Colouring complete bipartite graphs from random lists

Let $K_{n,n}$ be the complete bipartite graph with $n$ vertices in each side. For each vertex draw uniformly at random a list of size $k$ from a base set $S$ of size $s=s(n)$. In this paper we estimate the asymptotic probability of the existence of a proper colouring from the random lists for all fixed values of $k$ and growing $n$. We show that this property exhibits a sharp threshold for $k\geq 2$ and the location of the threshold is precisely $s(n)=2n$ for $k=2$, and approximately $s(n)=\frac{n}{2^{k-1}\ln 2}$ for $k\geq 3$.