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

Francesco Caravenna; Philippe Carmona; Nicolas Pétrélis

Annales de l'I.H.P. Probabilités et statistiques (2012)

  • Volume: 48, Issue: 4, page 1049-1080
  • ISSN: 0246-0203

Abstract

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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.

How to cite

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Caravenna, Francesco, Carmona, Philippe, and Pétrélis, Nicolas. "The discrete-time parabolic Anderson model with heavy-tailed potential." Annales de l'I.H.P. Probabilités et statistiques 48.4 (2012): 1049-1080. <http://eudml.org/doc/271948>.

@article{Caravenna2012,
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.},
author = {Caravenna, Francesco, Carmona, Philippe, Pétrélis, Nicolas},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {parabolic Anderson model; directed polymer; Heavy tailed potential; random environment; localization; heavy tailed potential},
language = {eng},
number = {4},
pages = {1049-1080},
publisher = {Gauthier-Villars},
title = {The discrete-time parabolic Anderson model with heavy-tailed potential},
url = {http://eudml.org/doc/271948},
volume = {48},
year = {2012},
}

TY - JOUR
AU - Caravenna, Francesco
AU - Carmona, Philippe
AU - Pétrélis, Nicolas
TI - The discrete-time parabolic Anderson model with heavy-tailed potential
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2012
PB - Gauthier-Villars
VL - 48
IS - 4
SP - 1049
EP - 1080
AB - 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.
LA - eng
KW - parabolic Anderson model; directed polymer; Heavy tailed potential; random environment; localization; heavy tailed potential
UR - http://eudml.org/doc/271948
ER -

References

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  2. [2] P. Carmona and Y. Hu. On the partition function of a directed polymer in a Gaussian random environment. Probab. Theory Related Fields124 (2002) 431–457. Zbl1015.60100MR1939654
  3. [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. Zbl1114.82017MR2073332
  4. [4] J. Gärtner and W. König. The parabolic Anderson model. In Interacting Stochastic Systems 153–179. Springer, Berlin, 2005. Zbl1111.82011MR2118574
  5. [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. Zbl0711.60055MR1069840
  6. [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. Zbl1115.82030
  7. [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]. Zbl1251.82070
  8. [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. Zbl1183.60024
  9. [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. Zbl1227.82098

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