Events Archive

Abstract: I review some recent results on Erdos-Renyi graphs G(N,p) near and above the critical scale pN = log N, where the graph undergoes a connectivity crossover. For pN >> log N, the graph G(N,p) is with high probability connected, while for pN << log N it has with high probability isolated vertices. In the supercritical regime pN >> log N, the eigenvalues stick to the bulk spectrum, a local law holds down to optimal scales, and the eigenvectors are completely delocalized. All three statements are false in the subcritical regime pN << log N. Based on joint work with F. Benaych-Georges, C. Bordenave, Y. He, and M. Marcozzi.

Abstract: Stochastic systems can be parametrised by time (like Markov chains), in which conditioning is one-sided(the past) or by one-dimensional space (like Markov fields), where conditioning is two-sided (right and left). I will discuss some examples, in particular generalising this to g-measures versus Gibbs measures, when the two descriptions are the same and when they are different. We show the role one-dimensional entropic repulsion plays in this setting. Joint work with R. Bissacot, E. Endo and A. Le Ny