### Matthew Tointon

University of Bristol

Thursday 07 March 2024, 4.00pm – 5.00pm

Lecture Theatre D (Mathematical Institute)

** Structure versus expansion, and probability on transitive graphs **

Classical results of Gromov, Trofimov and Coulhonâ€”Saloff-Coste show that vertex-transitive graphs exhibit a rather powerful dichotomy: every such graph exhibits either 'structure' (roughly, it looks like a low-complexity nilpotent group) or 'expansion' (in the sense of an isoperimetric inequality, i.e. a lower bound on the size of the boundary of an arbitrary set of given size). This in turn has had some remarkable applications to probability on such graphs. In particular, Varopoulos used it to characterise those transitive graphs on which the random walk is recurrent, and Duminil-Copin et al used it to characterise those in which there is a non-trivial percolation phase. In this talk I will give a gentle introduction to this theory and these applications, and talk about some finitary refinements that have emerged over the past five years.

series: Pure Mathematics Colloquium

organiser: Scott Harper