Almost sure rates of mixing for random intermittent maps

We consider a family $\mathcal F$ of maps with two branches and a common neutral fixed point $0$ such that the order of tangency at $0$ belongs to some interval $[\alpha_0, \alpha_1]\subset (0, 1)$. Maps in $\mathcal F$ do not necessarily share a common Markov partition. At each step a member of $\mathcal F$ is chosen independently with respect to the uniform distribution on $[\alpha_0, \alpha_1]$. We show that the construction of the random tower in Bahsoun-Bose-Ruziboev \cite{BBR} with \emph{general return time} can be carried out for random compositions of such maps. Thus their general results are applicable and gives upper bounds for the quenched decay of correlations of form $n^{1-1/\alpha_0+\delta}$ for any $\delta>0$.