No abstract

We prove a local limit theorem for the probability of a site to be connected by disjoint paths to three points in subcritical Bernoulli percolation on Z d , d ≥ 2 in the limit where their distances tend to infinity.

We prove Ornstein-Zernike behaviour in every direction for finite connection functions of the random cluster model on Z d , d ≥ 3, for q ≥ 1, when occupation probabilities of the bonds are close to 1. Moreover, we prove that equi-decay surfaces are locally analytic, strictly convex, with positive Gaussian curvature.

Lie-Poisson structure of the Lorenz'63 system gives a physical insight on its dynamical and statistical behavior considering the evolution of the associated Casimir functions. We study the invariant density and other recurrence features of a Markov expanding Lorenz-like map of the interval arising in the analysis of the predictability of the extreme values reached by particular physical observables evolving in time under the Lorenz'63 dynamics with the classical set of parameters. Moreover, we prove the statistical stability of such an invariant measure. This will allow us to further characterize the SRB measure of the system.

We review results of two previous papers on the asymptotic behavior of finite connection probabilities in three or more dimensions for Bernoulli percolation and the Fortuin-Kasteleyn random-cluster model. In the introduction, we prove a multidimensional renewal theorem that is needed for these results and previous results on Ornstein-Zernike behavior; the proof is significantly simpler than that originally derived by Doney (1966) and those of other subsequent works on this subject. Communicated by Raffaele Esposito. Campanino and Gianfelice are partially supported by the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) and thank the referee for useful suggestions.