Exponential distribution for the occurrence of rare patterns in Gibbsian random fields

We study the distribution of the occurrence of rare patterns in sufficiently mixing Gibbs random fields on the lattice $\mathbb{Z}^d$, $d\geq 2$. A typical example is the high temperature Ising model. This distribution is shown to converge to an exponential law as the size of the pattern diverges. Our analysis not only provides this convergence but also establishes a precise estimate of the distance between the exponential law and the distribution of the occurrence of finite patterns. A similar result holds for the repetition of a rare pattern. We apply these results to the fluctuation properties of occurrence and repetition of patterns: We prove a central limit theorem and a large deviation principle.Comment: To appear in Commun. Math. Phy

Sharp error terms for return time statistics under mixing conditions

We describe the statistics of repetition times of a string of symbols in a stochastic process. Denote by T(A) the time elapsed until the process spells the finite string A and by S(A) the number of consecutive repetitions of A. We prove that, if the length of the string grows unbondedly, (1) the distribution of T(A), when the process starts with A, is well aproximated by a certain mixture of the point measure at the origin and an exponential law, and (2) S(A) is approximately geometrically distributed. We provide sharp error terms for each of these approximations. The errors we obtain are point-wise and allow to get also approximations for all the moments of T(A) and S(A). To obtain (1) we assume that the process is phi-mixing while to obtain (2) we assume the convergence of certain contidional probabilities

Exponential distribution for the occurrence of rare patterns in Gibbsian random fields

Abstract: We study the distribution of the occurrence of rare patterns in sufficiently mixing Gibbs random fields on the lattice A typical example is the high temperature Ising model. This distribution is shown to converge to an exponential law as the size of the pattern diverges. Our analysis not only provides this convergence but also establishes a precise estimate of the distance between the exponential law and the distribution of the occurrence of finite patterns. A similar result holds for the repetition of a rare pattern. We apply these results to the fluctuation properties of occurrence and repetition of patterns: We prove a central limit theorem and a large deviation principle

The optimal sink and the best source in a Markov chain

It is well known that the distributions of hitting times in Markov chains are quite irregular, unless the limit as time tends to infinity is considered. We show that nevertheless for a typical finite irreducible Markov chain and for nondegenerate initial distributions the tails of the distributions of the hitting times for the states of a Markov chain can be ordered, i.e., they do not overlap after a certain finite moment of time. If one considers instead each state of a Markov chain as a source rather than a sink then again the states can generically be ordered according to their efficiency. The mechanisms underlying these two orderings are essentially different though.Comment: 12 pages, 1 figur

Return-time Lq -spectrum for equilibrium states with potentials of summable variation

Let $(X_k)_{k\geq 0}$ be a stationary and ergodic process with joint distribution $\mu$ where the random variables $X_k$ take values in a finite set $\mathcal{A}$. Let $R_n$ be the first time this process repeats its first $n$ symbols of output. It is well-known that $\frac{1}{n}\log R_n$ converges almost surely to the entropy of the process. Refined properties of $R_n$ (large deviations, multifractality, etc) are encoded in the return-time $L^q$-spectrum defined as\EuScript{R}(q)=\lim_n\frac{1}{n}\log\int R_n^q \dd\mu\quad (q\in\R)provided the limit exists.We consider the case where $(X_k)_{k\geq 0}$ is distributed according to the equilibrium state of a potential $\varphi:\mathcal{A}^{\N}\to\R$ with summable variation,and we prove that \EuScript{R}(q)=\begin{cases}P((1-q)\varphi) & \text{for}\;\; q\geq q_\varphi^*\\\sup_\eta \int \varphi \dd\eta & \text{for}\;\; qwhere $P((1-q)\varphi)$ is the topological pressure of $(1-q)\varphi$, the supremum is taken over all shift-invariant measures, and $q_\varphi^*$ is the unique solution of P((1-q)\varphi) =\sup_\eta \int \varphi \dd\eta.Unexpectedly, this spectrum does not coincide with the $L^q$-spectrum of $\mu_\varphi$, which is $P((1-q)\varphi)$,and does not coincide with the waiting-time $L^q$-spectrum in general.In fact, the return-time $L^q$-spectrum coincides with the waiting-time $L^q$-spectrum if and only if the equilibrium state of $\varphi$ is the measure of maximal entropy.As a by-product, we also improve the large deviation asymptotics of $\frac{1}{n}\log R_n$