Quenched decay of correlations for slowly mixing systems
journal contributionposted on 15.11.2019, 10:48 by Wael BahsounWael Bahsoun, Christopher Bose, Marks Ruziboev
We study random towers that are suitable to analyse the statistics of slowly mixing random systems. We obtain upper bounds on the rate of quenched correlation decay in a general setting. We apply our results to the random family of LiveraniSaussol-Vaienti maps with parameters in [α0, α1] ⊂ (0, 1) chosen independently with respect to a distribution ν on [α0, α1] and show that the quenched decay of correlation is governed by the fastest mixing map in the family. In particular, we prove that for every δ > 0, for almost every ω ∈ [α0, α1] Z, the upper bound n 1− 1 α0 +δ holds on the rate of decay of correlation for Holder observables on the fibre over ¨ ω. For three different distributions ν on [α0, α1] (discrete, uniform, quadratic), we also derive sharp asymptotics on the measure of return-time intervals for the quenched dynamics, ranging from n − 1 α0 to (log n) 1 α0 · n − 1 α0 to (log n) 2 α0 · n − 1 α0 respectively.
WB and MR would like to thank The Leverhulme Trust for supporting their research through the research grant RPG-2015-346. CB’s research is supported by a research grant from the National Sciences and Engineering Research Council of Canada.
- Mathematical Sciences