Non-trivial linear bounds for a random walk driven by a simple symmetric exclusion process

Non-trivial linear bounds are obtained for the displacement of a random walk in a dynamic random environment given by a one-dimensional simple symmetric exclusion process in equilibrium. The proof uses an adaptation of multiscale renormalization methods of Kesten and Sidoravicius.Comment: 20 pages, 3 figure

Brownian motion in attenuated or renormalized inverse-square Poisson potential

We consider the parabolic Anderson problem with random potentials having inverse-square singularities around the points of a standard Poisson point process in $\mathbb{R}^d$, $d \geq 3$. The potentials we consider are obtained via superposition of translations over the points of the Poisson point process of a kernel $\mathfrak{K}$ behaving as $\mathfrak{K}(x) \approx \theta |x|^{-2}$ near the origin, where $\theta \in (0,(d-2)^2/16]$. In order to make sense of the corresponding path integrals, we require the potential to be either attenuated (meaning that $\mathfrak{K}$ is integrable at infinity) or, when $d=3$, renormalized, as introduced by Chen and Kulik in [8]. Our main results include existence and large-time asymptotics of non-negative solutions via Feynman-Kac representation. In particular, we settle for the renormalized potential in $d=3$ the problem with critical parameter $\theta = 1/16$, left open by Chen and Rosinski in [arXiv:1103.5717].Comment: 36 page

Zero-one law for directional transience of one-dimensional random walks in dynamic random environments

We prove the trichotomy between transience to the right, transience to the left and recurrence of one-dimensional nearest-neighbour random walks in dynamic random environments under fairly general assumptions, namely: stationarity under space-time translations, ergodicity under spatial translations, and a mild ellipticity condition. In particular, the result applies to general uniformly elliptic models and also to a large class of non-uniformly elliptic cases that are i.i.d. in space and Markovian in time. An immediate consequence is the recurrence of models that are symmetric with respect to reflection through the origin.Comment: 14 pages, 1 figure. Added Corollary 2.

Brownian motion in attenuated or renormalized inverse-square Poisson potential

We consider the parabolic Anderson problem with random potentials having inverse-square singularities around the points of a standard Poisson point process in ℝ d, d ≥3. The potentials we consider are obtained via superposition of translations over the points of the Poisson point process of a kernel behaving as (x)≈ Θ x -2 near the origin, where Θ ∈(0,(d-2)2/16]. In order to make sense of the corresponding path integrals, we require the potential to be either attenuated (meaning that is integrable at infinity) or, when d=3, renormalized, as introduced by Chen and Kulik in [8]. Our main results include existence and large-time asymptotics of non-negative solutions via Feynman-Kac representation. In particular, we settle for the renormalized potential in d=3 the problem with critical parameter Θ = 1/16, left open by Chen and Rosinski in [9]

Brownian motion in attenuated or renormalized inverse-square Poisson potential

We consider the parabolic Anderson problem with random potentials having inverse-square singularities around the points of a standard Poisson point process in ℝ d, d ≥3. The potentials we consider are obtained via superposition of translations over the points of the Poisson point process of a kernel 픎 behaving as 픎 (x)≈ Θ x -2 near the origin, where Θ ∈(0,(d-2)2/16]. In order to make sense of the corresponding path integrals, we require the potential to be either attenuated (meaning that 픎 is integrable at infinity) or, when d=3, renormalized, as introduced by Chen and Kulik in [8]. Our main results include existence and large-time asymptotics of non-negative solutions via Feynman-Kac representation. In particular, we settle for the renormalized potential in d=3 the problem with critical parameter Θ = 1/16, left open by Chen and Rosinski in [9]

The quenched limiting distributions of a charged-polymer model

The limit distributions of the charged-polymer Hamiltonian of Kantor and Kardar [Bernoulli case] and Derrida, Griffiths and Higgs [Gaussian case] are considered. Two sources of randomness enter in the definition: a random field $q= (q_i)_{i\geq 1}$ of i.i.d. random variables (called random charges) and a random walk $S = (S_n)_{n \in \mathbb{N}}$ evolving in $\mathbb{Z}^d$, independent of the charges. The energy or Hamiltonian $K = (K_n)_{n \geq 2}$ is then defined as $$K_n := \sum_{1\leq i < j\leq n} q_i q_j {\bf 1}_{\{S_i=S_j\}}.$$ The law of $K$ under the joint law of $q$ and $S$ is called "annealed", and the conditional law given $q$ is called "quenched". Recently, strong approximations under the annealed law were proved for $K$. In this paper we consider the limit distributions of $K$ under the quenched law.Comment: 23 pages. v2->v3: Title corrected, some improvements for readability adde

A quenched functional central limit theorem for random walks in random environments under (T)[gamma]

We prove a quenched central limit theorem for random walks in i.i.d. weakly elliptic random environments in the ballistic regime. Such theorems have been proved recently under the assumption of large finite moments for the regeneration times. In this paper, we relax these moment assumptions under Sznitman's (T)γ ballisticity condition, which allows the inclusion of new non-uniformly elliptic examples such as Dirichlet random environments

A Quenched Functional Central Limit Theorem for Random Walks in Random Environments under (T)γ(T)_\gamma

International audienceWe prove a quenched central limit theorem for random walks in i.i.d. weakly elliptic random environments in the ballistic regime. Such theorems have been proved recently by Rassoul-Agha and Seppäläinen in [10] and Berger and Zeitouni in [2] under the assumption of large finite moments for the regeneration time. In this paper, with the extra $(T)_{\gamma}$ condition of Sznitman we reduce the moment condition to ${\Bbb E}(\tau^2(\ln \tau)^{1+m})1+1/\gamma$, which allows the inclusion of new non-uniformly elliptic examples such as Dirichlet random environments

A quenched functional central limit theorem for random walks in random environments under (T)gamma(T)_gamma

We prove a quenched central limit theorem for random walks in i.i.d. weakly elliptic random environments in the ballistic regime. Such theorems have been proved recently under the assumption of large finite moments for the regeneration times. In this paper, we relax these moment assumptions under Sznitman's (T)γ ballisticity condition, which allows the inclusion of new non-uniformly elliptic examples such as Dirichlet random environments

Mass concentration and aging in the parabolic Anderson model with doubly-exponential tails

We study the solutions to the Cauchy problem on the with random potential and localised initial data. Here we consider the random Schr?dinger operator, i.e., the Laplace operator with random field, whose upper tails are doubly exponentially distributed in our case. We prove that, for large times and with large probability, a majority of the total mass of the solution resides in a bounded neighborhood of a site that achieves an optimal compromise between the local Dirichlet eigenvalue of the Anderson Hamiltonian and the distance to the origin. The processes of mass concentration and the rescaled total mass are shown to converge in distribution under suitable scaling of space and time. Aging results are also established. The proof uses the characterization of eigenvalue order statistics for the random Schr"odinger operator in large sets recently proved by the first two authors