Equality of critical points for polymer depinning transitions with loop exponent one

Abstract

We consider a polymer with configuration modelled by the trajectory of a Markov chain, interacting with a potential of form $u+V_n$ when it visits a particular state 0 at time $n$, with $\{V_n\}$ representing i.i.d. quenched disorder. There is a critical value of $u$ above which the polymer is pinned by the potential. A particular case not covered in a number of previous studies is that of loop exponent one, in which the probability of an excursion of length $n$ takes the form $\varphi (n)/n$ for some slowly varying $\varphi$; this includes simple random walk in two dimensions. We show that in this case, at all temperatures, the critical values of $u$ in the quenched and annealed models are equal, in contrast to all other loop exponents, for which these critical values are known to differ, at least at low temperatures.

Published in at http://dx.doi.org/10.1214/09-AAP621 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• Disorder and wetting transition: the pinned harmonic crystal in dimension three or larger
• Subcritical percolation with a line of defects
• Weak coupling limits for directed polymers in tube environments