The sharp random-graph threshold conjecture for friends-and-strangers graphs

Let $X,Y\sim\mathcal{G}(n,p)$ denote independent Erdős–Rényi random graphs on $n$ vertices, and let $X,Y\sim\mathcal{G}(K_{n,n},p)$ denote independent random bipartite graphs with parts of size $n$. Let $\operatorname{\mathsf{FS}}(X,Y)$ be their friends-and-strangers graph, and let $\omega(\cdot)$ denote a function tending to infinity. The sharp random-graph threshold conjecture. For

$$p=\omega\left(\frac{\log^{1/2} n}{n^{1/2}}\right)$$

and sufficiently large $n$, both of the following hold: if $X,Y\sim\mathcal{G}(n,p)$, then $\operatorname{\mathsf{FS}}(X,Y)$ is connected with high probability; and if $X,Y\sim\mathcal{G}(K_{n,n},p)$, then $\operatorname{\mathsf{FS}}(X,Y)$ has exactly two connected components with high probability. This conjecture would determine the threshold probabilities precisely; the paper establishes the relevant threshold dependence only up to a factor of $n^{o(1)}$, so the conjecture remains open.