Maximum independent sets on random regular graphs

Abstract

We determine the asymptotics of the independence number of the random d-regular graph for all $${d\geq d_0}$$d≥d0. It is highly concentrated, with constant-order fluctuations around $${n\alpha_*-c_*\log n}$$nα∗-c∗logn for explicit constants $${\alpha_*(d)}$$α∗(d) and $${c_*(d)}$$c∗(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.