We discuss various properties of Probabilistic Cellular Automata, such as the structure of the set of stationary measures and multiplicity of stationary measures (or phase transition) for reversible models.

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,...

Mathematics Subject Classification: 26A33, 76M35, 82B31A stochastic solution is constructed for a fractional generalization of the KPP (Kolmogorov, Petrovskii, Piskunov) equation. The solution uses a fractional generalization of the branching exponential process and propagation processes which are spectral integrals of Levy processes.

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...

By using a supersymmetric gaussian representation, we transform the averaged Green's function for random walks in random potentials into a 2-point correlation function of a corresponding lattice field theory. We study the resulting lattice field theory using the Witten laplacian formulation. We obtain the asymptotics for the directional Lyapunov exponents.

We study an atomistic pair potential-energy E(n)(y) that describes the elastic behavior of two-dimensional crystals with natoms where $y\in {ℝ}^{2\times n}$ characterizes the particle positions. The main focus is the asymptotic analysis of the ground state energy asn tends to infinity. We show in a suitable scaling regime where the energy is essentially quadratic that the energy minimum of E(n) admits an asymptotic expansion involving fractional powers of n: ${\min }_y{E}^{\left(n\right)}\left(y\right)=n{E}_{\mathrm{bulk}}+\sqrt{n}{E}_{\mathrm{surface}}+o\left(\sqrt{n}\right),n\to \infty .$ The bulk energy densityEbulk is given by an explicit expression...

We study an atomistic pair potential-energy E(n)(y) that describes the elastic behavior of two-dimensional crystals with n atoms where $y\in {ℝ}^{2\times n}$ characterizes the particle positions. The main focus is the asymptotic analysis of the ground state energy as n tends to infinity. We show in a suitable scaling regime where the energy is essentially quadratic that the energy minimum of E(n) admits an asymptotic expansion involving fractional powers of n: ${\min }_y{E}^{\left(n\right)}\left(y\right)=n{E}_{\mathrm{bulk}}+\sqrt{n}{E}_{\mathrm{surface}}+o\left(\sqrt{n}\right),n\to \infty .$ The bulk energy density Ebulk is given by an explicit expression...

The Hamiltonian for an extended Hubbard model with phonons as introduced by A. Montorsi and M. Rasetti is considered on a D-dimensional lattice. The symmetries of the model are studied in various cases. It is shown that for a certain choice of the parameters a superconducting $S{U}_q\left(2\right)$ holds as a true quantum symmetry, but only for D=1.

Systems with Coulomb and logarithmic interactions arise in various settings: an instance is the classical Coulomb gas which in some cases happens to be a random matrix ensemble, another is vortices in the Ginzburg-Landau model of superconductivity, where one observes in certain regimes the emergence of densely packed point vortices forming perfect triangular lattice patterns named Abrikosov lattices, a third is the study of Fekete points which arise in approximation theory. In this review, we describe...