Mixing and decorrelation in infinite measure: the case of the periodic sinai billiard

We investigate the question of the rate of mixing for observables of a Z d-extension of a probability preserving dynamical system with good spectral properties. We state general mixing results, including expansions of every order. The main part of this article is devoted to the study of mixing rate for smooth observables of the Z 2-periodic Sinai billiard, with different kinds of results depending on whether the horizon is finite or infinite. We establish a first order mixing result when the horizon is infinite. In the finite horizon case, we establish an asymptotic expansion of every order, enabling the study of the mixing rate even for observables with null integrals

The Nagaev-Guivarc'h method via the Keller-Liverani theorem

The Nagaev-Guivarc'h method, via the perturbation operator theorem of Keller and Liverani, has been exploited in recent papers to establish local limit and Berry-Essen type theorems for unbounded functionals of strongly ergodic Markov chains. The main difficulty of this approach is to prove Taylor expansions for the dominating eigenvalue of the Fourier kernels. This paper outlines this method and extends it by proving a multi-dimensional local limit theorem, a first-order Edgeworth expansion, and a multi-dimensional Berry-Esseen type theorem in the sense of Prohorov metric. When applied to uniformly or geometrically ergodic chains and to iterative Lipschitz models, the above cited limit theorems hold under moment conditions similar, or close, to those of the i.i.d. case

Persistence exponent for random walk on directed versions of Z2Z^2

We study the persistence exponent for random walks in random sceneries (RWRS) with integer values and for some special random walks in random environment in $\mathbb Z^2$ including random walks in $\mathbb Z^2$ with random orientations of the horizontal layers.Comment: 19 page

Mixing rate in infinite measure for Z^d-extension, application to the periodic Sinai billiard

We study the rate of mixing of observables of Z^d-extensions of probability preserving dynamical systems. We explain how this question is directly linked to the local limit theorem and establish a rate of mixing for general classes of observables of the Z^2-periodic Sinai billiard. We compare our approach with the induction method

Empirical central limit theorems for ergodic automorphisms of the torus

Let T be an ergodic automorphism of the d-dimensional torus T^d, and f be a continuous function from T^d to R^l. On the probability space T^d equipped with the Lebesgue-Haar measure, we prove the weak convergence of the sequential empirical process of the sequence (f o T^i)_{i \geq 1} under some mild conditions on the modulus of continuity of f. The proofs are based on new limit theorems and new inequalities for non-adapted sequences, and on new estimates of the conditional expectations of f with respect to a natural filtration.Comment: 32 page

Multiplicative ergodicity of Laplace transforms for additive functional of Markov chains

We study properties of the Laplace transforms of non-negative additive functionals of Markov chains. We are namely interested in a multiplicative ergodicity property used in [18] to study bifurcating processes with ancestral dependence. We develop a general approach based on the use of the operator perturbation method. We apply our general results to two examples of Markov chains, including a linear autoregressive model. In these two examples the operator-type assumptions reduce to some expected finite moment conditions on the functional (no exponential moment conditions are assumed in this work)

Dynamical lattice computation of the Isgur-Wise functions τ1/2 and τ3/2

We perform a two-flavor dynamical lattice computation of the Isgur-Wise functions t1/2 and t3/2 at zero recoil in the static limit. We find t1/2(1) = 0.297(26) and t3/2(1) = 0.528(23) fulfilling Uraltsev’s sum rule by around 80%. We also comment on a persistent conflict between theory and experiment regarding semileptonic decays of B mesons into orbitally excited P wave D mesons, the so-called “1/2 versus 3/2 puzzle”, and we discuss the relevance of lattice results in this context

Rate of convergence in the multidimensional central limit theorem for stationary processes. Application to the Knudsen gas and to the Sinai billiard

We show how Rio's method [Probab. Theory Related Fields 104 (1996) 255--282] can be adapted to establish a rate of convergence in ${\frac{1}{\sqrt{n}}}$ in the multidimensional central limit theorem for some stationary processes in the sense of the Kantorovich metric. We give two applications of this general result: in the case of the Knudsen gas and in the case of the Sinai billiard.Comment: Published at http://dx.doi.org/10.1214/105051605000000476 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Relativistic Quantum Scattering of High Energy Fermions in the Presence of Phase Transition

We study the high energy behaviour of fermions hitting a general wall caused by a first-order phase transition. The wall profile is introduced through a general analytic function. The reflection coefficient is computed in the high energy limit and expressed in terms of the poles of the wall profile function. It is shown that the leading singularity gives the high energy behaviour.Comment: 10 page