-We establish a Mermin-Wagner type theorem for Gibbs states on infinite random Lorentzian triangulation arizing in models

of quantum gravity. Such a triangulation is naturally related to the distribution of a critical Galton-Watson tree, conditional upon

non-extinction. As the vertices of the triangles we place classical spins taking values in a d-dimensional torus, with a group

action of a torus. We analyze a generated quenched Gibbs measure and establish the absense of spontaneous continuous

symmetry-breaking.

We compute analytically the probability distribution function PP(*ε*) of the dissipation field *ε*=(∇*θ*)2 of a passive scalar *θ* advected by a *d*-dimensional random flow, in the limit of large Peclet and Prandtl numbers (Batchelor–Kraichnan regime). The tail of the distribution is a stretched exponential: for *ε*→∞, ln PP(*ε*)∼−(*d*2*ε*)1/3.

We show that beta ensembles in Random Matrix Theory with generic real analytic potential have the asymptotic equipartition property. In addition, we prove a Central Limit Theorem for the density of the eigenvalues of these ensembles.

We consider a simple class of fast-slow partially hyperbolic dynamical systems and show that the (properly rescaled) behaviour of the slow variable is very close to a Freidlin-Wentzell type random system for times that are rather long, but much shorter than the metastability scale. Also, we show the possibility of a "sink" with all the Lyapunov exponents positive, a phenomenon that turns out to be related to the lack of absolutely continuity of the central foliation.

Statistical properties of infinite products of random isotropically distributed matrices are investigated. Both for continuous processes with finite correlation time and discrete sequences of independent matrices, a formalism that allows to calculate easily the Lyapunov spectrum and generalized Lyapunov exponents is developed. This problem is of interest to probability theory, statistical characteristics of matrix T-exponentials are also needed for turbulent transport problems, dynamical chaos and other parts of statistical physics.

Statisticalpropertiesofinﬁniteproductsofrandomisotropicallydistributedmatrices are investigated. Both for continuous processes with ﬁnite correlation time and discrete sequencesofindependentmatrices,aformalismthatallowstocalculateeasilytheLyapunov spectrum and generalized Lyapunov exponents is developed. This problem is of interest to probability theory, statistical characteristics of matrix T-exponentials are also needed for turbulent transport problems, dynamical chaos and other parts of statistical physics.

We consider a finite region of a d-dimensional lattice of nonlinear Hamiltonian rotators, where neighbouring rotators have opposite (alternated) spins and are coupled by a small potential of order $\epsilon^a,\, a\geq 1/2$. We weakly stochastically perturb the system in such a way that each rotator interacts with its own stochastic thermostat with a force of order $\epsilon$. Then we introduce action-angle variables for the system of uncoupled rotators ($\epsilon=0$) and note that the sum of actions over all nodes is conserved by the purely Hamiltonian dynamics of the system with $\epsilon>0$. We investigate the limiting (as $\epsilon \rightarrow 0$) dynamics of actions for solutions of the $\epsilon$-perturbed system on time intervals of order $\epsilon^{-1}$. It turns out that the limiting dynamics is governed by a certain stochastic equation for the vector of actions, which we call the transport equation. This equation has a completely non-Hamiltonian nature. This is a consequence of the fact that the system of rotators with alternated spins do not have resonances of the first order. The $\epsilon$-perturbed system has a unique stationary measure $\wid \mu^\epsilon$ and is mixing. Any limiting point of the family $\{\wid \mu^\epsilon\}$ of stationary measures as $\eps\ra 0$ is an invariant measure of the system of uncoupled integrable rotators. There are plenty of such measures. However, it turns out that only one of them describes the limiting dynamics of the $\epsilon$-perturbed system: we prove that a limiting point of $\{\wid\mu^\epsilon\}$ is unique, its projection to the space of actions is the unique stationary measure of the transport equation, which turns out to be mixing, and its projection to the space of angles is the normalized Lebesque measure on the torus $\mathbb{T}^N$. The results and convergences, which concern the behaviour of actions on long time intervals, are uniform in the number $N$ of rotators. Those, concerning the stationary measures, are uniform in $N$ in some natural case.

A functional method for calculating averages of the time-ordered exponential of a continuous isotropic random N×NN×N matrix process is presented. The process is not assumed to be Gaussian. In particular, the Lyapunov exponents and higher correlation functions of the T-exponent are derived from the statistical properties of the process. The approach may be of use in a wide range of physical problems. For example, in theory of turbulence the account of non-gaussian statistics is very important since the non-Gaussian behavior is responsible for the time asymmetry of the energy flow.

The set of moments and the distribution function of the one-electron current in a one-dimensional disordered ring with arbitrary magnetic flux are calculated.

We present a functional integration method for the averaging of continuous products*P**t* of*N×N* random matrices. As an application, we compute exactly the statistics of the Lyapunov spectrum of*P**t*. This problem is relevant to the study of the statistical properties of various disordered physical systems, and specifically to the computation of the multipoint correlators of a passive scalar advected by a random velocity field. Apart from these applications, our method provides a general setting for computing statistical properties of linear evolutionary systems subjected to a white-noise force field.