2005/07/31 by Jürgen Gärtner, Wolfgang König, Stanislav Molchanov · 2 citations
Mathematics · #Markov Chains and Monte Carlo Methods #Random Matrices and Applications #Spectral Theory in Mathematical Physics #math.PR #msc:35B40 #msc:60F10 #msc:60H25 #msc:82C44
paper · pdf · doi:10.1214/009117906000000764
published as Annals of Probability 2007, Vol. 35, No. 2, 439-499 · Published at http://dx.doi.org/10.1214/009117906000000764 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2007/03/01 · arxiv created 2007/07/25 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/31
We consider the parabolic Anderson problem ∂tu=Δu+ξ(x)u on ℝ+×ℤd with localized initial condition u(0,x)=δ0(x) and random i.i.d. potential ξ. Under the assumption that the distribution of ξ(0) has a double-exponential, or slightly heavier, tail, we prove the following geometric characterization of intermittency: with probability one, as t→∞, the overwhelming contribution to the total mass ∑xu(t,x) comes from a slowly increasing number of ``islands'' which are located far from each other. These ``islands'' are local regions of those high exceedances of the field ξ in a box of side length 2tlog2t for which the (local) principal Dirichlet eigenvalue of the random operator Δ+ξ is close to the top of the spectrum in the box. We also prove that the shape of ξ in these regions is nonrandom and that u(t,⋅) is close to the corresponding positive eigenfunction. This is the geometric picture suggested by localization theory for the Anderson Hamiltonian.