We give an accurate asymptotic estimate for the gap of the generator of a particular interacting particle system. The model we consider may be informally described as follows. A certain number of charged particles moves on the segment [1,L] according to a Markovian law. One unitary charge, positive or negative, jumps from a site k to another site k'=k+1 or k'=k-1 at a rate which depends on the charge at site k and at site k'. The total charge of the system is preserved by the dynamics, in...

We study various statistics related to the eigenvalues and eigenfunctions of random Hamiltonians in the localized regime. Consider a random Hamiltonian at an energy $E$ in the localized phase. Assume the density of states function is not too flat near $E$. Restrict it to some large cube $\Lambda$. Consider now $I_{\Lambda }$, a small energy interval centered at $E$ that asymptotically contains infintely many eigenvalues when the volume of the cube $\Lambda$ grows to infinity. We prove that, with probability one in the large volume...

We discuss spectral and scattering theory of the discrete laplacian limited to a half-space. The interesting properties of such operators stem from the imposed boundary condition and are related to certain phenomena in surface physics.

Let $h$ be a three times partially differentiable function on ${ℝ}^n$, let $X=(X_1,...,X_n)$ be a collection of real-valued random variables and let $Z=(Z_1,...,Z_n)$ be a multivariate Gaussian vector. In this article, we develop Stein’s method to give error bounds on the difference $𝔼h\left(X\right)-𝔼h\left(Z\right)$ in cases where the coordinates of $X$ are not necessarily independent, focusing on the high dimensional case $n\to \infty$. In order to express the dependency structure we use Stein couplings, which allows for a broad range of applications, such as classic occupancy,...

We consider a random walk in a random potential, which models a situation of a random polymer and we study the annealed and quenched costs to perform long crossings from a point to a hyperplane. These costs are measured by the so called Lyapounov norms. We identify situations where the point-to-hyperplane annealed and quenched Lyapounov norms are different. We also prove that in these cases the polymer path exhibits localization.

We study trajectories of $d$-dimensional Brownian Motion in Poissonian potential up to the hitting time of a distant hyper-plane. Our Poissonian potential $V$ is constructed from a field of traps whose centers location is given by a Poisson Point Process and whose radii are IID distributed with a common distribution that has unbounded support; it has the particularity of having long-range correlation. We focus on the case where the law of the trap radii $\nu$ has power-law decay and prove that superdiffusivity...

This paper continues a study on trajectories of Brownian Motion in a field of soft trap whose radius distribution is unbounded. We show here that for both point-to-point and point-to-plane model the volume exponent (the exponent associated to transversal fluctuation of the trajectories) $\xi$ is strictly less than $1$ and give an explicit upper bound that depends on the parameters of the problem. In some specific cases, this upper bound matches the lower bound proved in the first part of this work and...