Before revision:  As an application, } we show how to leverage down-up random walks to approximately sample  spanning trees in a graph with n edges in time O(n\log^2 n) , improving on the almost-linear time algorithm by Schild [Sch18].
After revision:  As an application, } we show how to leverage down-up random walks to approximately sample spanning trees in a graph with n edges in time O(n\log^2 n) . The best known result for sampling random forest was a FPAUS with high polynomial runtime recently found by \mbox{%DIFAUXCMD ALOV19, CGM19 on the almost-linear time algorithm by Schild [Sch18].