Quenched and Annealed Critical Points in Polymer Pinning Models

Abstract

We consider a polymer with configuration modeled by the path of a Markov chain, interacting with a potential $u+V_n$ which the chain encounters when it visits a special state 0 at time $n$. The disorder $(V_n)$ is a fixed realization of an i.i.d. sequence. The polymer is pinned, i.e. the chain spends a positive fraction of its time at state 0, when $u$ exceeds a critical value. We assume that for the Markov chain in the absence of the potential, the probability of an excursion from 0 of length $n$ has the form $n^{-c}\varphi (n)$ with $c\ge 1$ and $\varphi$ slowly varying. Comparing to the corresponding annealed system, in which the $V_n$ are effectively replaced by a constant, it is known that the quenched and annealed critical points differ at all temperatures for $3/2<c<2$ and $c>2$, but only at low temperatures for $c<3/2$. For high temperatures and $3/2<c<2$ we establish the exact order of the gap between critical points, as a function of temperature. For the borderline case $c=3/2$ we show that the gap is positive provided $\varphi (n)\to 0$ as $n\to \infty$, and for $c>3/2$ with arbitrary temperature we provide a new proof that the gap is positive, and extend it to $c=2$.

33 pages

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• Solid-On-Solid interfaces with disordered pinning
• The critical disordered pinning measure