The evolution of random graphs on surfaces

Chris Dowden, Mihyun Kang, Philipp Sprüssel

For integers $g,m\ge 0$ and $n>0$, let $S_g(n,m)$ denote the graph taken uniformly at random from the set of all graphs on $\{1,2,\dots ,n\}$ with exactly $m=m(n)$ edges and…