We determine the asymptotics of the independence number of the random d-regular graph for all . It is highly concentrated, with constant-order fluctuations around for explicit constants and . 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.
Research supported by NSF grant DMS-1313596 (J. D.), Sloan Research Fellowship (A. S.), NDSEG and NSF GRF (N. S.).
Jian Ding. Allan Sly. Nike Sun. "Maximum independent sets on random regular graphs." Acta Math. 217 (2) 263 - 340, 2016. https://doi.org/10.1007/s11511-017-0145-9