# Lecturer/Senior Lecturer Seminars - Dr Sanchayan Sen

Abstract: One key focus of probability theory is understanding asymptotics in the context of high dimensional structures. The aim of this talk is to give an overview of two interconnected themes that have arisen in my work. More precisely:

a) Random graphs and complex networks: 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 $\tau \gt 3$, 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 scales like $n^{\frac{\tau \wedge4-3}{\tau \wedge4-1}}$. In other words, the degree exponent of a complex network 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.

Based on joint work with Shankar Bhamidi, Nicolas Broutin, Sourav Chatterjee, Remco van der Hofstad, and Xuan Wang.