MARK KAC SEMINAR

We discuss anomalous behavior of simple random walk on the two-dimensional uniform spanning tree.

The first part of my talk is a overview of the area. I will present results concerning the behavior of random walk and diffusion on disordered media. Examples treated include fractals and various models of random graphs, such as percolation clusters and uniform spanning trees. I will describe how the techniques have developed from those introduced for exactly self-similar fractals to more robust arguments required for random graphs.

In the second part of my talk, I will focus on simple random walk on the two-dimensional uniform spanning tree. It has been studied by obtaining its heat kernel estimates that this random walk behaves anomalously, and the scaling limit of the random walk has been investigated. After summarizing these, I will explain further details of the heat kernel estimates for the random walk, including log-log fluctuations for the on-diagonal part of the quenched heat kernel, and two-sided estimates for the averaged heat kernel. These demonstrate a discrepancy between the exponents that appear in the off-diagonal parts of the quenched and averaged versions of the heat kernel. This part of the talk is based on joint work with M.T. Barlow and D.A. Croydon.

The structure of triangle-free graphs has played an influential role in the development of combinatorial mathematics. Both of the fundamental results of Mantel (1907) and of Ramsey (1930) yield global structure from the local property of having no edges in any neighbourhood.

I recently began some basic explorations in this classic and well-studied area. This has led to novel questions and developments, especially with respect to independent sets and colourings in graphs. To begin I will give an overview of the history including the focus of current/recent activities.

Then I will present a new general framework for locally sparse graphs. This generalises and strengthens many notable results in the area, including those of Ajtai, Komlós, Szemerédi (1981), Shearer (1983/1996), Johansson (1996), Alon (1996), Alon, Krivelevich, Sudakov (1999), Molloy (2019), Bernshteyn (2019), and Achlioptas, Iliopoulos, Sinclair (2019). The methodology behind this asymptotically cannot be improved in general, by a comparison to the ultimate outcome of the triangle-free process. The framework is built around a technique inspired by statistical physics --namely, a local analysis of the hard-core model-- as well as the suitable application of the Lovász local lemma.

This covers joint work with, variously, Wouter Cames van Batenburg, Ewan Davies, Louis Esperet, Rémi de Joannis de Verclos, François Pirot, Jean-Sébastien Sereni, and Stéphan Thomassé.