M. Biskup et W. Konig, Screening effect due to heavy lower tails in one-dimensional parabolic Anderson model, J STAT PHYS, 102(5-6), 2001, pp. 1253-1270
We consider the large-time behavior of the solution u: [0, proportional to
) x Z --> [0, proportional to) to the parabolic Anderson problem partial de
rivative ,u = kappa Deltau + xiu with initial data u(0, proportional to) =
1 and non-positive finite i.i.d. potentials (xi (z))(z is an element ofZ).
Unlike in dimensions d greater than or equal to 2, the almost-sure decay ra
te u(t,0) as t --> proportional to is not determined solely by the upper ta
ils of xi (0); too heavy lower tails of xi (0) accelerate the decay. The in
terpretation is that sites x with large negative xi (x) hamper the mass flo
w and hence screen off the influence of more favorable regions of the poten
tial. The phenomenon is unique to d = 1. The result answers an open questio
n from our previous study [BK00] of this model in general dimension.