2009/12/09 by Matthias Birkner, Birkner, Matthias, Rongfeng Sun +1 · 3 citations
Mathematics · Physics and Astronomy · #60K35 #82B44 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.0912.1663
openalex publication_date 2009/12/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the continuous time version of the random walk pinning model, where conditioned on a continuous time random walk Y 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 X with jump rate 1 is Gibbs transformed with weight eβLt(X,Y), where Lt(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 βc>=0. A natural question is whether or not there is disorder relevance, namely whether or not βc differs from the critical point βcann 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 βc-βannc 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.