STOR Colloquium: Sanchayan Sen, Eindhoven University of Technology

Asymptotics of high dimensional random structures: Probabilistic combinatorial optimization, stochastic geometry and random matrices

One key focus of probability theory over the last few years has been on understanding asymptotics in the context of high dimensional structures. The aim of this talk is to give an overview of three major themes that have arisen in my work. More precisely:

a) Probabilistic combinatorial optimization: One major conjecture in probabilistic combinatorics, formulated by statistical physicists using non-rigorous arguments and enormous simulations in the early 2000s, is as follows:  for a wide array of random graph models  on  n vertices and degree exponent tau>3, typical distance both within maximal components in the critical regime as well as on the minimal spanning tree on the giant component in the supercritical regime scale like n^{(min(tau,4) -3)/(min(tau,4) -1)}. In other words, the degree exponent determines the universality class the network belongs to. The mathematical machinery available at the time was insufficient for providing a rigorous justification of this conjecture. We report on recent progress in proving this conjecture and characterizing these universality classes in a broader sense.

b) Stochastic geometry:  In the case of spatial systems, less precise results are known. We discuss a recent result concerning the length of spatial minimal spanning trees that answers a question raised by Kesten and Lee in the 90’s, the proof of which relies on a variation of Stein’s method and a quantification of a classical argument in percolation theory.

c) Random matrix theory: Many random matrix ensembles arise naturally in statistics and physics. In the final part of the talk, we discuss properties of empirical spectral distributions of two such ensembles.