Screening effect due to heavy lower tails in one-dimensional parabolic Anderson model

We consider the large-time behavior of the solution $u\colon [0,\infty)\times\Z\to[0,\infty)$totheparabolicAndersonproblem$\partial_t u=\kappa\Delta u+\xi u$withinitialdata$u(0,\cdot)=1$ and non-positive finite i.i.d. potentials $(\xi (z){)}_{z\in \mathbb{Z}}$. Unlike in dimensions $d\ge 2$, the almost-sure decay rate of $u(t,0)$ as $t\to \mathrm{\infty }$ is not determined solely by the upper tails of $\xi (0)$; too heavy lower tails of $\xi (0)$ accelerate the decay. The interpretation is that sites $x$ with large negative $\xi (x)$ hamper the mass flow and hence screen off the influence of more favorable regions of the potential. The phenomenon is unique to $d=1$. The result answers an open question from our previous study \cite{BK00} of this model in general dimension.