Statistics Papers

Document Type

Journal Article

Date of this Version


Publication Source

Electronic Journal of Probability



Start Page

paper no. 68




Consider a Markov process ωt at stationarity and some event C (a subset of the state-space of the process). A natural measure of correlations in the process is the pairwise correlation P[ω0,ωtC]−P[ω0C]2. A second natural measure is the probability of the continual occurrence event {ωsC,∀s∈[0,t]}. We show that for reversible Markov chains, and any event C, pairwise decorrelation of the event C implies a decay of the probability of the continual occurrence event {ωsCs∈[0,t]} as t→∞. We provide examples showing that our results are often sharp.

Our main applications are to dynamical critical percolation. Let C be the left-right crossing event of a large box, and let us scale time so that the expected number of changes to C is order 1 in unit time. We show that the continual connection event has superpolynomial decay. Furthermore, on the infinite lattice without any time scaling, the first exceptional time with an infinite cluster appears with an exponential tail.


This work is licensed under a Creative Commons Attribution 3.0 License.


decorrelation, hidden Markov chains, hitting and exit times, spectral gap, dynamical percolation, exceptional times, scaling limits



Date Posted: 27 November 2017

This document has been peer reviewed.