Random Laplacian matrices and convex relaxations

The largest eigenvalue of a matrix is always larger or equal than its largest diagonal entry. We show that for a large class of random Laplacian matrices, this bound is essentially tight: the largest eigenvalue is, up to lower order terms, often the size of the largest diagonal entry. Besides being a simple tool to obtain precise estimates on the largest eigenvalue of a large class of random Laplacian matrices, our main result settles a number of open problems related to the tightness of certain convex relaxation-based algorithms. It easily implies the optimality of the semidefinite relaxation approaches to problems such as mathbbZ2 Synchronization and Stochastic Block Model recovery. Interestingly, this result readily implies the connectivity threshold for ErdHos-R'enyi graphs and suggests that these three phenomena are manifestations of the same underlying principle. The main tool is a recent estimate on the spectral norm of matrices with independent entries by van Handel and the author.