The effect of disorder on the free-energy for the Random Walk Pinning Model: smoothing of the phase transition and low temperature asymptotics

TL;DR

This paper studies how disorder affects the phase transition and low temperature behavior of the continuous-time Random Walk Pinning Model, showing disorder smooths the transition in higher dimensions and alters low temperature asymptotics.

Contribution

It proves that disorder makes the phase transition at least second order in dimensions ≥3 and changes the low temperature asymptotics of the free energy in all dimensions.

Findings

Disorder smooths the phase transition to at least second order in dimensions ≥3.

In dimensions ≥4, the transition differs from the first-order annealed case.

Disorder modifies the low temperature asymptotic behavior of the free energy.

Abstract

We consider the continuous time version of the Random Walk Pinning Model (RWPM), studied in [5,6,7]. Given a fixed realization of a random walk Y$ on Z^d with jump rate rho (that plays the role of the random medium), we modify the law of a random walk X on Z^d with jump rate 1 by reweighting the paths, giving an energy reward proportional to the intersection time L_t(X,Y)=\int_0^t \ind_{X_s=Y_s}\dd s: the weight of the path under the new measure is exp(beta L_t(X,Y)), beta in R. As beta increases, the system exhibits a delocalization/localization transition: there is a critical value beta_c, such that if beta>beta_c the two walks stick together for almost-all Y realizations. A natural question is that of disorder relevance, that is whether the quenched and annealed systems have the same behavior. In this paper we investigate how the disorder modifies the shape of the free energy curve: (1) We prove that, in dimension d larger or equal to three 3, the presence of disorder makes the phase transition at least of second order. This, in dimension larger or equal to 4, contrasts with the fact that the phase transition of the annealed system is of first order. (2) In any dimension, we prove that disorder modifies the low temperature asymptotic of the free energy.