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Register a new userScreening effect due to heavy lower tails in one-dimensional parabolic Anderson model
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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.
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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 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.
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 consider the system of particles with equal charges and nearest neighbour Coulomb interaction on the interval. We study local properties of this system, in particular the distribution of distances between neighbouring charges. For zero temperature case there is sufficiently complete picture and we give a short review. For Gibbs distribution the situation is more difficult and we present two related results.
The hard disk model is a 2D Gibbsian process of particles interacting via pure hard core repulsion. At high particle density the model is believed to show orientational order, however, it is known not to exhibit positional order. Here we investigate to what extent particle positions may fluctuate. We consider a finite volume version of the model in a box of dimensions $2n times 2n$ with arbitrary boundary configuration,and we show that the mean square displacement of particles near the center of the box is bounded from below by $c log n$. The result generalizes to a large class of models with fairly arbitrary interaction.
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