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We consider the parabolic Anderson problem $partial_t u=kappaDelta u+xi u$ on $(0,infty)times Z^d$ with random i.i.d. potential $xi=(xi(z))_{zinZ^d}$ and the initial condition $u(0,cdot)equiv1$. Our main assumption is that $esssupxi(0)=0$. Depending on the thickness of the distribution $prob(xi(0)incdot)$ close to its essential supremum, we identify both the asymptotics of the moments of $u(t,0)$ and the almost-sure asymptotics of $u(t,0)$ as $ttoinfty$ in terms of variational problems. As a by-product, we establish Lifshitz tails for the random Schrodinger operator $-kappaDelta-xi$ at the bottom of its spectrum. In our class of $xi$ distributions, the Lifshitz exponent ranges from $d/2$ to $infty$; the power law is typically accompanied by lower-order corrections.
We consider the large-time behavior of the solution $ucolon [0,infty)timesZto[0,infty)$ to the parabolic Anderson problem $partial_t u=kappaDelta u+xi u$ with initial data $u(0,cdot)=1$ and non-positive finite i.i.d. potentials $(xi(z))_{zinZ}$. Unli
We consider a one-dimensional continuum Anderson model where the potential decays in average like $|x|^{-alpha}$, $alpha>0$. We show dynamical localization for $0<alpha<frac12$ and provide control on the decay of the eigenfunctions.
We study the solutions $u=u(x,t)$ to the Cauchy problem on $mathbb Z^dtimes(0,infty)$ for the parabolic equation $partial_t u=Delta u+xi u$ with initial data $u(x,0)=1_{{0}}(x)$. Here $Delta$ is the discrete Laplacian on $mathbb Z^d$ and $xi=(xi(z))_
We consider a one-dimensional Anderson model where the potential decays in average like $n^{-alpha}$, $alpha>0$. This simple model is known to display a rich phase diagram with different kinds of spectrum arising as the decay rate $alpha$ varies. W
This is a short review of the two papers on the $x$-space asymptotics of the critical two-point function $G_{p_c}(x)$ for the long-range models of self-avoiding walk, percolation and the Ising model on $mathbb{Z}^d$, defined by the translation-invari