Random structures: Phase transitions, scaling limits, and universality
Speaker 
Dr. Sanchayan Sen, McGill University, Canada


When 
Jun 07, 2017
from 11:30 AM to 12:30 PM 
Where  LH 006 
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{Abstract:} The aim of this talk is to give an overview of some recent results in two interconnected areas:
a) Random discrete structures: One major conjecture in probabilistic combinatorics, formulated by statistical physicists using nonrigorous 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 in the strong disorder regime scale like \([n^{\frac{\tau\wedge 4 3}{\tau\wedge 4 1}}]\). In other words, the degree
exponent determines the universality class the random graph belongs to. The mathematical machinery available at the time was insufficient for providing a rigorous justification of this conjectur
More generally, recent research has provided strong evidence to believe
that several objects, including
(i) components under critical percolation,
(ii) the vacant set left by a random walk, and
(iii) the minimal spanning tree,
constructed on a wide class of random discrete structures converge, when viewed as metric measure spaces, to some random fractals in the GromovHausdorff
sense, and these limiting objects are universal under some general assumptions. We will discuss recent developments in a larger program aimed
at a complete resolution of these conjectures.
b) Stochastic geometry: In contrast, less precise results are known in the case of spatial systems. 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 Louigi AddarioBerry, Shankar Bhamidi, Nicolas Broutin, Sourav Chatterjee, Remco van der Hofstad, and Xuan Wang.