  Let $G$ be a graph of minimum degree at least $k$ and let $G_p$ 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 $G_p$
contains when $p = \frac{1+\varepsilon}{k}$ with $\varepsilon >0$. We show that
with high probability $G_p$ contains a complete minor of order
$\tilde{\Omega}(\sqrt{k})$, where the $\sim$ 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.
