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Contents § 1. Introduction § 2. Proof of Theorem 1 § 3. Anderson localization of the operator Hα § 4. Exponential decay of the approximate eigenfunctions and other asymptotic relations Bibliography

The joint limiting probability distribution is studied for the two-dimensional random walk with topological constraints, omega(2ns), on Z(2) lattice, where 2n is its total length and (0/=s/=1). The expression for the density of finite-dimensional limiting probability distribution pi{xi(n)(s(1)),xi(n)(s(2)),.,xi(n)(s(r))} where xi(n)(s) = omega(2ns)/n(1/4), is described.

We consider the dynamical system (\(\mathfrak{X}\),μ,Tt) where (\(\mathfrak{X}\),μ) is the Gibbs ensemble at some fixed temperature and density for a semi-infinite one-dimensional ideal gas of point particles. The first particle has massM, all the other particles massm

The depinning phase transition in the discrete Frenkel-Kontorova model is considered. It is found that there exists a set of irrational periodicities such that the chain in pinned at any periodic potential strenght. Every Cantorus describin the pinned phase is shown to belong to the stochastic layer Etude de la transition d'ancrage dans le modele discret de Frenkel-Kontorova: il existe un ensemble de periodes irrationnelles ou la chaine est ancree pour toute valeur du potentiel periodique. Chaque ensemble de Cantor decrivant la phase ancree appartient a la couche stochastique

We consider the Boltzmann-Grad limit for the Lorentz, or wind-tree, model. We prove that if ω is a fixed configuration of scatterer centers belonging to a set of full measure with respect to the Poisson distribution with parameter λ>0, then the evolution of an initial a.c. particle density tends in the Boltzmann-Grad limit to the solution of the Boltzmann equation for the model. As an intermediate step we prove that the process of the free path lengths and impact parameters induced by the Lebesgue measure on a small region tends to a limiting independent process.

We consider a random walk on thed-dimensional lattice ℤ d where the transition probabilitiesp(x,y) are symmetric,p(x,y)=p(y,x), different from zero only ify−x belongs to a finite symmetric set including the origin and are random. We prove the convergence of the finite-dimensional probability distributions of normalized random paths to the finite-dimensional probability distributions of a Wiener process and find our an explicit expression for the diffusion matrix.