The Effect of Disorder on Polymer Depinning Transitions Article

First Online: 08 February 2008 Received: 20 December 2006 Accepted: 29 August 2007 DOI :
10.1007/s00220-008-0425-5

Cite this article as: Alexander, K.S. Commun. Math. Phys. (2008) 279: 117. doi:10.1007/s00220-008-0425-5
Abstract We consider a polymer, with monomer locations modeled by the trajectory of a Markov chain, in the presence of a potential that interacts with the polymer when it visits a particular site 0. We assume that probability of an excursion of length n is given by \(n^{-c}\varphi(n)\) for some 1 < c < 2 and slowly varying \(\varphi\) . Disorder is introduced by having the interaction vary from one monomer to another, as a constant u plus i.i.d. mean-0 randomness. There is a critical value of u above which the polymer is pinned, placing a positive fraction (called the contact fraction) of its monomers at 0 with high probability. To see the effect of disorder on the depinning transition, we compare the contact fraction and free energy (as functions of u ) to the corresponding annealed system. We show that for c > 3/2, at high temperature, the quenched and annealed curves differ significantly only in a very small neighborhood of the critical point—the size of this neighborhood scales as \(\beta^{1/(2c-3)}\) , where β is the inverse temperature. For c < 3/2, given \(\epsilon > 0\) , for sufficiently high temperature the quenched and annealed curves are within a factor of \(1-\epsilon\) for all u near the critical point; in particular the quenched and annealed critical points are equal. For c = 3/2 the regime depends on the slowly varying function \(\varphi\) .

Communicated by F. Toninelli

Research supported by NSF grant DMS-0405915.

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Authors and Affiliations 1. Department of Mathematics KAP108 University of Southern California Los Angeles USA