The Effect of Disorder on Polymer Depinning Transitions

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 $\varphi$.