Quenched and Annealed Critical Points in Polymer Pinning Models

TL;DR

This paper investigates the difference between quenched and annealed critical points in polymer pinning models with Markov chain configurations, providing precise results on the gap's order across various parameters and temperature regimes.

Contribution

It establishes the exact order of the critical point gap for high temperatures and various tail behaviors of excursion lengths, extending previous results to new parameter ranges.

Findings

The gap between quenched and annealed critical points is characterized for different values of c.

For c=3/2, the gap is positive if the slowly varying function tends to zero.

A new proof shows the gap is positive for c>3/2 at arbitrary temperatures.

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 $n^{-c}\phi(n)$ with $c \geq 1$ and $\phi$ 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 $\phi(n) \to 0$ 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 $c=2$.