Zhongjian WangDepartment of Mathematics, The University of Hong KongXue LuoSchool of Mathematical Sciences, Beihang University (Shahe campus)Stephen S.-T. YauDepartment of Mathematical Sciences, Tsinghua UniversityZhiwen ZhangDepartment of Mathematics, The University of Hong Kong
Numerical Analysis and Scientific ComputingOptimization and ControlProbabilitymathscidoc:2004.25001
We show that complex hypercontractivity gives better constants than real hypercontractivity in comparison inequalities for (low) moments of Rademacher chaoses (homogeneous polynomials on the discrete cube).
We determine the asymptotics of the independence number of the random d-regular graph for all d≥d0. It is highly concentrated, with constant-order fluctuations around nα∗−c∗logn for explicit constants α∗(d) and 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.