### A central limit theorem for weighted averages of spins in the high temperature region of the Sherrington-Kirkpatrick model.

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We study continuous Anderson Hamiltonians with non-degenerate single site probability distribution of bounded support, without any regularity condition on the single site probability distribution. We prove the existence of a strong form of localization at the bottom of the spectrum, which includes Anderson localization (pure point spectrum with exponentially decaying eigenfunctions) with finite multiplicity of eigenvalues, dynamical localization (no spreading of wave packets under the time evolution),...

In this article the linear Boltzmann equation is derived for a particle interacting with a Gaussian random field, in the weak coupling limit, with renewal in time of the random field. The initial data can be chosen arbitrarily. The proof is geometric and involves coherent states and semi-classical calculus.

This article is dedicated to localization of the principal eigenvalue (PE) of the Stokes operator acting on solenoidal vector fields that vanish outside a large random domain modeling the pore space in a cubic block of porous material with disordered micro-structure. Its main result is an asymptotically deterministic lower bound for the PE of the sum of a low compressibility approximation to the Stokes operator and a small scaled random potential term, which is applied to produce a similar bound...

We consider the continuous time, one-dimensional random walk in random environment in Sinai’s regime. We show that the probability for the particle to be, at time $t$ and in a typical environment, at a distance larger than ${t}^{a}$ ($0\<a\<1$) from its initial position, is $exp\{-\mathrm{Const}\xb7{t}^{a}/[(1-a)lnt](1+o\left(1\right))\}$.

We consider the continuous time, one-dimensional random walk in random environment in Sinai's regime. We show that the probability for the particle to be, at time t and in a typical environment, at a distance larger than ta (0<a<1) from its initial position, is exp{-Const ⋅ ta/[(1 - a)lnt](1 + o(1))}.

We study the discrete Schrödinger operator $H$ in ${\mathbf{Z}}^{d}$ with the surface quasi periodic potential $V\left(x\right)=g\delta \left({x}_{1}\right)tan\pi (\alpha \xb7{x}_{2}+\omega )$, where $x=({x}_{1},{x}_{2}),\phantom{\rule{4pt}{0ex}}{x}_{1}\in {\mathbf{Z}}^{{d}_{1}},\phantom{\rule{4pt}{0ex}}{x}_{2}\in {\mathbf{Z}}^{{d}_{2}},\phantom{\rule{4pt}{0ex}}\alpha \in {\mathbf{R}}^{{d}_{2}},\phantom{\rule{4pt}{0ex}}\omega \in [0,1)$. We first discuss a proof of the pure absolute continuity of the spectrum of $H$ on the interval $[-d,d]$ (the spectrum of the discrete laplacian) in the case where the components of $\alpha $ are rationally independent. Then we show that in this case the generalized eigenfunctions have the form of the “volume” waves, i.e. of the sum of the incident plane wave and reflected from the hyper-plane ${\mathbf{Z}}^{{d}_{1}}$ waves, the form...

We study a random walk pinning model, where conditioned on a simple random walk Y on ℤd acting as a random medium, the path measure of a second independent simple random walk X up to time t is Gibbs transformed with hamiltonian −Lt(X, Y), where Lt(X, Y) is the collision local time between X and Y up to time t. This model arises naturally in various contexts, including the study of the parabolic Anderson model with moving catalysts, the parabolic Anderson model with brownian noise, and the directed...

A new class of CED systems, providing insight into behaviour of physical disordered materials, is introduced. It includes systems in which the conditionally exponential decay property can be attached to each entity. A limit theorem for the normalized minimum of a CED system is proved. Employing different stable schemes the universal characteristics of the behaviour of such systems are derived.

We prove a number of results concerning the large $N$ asymptotics of the free energy of a random matrix model with a polynomial potential. Our approach is based on a deformation of potential and on the use of the underlying integrable structures of the matrix model. The main results include the existence of a full asymptotic expansion in even powers of $N$ of the recurrence coefficients of the related orthogonal polynomials for a one-cut regular potential and the double scaling asymptotics of the free...