Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Disorder relevance for the random walk pinning model in dimension 3

Matthias Birkner and Rongfeng Sun

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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≤st 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 βcβcann is at least of the order eC(ζ)/ρζ, C(ζ) > 0, for any ζ > 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.


Nous étudions la version à temps continu du modèle de marche aléatoire avec accrochage, où conditionné sur une marche aléatoire à temps continu (Ys)s≥0 sur ℤd avec taux de saut ρ > 0, qui joue le rôle de désordre, la loi jusqu’au temps t d’une seconde marche aléatoire indépendante (Xs)0≤st avec taux de saut 1 est la transformée de Gibbs avec poids eβLt(X,Y), où Lt(X, Y) est le temps local de collision entre X et Y jusqu’au temps t. Lorsque la température inverse β varie, le modèle subit une transition de localisation-délocalisation à un βc ≥ 0 critique. Une question naturelle est de savoir s’il y a pertinence du désordre ou pas, i.e., si βc diffère ou pas du point critique βcann pour le modèle moyenné. Dans [3], il a été montré qu’il y avait non pertinence du désordre en dimensions d = 1 et 2, et pertinence du désordre lorsque d ≥ 4. Pour d ≥ 5, la pertinence du désordre fût d’abord prouvée dans [2]. Dans ce papier, nous prouvons que si X et Y ont le même noyau de probabilité de saut, qui est irréductible et symétrique avec des moments du second ordre finis, alors il y a également pertinence du désordre en dimension critique d = 3, et βcβcann est au moins de l’ordre eC(ζ)/ρζ, C(ζ) > 0, pour tout ζ > 2. Notre preuve utilise des techniques de coarse graining et de moment fractionnaire, qui ont été récemment appliquées par Lacoin [13] au modèle de polymère dirigé en milieu aléatoire, et par Giacomin, Lacoin et Toninelli [10] pour établir la pertinence du désordre pour le modèle d’accrochages aléatoires en dimension critique. En chemin, nous prouvons également une version en temps continu du théorème limite local de Doney [5] pour des processus de renouvellement avec moyenne infinie.

Article information

Ann. Inst. H. Poincaré Probab. Statist. Volume 47, Number 1 (2011), 259-293.

First available in Project Euclid: 4 January 2011

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Collision local time Disordered pinning models Fractional moment method Local limit theorem Marginal disorder Random walks Renewal processes with infinite mean


Birkner, Matthias; Sun, Rongfeng. Disorder relevance for the random walk pinning model in dimension 3. Ann. Inst. H. Poincaré Probab. Statist. 47 (2011), no. 1, 259--293. doi:10.1214/10-AIHP374.

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