Disorder relevance for the random walk pinning model in dimension 3

Birkner, Matthias, and Sun, Rongfeng. "Disorder relevance for the random walk pinning model in dimension 3." Annales de l'I.H.P. Probabilités et statistiques 47.1 (2011): 259-293. <http://eudml.org/doc/240888>.

@article{Birkner2011,
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 &gt; 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 eβLt(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 βcann 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 − βcann is at least of the order e−C(ζ)/ρζ, C(ζ) &gt; 0, for any ζ &gt; 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.},
author = {Birkner, Matthias, Sun, Rongfeng},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {collision local time; disordered pinning models; fractional moment method; local limit theorem; marginal disorder; random walks; renewal processes with infinite mean; fractional moment methods; local limit theorems; renewal process with infinite mean},
language = {eng},
number = {1},
pages = {259-293},
publisher = {Gauthier-Villars},
title = {Disorder relevance for the random walk pinning model in dimension 3},
url = {http://eudml.org/doc/240888},
volume = {47},
year = {2011},
}

TY - JOUR
AU - Birkner, Matthias
AU - Sun, Rongfeng
TI - Disorder relevance for the random walk pinning model in dimension 3
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2011
PB - Gauthier-Villars
VL - 47
IS - 1
SP - 259
EP - 293
AB - 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 &gt; 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 eβLt(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 βcann 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 − βcann is at least of the order e−C(ζ)/ρζ, C(ζ) &gt; 0, for any ζ &gt; 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.
LA - eng
KW - collision local time; disordered pinning models; fractional moment method; local limit theorem; marginal disorder; random walks; renewal processes with infinite mean; fractional moment methods; local limit theorems; renewal process with infinite mean
UR - http://eudml.org/doc/240888
ER -