Mass concentration and aging in the parabolic Anderson model with
doublyexponential tails
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by
Marek Biskup, Wolfgang Koenig, Renato Soares dos Santos
(2016)
Abstract
We study the solutions u=u(x,t) to the Cauchy problem on
Z^d×(0,∞) for the parabolic equation ∂_t u=Δ u+ξ u
with initial data u(x,0)=1_{0}(x). Here Δ is the discrete
Laplacian on Z^d and ξ=(ξ(z))_z∈ Z^d is an i.i.d.random field with doublyexponential upper tails. We prove that, for large t
and with large probability, a majority of the total mass U(t):=∑_x u(x,t)
of the solution resides in a bounded neighborhood of a site Z_t that achieves
an optimal compromise between the local Dirichlet eigenvalue of the Anderson
Hamiltonian Δ+ξ and the distance to the origin. The processes
t Z_t and t 1t U(t) are shown to converge in
distribution under suitable scaling of space and time. Aging results for Z_t,
as well as for the solution to the parabolic problem, are also established. The
proof uses the characterization of eigenvalue order statistics for Δ+ξ
in large sets recently proved by the first two authors.
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Stage
submitted
Date 20160904
Version
v1
Language
en
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1609.00989v1
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