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

Caravenna, Francesco; Carmona, Philippe; Pétrélis, Nicolas

Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 4, pp. 1049-1080.

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.

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.

[1] A. Auffinger and O. Louidor. Directed polymers in random environment with heavy tails. Comm. Pure Appl. Math. 64 (2010) 183-204. | MR | Zbl

[2] P. Carmona and Y. Hu. On the partition function of a directed polymer in a Gaussian random environment. Probab. Theory Related Fields 124 (2002) 431-457. | MR | Zbl

[3] F. Comets, T. Shiga and N. Yoshida. Probabilistic analysis of directed polymers in a random environment: a review. In Stochastic Analysis on Large Scale Interacting Systems 115-142. Adv. Stud. Pure Math. 39. Math. Soc. Japan, Tokyo, 2004. | MR | Zbl

[4] J. Gärtner and W. König. The parabolic Anderson model. In Interacting Stochastic Systems 153-179. Springer, Berlin, 2005. | MR | Zbl

[5] J. Gärtner and S. A. Molchanov. Parabolic problems for the Anderson model. I. Intermittency and related topics. Comm. Math. Phys. 132 (1990) 613-655. | MR | Zbl

[6] R. Van Der Hofstad, W. König and P. Mörters. The universality classes in the parabolic Anderson model. Comm. Math. Phys. 267 (2006) 307-353. | Zbl

[7] D. Ioffe and Y. Velenik. Stretched polymers in random environment. In Probability in Complex Physical Systems, in honour of E. Bolthausen and J. Gärtner 339-369. J.-D. Deuschel et al. (Eds). Springer Proceedings in Mathematics 11. Springer, Berlin, 2012. Available at arXiv.org:1011.0266 [math.PR]. | Zbl

[8] W. König, H. Lacoin, P. Mörters and N. Sidorova. A two cities theorem for the parabolic Anderson model. Ann. Probab. 37 (2009) 347-392. | Zbl

[9] H. Lacoin. New bounds for the free energy of directed polymers in dimension $1+1$ and $1+2$. Comm. Math. Phys. 294 (2010) 471-503. | Zbl