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Register a new userLong-time tails in the parabolic Anderson model with bounded potential
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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.
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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 $(xi(z))_{zinZ}$. Unlike in dimensions $dge2$, the almost-sure decay rate of $u(t,0)$ as $ttoinfty$ 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.
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 $xi=(xi(z))_{zinmathbb Z^d}$ is an i.i.d. random field with doubly-exponential upper tails. We prove that, for large $t$ and with large probability, a majority of the total mass $U(t):=sum_x u(x,t)$ 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 $Delta+xi$ and the distance to the origin. The processes $tmapsto Z_t$ and $t mapsto tfrac1t log U(t)$ 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 $Delta+xi$ in large sets recently proved by the first two authors.
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. We review an article of Kiselev, Last and Simon where the authors show a.c. spectrum in the super-critical case $alpha>frac12$, a transition from singular continuous to pure point spectrum in the critical case $alpha=frac12$, and dense pure point spectrum in the sub-critical case $alpha<frac12$. We present complete proofs of the cases $alphagefrac12$ and simplify some arguments along the way. We complement the above result by discussing the dynamical aspects of the model. We give a simple argument showing that, despite of the spectral transition, transport occurs for all energies for $alpha=frac12$. Finally, we discuss a theorem of Simon on dynamical localization in the sub-critical region $alpha<frac12$. This implies, in particular, that the spectrum is pure point in this regime.
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-invariant power-law step-distribution/coupling $D(x)propto|x|^{-d-alpha}$ for some $alpha>0$. Let $S_1(x)$ be the random-walk Green function generated by $D$. We have shown that $bullet~~S_1(x)$ changes its asymptotic behavior from Newton ($alpha>2$) to Riesz ($alpha<2$), with log correction at $alpha=2$; $bullet~~G_{p_c}(x)simfrac{A}{p_c}S_1(x)$ as $|x|toinfty$ in dimensions higher than (or equal to, if $alpha=2$) the upper critical dimension $d_c$ (with sufficiently large spread-out parameter $L$). The model-dependent $A$ and $d_c$ exhibit crossover at $alpha=2$. The keys to the proof are (i) detailed analysis on the underlying random walk to derive sharp asymptotics of $S_1$, (ii) bounds on convolutions of power functions (with log corrections, if $alpha=2$) to optimally control the lace-expansion coefficients $pi_p^{(n)}$, and (iii) probabilistic interpretation (valid only when $alphale2$) of the convolution of $D$ and a function $varPi_p$ of the alternating series $sum_{n=0}^infty(-1)^npi_p^{(n)}$. We outline the proof, emphasizing the above key elements for percolation in particular.
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