Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Annealed vs quenched critical points for a random walk pinning model

Matthias Birkner and Rongfeng Sun

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We study a random walk pinning model, where conditioned on a simple random walk Y on ℤd acting as a random medium, the path measure of a second independent simple random walk X up to time t is Gibbs transformed with Hamiltonian −Lt(X, Y), where Lt(X, Y) is the collision local time between X and Y up to time t. This model arises naturally in various contexts, including the study of the parabolic Anderson model with moving catalysts, the parabolic Anderson model with Brownian noise, and the directed polymer model. It falls in the same framework as the pinning and copolymer models, and exhibits a localization-delocalization transition as the inverse temperature β varies. We show that in dimensions d=1, 2, the annealed and quenched critical values of β are both 0, while in dimensions d≥4, the quenched critical value of β is strictly larger than the annealed critical value (which is positive). This implies the existence of certain intermediate regimes for the parabolic Anderson model with Brownian noise and the directed polymer model. For d≥5, the same result has recently been established by Birkner, Greven and den Hollander [Quenched LDP for words in a letter sequence (2008)] via a quenched large deviation principle. Our proof is based on a fractional moment method used recently by Derrida et al. [Comm. Math. Phys. 287 (2009) 867–887] to establish the non-coincidence of annealed and quenched critical points for the pinning model in the disorder-relevant regime. The critical case d=3 remains open.


Nous considérons le modèle de marche aléatoire avec pinning suivant : étant donné une marche aléatoire simple Y sur ℤd qui sert d’environnement aléatoire, on se donne une mesure de Gibbs sur les trajectoires d’une marche aléatoire X jusqu’au temps t de Hamiltonien −Lt(X, Y) où Lt(X, Y) est le temps local d’intersection entre X et Y jusqu’au temps t. Ce modèle apparaît naturellement dans des contextes variés tels que l’étude du modèle parabolique d’Anderson avec catalyseurs mouvants, l’étude du modèle parabolique d’Anderson avec bruit Brownien ainsi que dans le cadre de l’étude de polymères dirigés. Ce modèle appartient à la même classe que les modèles de pinning et copolymères et présente une transition localisation / délocalisation quand la température inverse β varie. Nous montrons qu’en dimension d=1, 2 les valeurs critiques annealed et quenched de β sont toutes deux 0 mais que en dimension d≥4 la valeur critique quenched de β est strictement supérieure à la valeur annealed (qui est positive). Ceci entraine l’existence de certains régimes intermédiaires pour le modèle parabolique de Anderson avec bruit Brownien et pour les polymères dirigés. Pour d≥5 des résultats similaires ont été récemment établis par Birkner, Greven et den Hollander [Quenched LDP for words in a letter sequence (2008)] via un principe de grandes déviations quenched. Notre preuve se fonde sur la méthode des moments fractionnaires utilisée récemment par Derrida, Giacomin, Lacoin et Toninelli [Comm. Math. Phys. 287 (2009) 867–887] pour établir la non-coïncidence des valeurs critiques quenched et annealed du modèle de pinning dans le régime lié au désordre. Le cas de la dimension critique d=3 reste ouvert.

Article information

Ann. Inst. H. Poincaré Probab. Statist. Volume 46, Number 2 (2010), 414-441.

First available in Project Euclid: 11 May 2010

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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.)

Random walks Pinning models Annealed and quenched critical points Collision local time Disordered system


Birkner, Matthias; Sun, Rongfeng. Annealed vs quenched critical points for a random walk pinning model. Ann. Inst. H. Poincaré Probab. Statist. 46 (2010), no. 2, 414--441. doi:10.1214/09-AIHP319.

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