The Random Walk Pinning Model II: Upper bounds on the free energy and disorder relevance

Abstract

This article investigates the question of disorder relevance for the continuous-time Random Walk Pinning Model (RWPM) and completes the results of our companion paper. The RWPM considers a continuous time random walk $X=(X_t{)}_{t\ge 0}$, whose law is modified by a Gibbs weight given by $\exp (\beta {\int }_0^T{1}_{\{X_t=Y_t\}}dt)$, where $Y=(Y_t{)}_{t\ge 0}$ is a quenched trajectory of a second (independent) random walk and $\beta \ge 0$ is the inverse temperature. The random walk $Y$ has the same distribution as $X$ but a jump rate $\rho \ge 0$, interpreted as the disorder intensity. For fixed $\rho \ge 0$, the RWPM undergoes a localization phase transition as $\beta$ crosses a critical threshold ${\beta }_c(\rho )$. The question of disorder relevance then consists in determining whether a disorder of arbitrarily small intensity $\rho$ changes the properties of the phase transition. We focus our analysis on the case of transient $\gamma$-stable walks on $\mathbb{Z}$, i.e. random walks in the domain of attraction of a $\gamma$-stable law, with $\gamma \in (0,1)$. In the present paper, we show that disorder is relevant when $\gamma \in (0,\frac{2}{3}]$, namely that ${\beta }_c(\rho )>{\beta }_c(0)$ for every $\rho >0$. We also provide lower bounds on the critical point shift, which are matching the upper bounds obtained in our companion paper. Interestingly, in the marginal case $\gamma =\frac{2}{3}$, disorder is always relevant, independently of the fine properties of the random walk distribution. When $\gamma \in (\frac{2}{3},1)$, our companion paper proves that disorder is irrelevant (in particular ${\beta }_c(\rho )={\beta }_c(0)$ for $\rho$ small enough). We provide here an upper bound on the free energy in the regime $\gamma \in (\frac{2}{3},1)$ that highlights the fact that although disorder is irrelevant, it still has a non-trivial effect on the phase transition, at any $\rho >0$.

31 pages