Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

The discrete-time parabolic Anderson model with heavy-tailed potential

Francesco Caravenna, Philippe Carmona, and Nicolas Pétrélis

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Abstract

We consider a discrete-time version of the parabolic Anderson model. This may be described as a model for a directed $(1+d)$-dimensional polymer interacting with a random potential, which is constant in the deterministic direction and i.i.d. in the $d$ orthogonal directions. The potential at each site is a positive random variable with a polynomial tail at infinity. We show that, as the size of the system diverges, the polymer extremity is localized almost surely at one single point which grows ballistically. We give an explicit characterization of the localization point and of the typical paths of the model.

Résumé

Nous considérons une version discrète du modèle parabolique d’Anderson. Ceci nous permet, par exemple, d’étudier un polymère dirigé en dimension $1+d$ qui interagit avec un potentiel constant dans la direction déterministe et i.i.d. dans l’hyperplan orthogonal. Le potentiel en chaque site est une variable aléatoire positive dont la queue décroît polynomialement. Nous prouvons que, lorsque la taille du système tend vers l’infini, l’extrémité du polymère se localise presque surement en un site unique, que nous caractérisons et qui s’éloigne balistiquement de l’origine. Nous donnons également une caractérisation du comportement typique des trajectoires de ce modèle.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 48, Number 4 (2012), 1049-1080.

Dates
First available in Project Euclid: 16 November 2012

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1353098440

Digital Object Identifier
doi:10.1214/11-AIHP465

Mathematical Reviews number (MathSciNet)
MR3052403

Zentralblatt MATH identifier
1266.60162

Subjects
Primary: 60K37: Processes in random environments 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.) 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]

Keywords
Parabolic Anderson model Directed polymer Heavy tailed potential Random environment Localization

Citation

Caravenna, Francesco; Carmona, Philippe; Pétrélis, Nicolas. The discrete-time parabolic Anderson model with heavy-tailed potential. Ann. Inst. H. Poincaré Probab. Statist. 48 (2012), no. 4, 1049--1080. doi:10.1214/11-AIHP465. https://projecteuclid.org/euclid.aihp/1353098440


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References

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