Potential confinement property of the parabolic Anderson model

Grüninger, Gabriela; König, Wolfgang

Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 3, pp. 840-863.

Nous considérons le modèle Anderson parabolique donné par le problème de Cauchy pour l'équation de la chaleur avec un potentiel aléatoire dans ℤ^d. Nous utilisons des potentiels i.i.d. ξ:ℤ^d→ℝ qui sont dans la troisième classe d'universalité de la classification de van der Hofstad, König and Mörters [Commun. Math. Phys. 267 (2006) 307-353]. Cette classe, les potentiels presque bornés, contient les potentiels dont le logarithme de la transformée de Laplace est de variation régulière avec paramètre γ=1, mais qui n'appartiennent pas à la classe des potentiels «double-exponentially distributed» étudiée par Gärtner et Molchanov dans [Probab. Theory Related Fields 111 (1998) 17-55]. Dans [Commun. Math. Phys. 267 (2006) 307-353], le comportement asymptotique de l'espérance de la masse totale est décrit par un problème variationnel qui est lié à l'inégalité de Sobolev logarithmique. La solution de ce problème, qui est unique à des translations spatiales près, est une parabole. Dans cet article, nous montrons que la contribution à la masse totale des potentiels qui ont (après un changement d'échelle) une forme différente est négligeable. Nous utilisons la topologie L¹ sur les compacts pour l'exponentielle du potentiel. Au cours de la preuve, nous montrons que n'importe quelle suite des solutions approximatives du problème variationnel converge vers une translation spatiale de la solution qui est une parabole.

We consider the parabolic Anderson model, the Cauchy problem for the heat equation with random potential in ℤ^d. We use i.i.d. potentials ξ:ℤ^d→ℝ in the third universality class, namely the class of almost bounded potentials, in the classification of van der Hofstad, König and Mörters [Commun. Math. Phys. 267 (2006) 307-353]. This class consists of potentials whose logarithmic moment generating function is regularly varying with parameter γ=1, but do not belong to the class of so-called double-exponentially distributed potentials studied by Gärtner and Molchanov [Probab. Theory Related Fields 111 (1998) 17-55]. In [Commun. Math. Phys. 267 (2006) 307-353] the asymptotics of the expected total mass was identified in terms of a variational problem that is closely connected to the well-known logarithmic Sobolev inequality and whose solution, unique up to spatial shifts, is a perfect parabola. In the present paper we show that those potentials whose shape (after appropriate vertical shifting and spatial rescaling) is away from that parabola contribute only negligibly to the total mass. The topology used is the strong L¹-topology on compacts for the exponentials of the potential. In the course of the proof, we show that any sequence of approximate minimisers of the above variational formula approaches some spatial shift of the minimiser, the parabola.

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