Disorder relevance for the random walk pinning model in dimension 3

Abstract

We study the continuous time version of the random walk pinning model, where conditioned on a continuous time random walk (Ys) s>0 on ℤd with jump rate ρ > 0, which plays the role of disorder, the law up to time t of a second independent random walk (Xs)0≤s≤t with jump rate 1 is Gibbs transformed with weight eβL t (X,y), where Lt(X, Y) is the collision local time between X and Y up to time t. As the inverse temperature β varies, the model undergoes a localization-delocalization transition at some critical βc > 0. A natural question is whether or not there is disorder relevance, namely whether or not βc differs from the critical point β c ann for the annealed model. In [3], it was shown that there is disorder irrelevance in dimensions d = 1 and 2, and disorder relevance in d > 4. For d > 5, disorder relevance was first proved in [2]. In this paper, we prove that if X and Y have the same jump probability kernel, which is irreducible and symmetric with finite second moments, then there is also disorder relevance in the critical dimension d = 3, and βc - βc ann is at least of the order e-C(ξ)/ρξ, C(ξ) > 0, for any ξ > 2. Our proof employs coarse graining and fractional moment techniques, which have recently been applied by Lacoin [13] to the directed polymer model in random environment, and by Giacomin, Lacoin and Toninelli [10] to establish disorder relevance for the random pinning model in the critical dimension. Along the way, we also prove a continuous time version of Doney's local limit theorem [5] for renewal processes with infinite mean. © Association des Publications de l'Institut Henri Poincaré, 2011.