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 [α 0 , α 1 ] ⊂ (0, 1) chosen independently with respect to a distribution ν on [α 0 , α 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 δ > 0, for almost every ω ∈ [α 0 , α 1 ] Z , the upper bound n 1− 1 α 0 +δ holds on Date: January 30, 2018. 1991 Mathematics Subject Classification. Primary 37A05, 37E05. Key words and phrases. Random dynamical systems, slowly mixing systems, quenched decay of correlations.