Abstract
We discuss the long time behaviour of the parabolic Anderson model, the Cauchy problem for the heat equation with random potential on\(\mathbb{Z}^{d}\). We consider general i.i.d. potentials and show that exactly four qualitatively different types of intermittent behaviour can occur. These four universality classes depend on the upper tail of the potential distribution: (1) tails at ∞ that are thicker than the double-exponential tails, (2) double-exponential tails at ∞ studied by Gärtner and Molchanov, (3) a new class called almost bounded potentials, and (4) potentials bounded from above studied by Biskup and König. The new class (3), which contains both unbounded and bounded potentials, is studied in both the annealed and the quenched setting. We show that intermittency occurs on unboundedly increasing islands whose diameter is slowly varying in time. The characteristic variational formulas describing the optimal profiles of the potential and of the solution are solved explicitly by parabolas, respectively, Gaussian densities. Our analysis of class (3) relies on two large deviation results for the local times of continuous-time simple random walk. One of these results is proved by Brydges and the first two authors in [BHK04], and is also used here to correct a proof in [BK01].
Similar content being viewed by others
References
Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Cambridge: Cambridge University Press 1987
Biskup M., König W. (2001): Long-time tails in the parabolic Anderson model with bounded potential. Ann. Probab. 29(2): 636–682
Biskup M., König W. (2001): Screening effect due to heavy lower tails in one-dimensional parabolic Anderson model. J. Stat. Phys. 102(5/6): 1253–1270
Brydges, D., van der Hofstad, R., König, W.: Joint density for the local times of Markov chains (2005) available at http://www.math.uni-leipzig.de/~koenig/www/
Carmona, R., Molchanov, S.A.: Parabolic Anderson problem and intermittency. Mem. Amer. Math. Soc. 108(518), (1994)
Chen X. (2004): Exponential asymptotics and law of the iterated logarithm for self-intersection local times of random walks. Ann. Probab. 32(4): 3248–3300
Dembo A., Zeitouni O. (1998): Large Deviations Techniques and Applications, 2nd Edition. Springer, New York
Donsker M., Varadhan S.R.S. (1979): On the number of distinct sites visited by a random walk. Comm. Pure Appl. Math. 32, 721–747
Gantert, N.,König, W., Shi, Z.: Annealed deviations for random walk in random scenery. Annales Inst. H. Poincaré: Probab. et Stat., to appear (2006).
Gärtner J., den Hollander F. (1999): Correlation structure of intermittency in the parabolic Anderson model. Probab. Theory Relat. Fields 114, 1–54
Gärtner J., König W. (2000): Moment asymptotics for the continuous parabolic Anderson model. Ann. Appl. Probab. 10(3): 192–217
Gärtner J., König W. (2005): The parabolic Anderson model. In: Deuschel J.-D., Greven A (eds). Interacting Stochastic Systems. Springer, Berlin Heidelberg Newyork, pp. 153–179
Gärtner J., König W., Molchanov S. (2000): Almost sure asymptotics for the continuous parabolic Anderson model. Probab. Theory Relat. Fields 118(4): 547–573
Gärtner, J., König, W., Molchanov, S.: Geometric characterization of intermittency in the parabolic Anderson model. Ann. Probab., to appear (2006)
Gärtner J., Molchanov S. (1990): Parabolic problems for the Anderson model I. Intermittency and related topics. Commun. Math. Phys. 132, 613–655
Gärtner J., Molchanov S. (1998): Parabolic problems for the Anderson model II. Second-order asymptotics and structure of high peaks. Probab. Theory Relat. Fields 111, 17–55
den Hollander, F.: Large Deviations. Fields Institute Monographs. Providence, RI: Amer. Math. Soc. 2000
König W., Mörters P. (2002): Brownian intersection local times: upper tail asymptotics and thick points. Ann. Probab. 30, 1605–1656
Lieb, E.H., Loss, M.: Analysis, 2nd Edition. AMS Graduate Studies, Vol. 14, Providence, RI: Amer. Math. Soc., 2001
Molchanov, S.: Lectures on random media. In: D. Bakry, R.D. Gill, S. Molchanov, Lectures on Probability Theory, Ecole d’Eté de Probabilités de Saint-Flour XXII-1992, LNM 1581, Berlin: Springer, 1994, pp. 242–411
Molchanov, S., Ruzmaikin, A.: Lyapunov exponents and distributions of magnetic fields in dynamo models. In The Dynkin Festschrift: Markov Processes and their Applications. Mark Freidlin, (ed.) Basel: Birkhäuser, (1994)
Sznitman A.-S. (1998): Brownian motion, Obstacles and Random Media. Springer, Berlin
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by H. Spohn
Rights and permissions
About this article
Cite this article
van der Hofstad, R., König, W. & Mörters, P. The Universality Classes in the Parabolic Anderson Model. Commun. Math. Phys. 267, 307–353 (2006). https://doi.org/10.1007/s00220-006-0075-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-006-0075-4