, Yvan Velenik
Ann. Probab. 40(2): 714-742 (March 2012). DOI: 10.1214/10-AOP625


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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Dmitry Ioffe. Yvan Velenik. "Crossing random walks and stretched polymers at weak disorder." Ann. Probab. 40 (2) 714 - 742, March 2012.


Published: March 2012
First available in Project Euclid: 26 March 2012

zbMATH: 1251.60074
MathSciNet: MR2952089
Digital Object Identifier: 10.1214/10-AOP625

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

Rights: Copyright © 2012 Institute of Mathematical Statistics


Vol.40 • No. 2 • March 2012
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