# Quenched decay of correlations for slowly mixing systems

## TL;DR

This paper investigates the decay of correlations in slowly mixing random dynamical systems using random towers, providing bounds and asymptotics that depend on the fastest mixing map in the family.

## Contribution

It introduces a general framework for analyzing quenched correlation decay in slowly mixing systems and applies it to Liverani-Saussol-Vaienti maps with various distributions.

## Key findings

- Upper bounds on quenched correlation decay rates.
- Decay governed by the fastest mixing map in the family.
- Sharp asymptotics for return-time intervals for different distributions.

## Abstract

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 Liverani-Saussol-Vaienti maps with parameters in $[\alpha_0,\alpha_1]\subset (0,1)$ chosen independently with respect to a distribution $\nu$ on $[\alpha_0,\alpha_1]$ 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 $\delta >0$, for almost every $\omega \in [\alpha_0,\alpha_1]^\mathbb Z$, the upper bound $n^{1-\frac{1}{\alpha_0}+\delta}$ holds on the rate of decay of correlation for H\"older observables on the fibre over $\omega$. For three different distributions $\nu$ on $[\alpha_0,\alpha_1]$ (discrete, uniform, quadratic), we also derive sharp asymptotics on the measure of return-time intervals for the quenched dynamics, ranging from $n^{-\frac{1}{\alpha_0}}$ to $(\log n)^{\frac{1}{\alpha_0}}\cdot n^{-\frac{1}{\alpha_0}}$ to $(\log n)^{\frac{2}{\alpha_0}}\cdot n^{-\frac{1}{\alpha_0}}$ respectively.