We show that on an arbitrary, finitely generated, non-virtually-solvable linear group, any two independent random walks will eventually generate a free subgroup. In fact, this will hold for an exponential number of independent random walks.
"Random subgroups of linear groups are free." Duke Math. J. 160 (1) 117 - 173, 1 October 2011. https://doi.org/10.1215/00127094-1443493