Disorder relevance for the random walk pinning model in dimension 3

We study the continuous time version of the random walk pinning model, where conditioned on a continuous time random walk Y on Z^d with jump rate ρ>0, which plays the role of disorder, the law up to time t of a second independent random walk X with jump rate 1 is Gibbs transformed with weight e^{βL_t(X,Y)}, where L_t(X,Y) is the collision local time between X and Y up to time t. As the inverse temperature βvaries, the model undergoes a localization-delocalization transition at some critical β_c>=0. A natural question is whether or not there is disorder relevance, namely whether or not β_c differs from the critical point β_c^{ann} for the annealed model. In Birkner and Sun [BS09], 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 by Birkner, Greven and den Hollander [BGdH08]. 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 β_c-β^{ann}_c is at least of the order e^{-C(ζ)ρ^{-ζ}}, C(ζ)>0, for any ζ>2. Our proof employs coarse graining and fractional moment techniques, which have recently been applied by Lacoin [L09] to the directed polymer model in random environment, and by Giacomin, Lacoin and Toninelli [GLT09] 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 [D97] for renewal processes with infinite mean.

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