Large complete minors in random subgraphs

Let G be a graph of minimum degree at least k and let Gp be the random subgraph of G obtained by keeping each edge independently with probability p. We are interested in the size of the largest complete minor that Gp contains when p = (1 + ϵ)/k with ϵ > 0. We show that with high probability Gp contains a complete minor of order <![CDATA[ \tilde{\Omega}(\sqrt{k})[]>, where the hides a polylogarithmic factor. Furthermore, in the case where the order of G is also bounded above by a constant multiple of k, we show that this polylogarithmic term can be removed, giving a tight bound.