Maximum independent sets on random regular graphs

Abstract

We determine the asymptotics of the independence number of the random d-regular graph for all $d\ge d_0$. It is highly concentrated, with constant-order fluctuations around $n{\mathit{\alpha }}_{\ast }-c_{\ast }logn$ for explicit constants ${\mathit{\alpha }}_{\ast }(d)$ and $c_{\ast }(d)$. Our proof rigorously confirms the one-step replica symmetry breaking heuristics for this problem, and we believe the techniques will be more broadly applicable to the study of other combinatorial properties of random graphs.