Leírás
Abstract: The study of random walks on random planar maps was initiated in a series of seminal papers of Benjamini and Schramm at the end of the 90s, motivated by contemporary (non-rigourous) works in the study of Liouville Quantum Gravity (LQG). Both topics have been the subject of intense research following remarkable breakthroughs in the last few years. After reviewing some of the recent developments in these fields - including Liouville Brownian motion, a canonical notion of diffusion on LQG surfaces - I will describe some joint work in progress with Ewain Gwynne. In this work we show that random walks on certain models of random planar maps (known as mated-CRT planar maps) have a scaling limit given by Liouville Brownian motion. This is true whether the maps are embedded using SLE/LQG theory or more intrinsically using the Tutte embedding. This is the first result confirming that Liouville Brownian motion is the scaling limit of random walks on planar maps. The proof relies on some earlier work of Gwynne, Miller and Sheffield which proves convergence to Brownian motion, modulo time-parametrisation. As an intermediate result of independent interest, we derive an axiomatic characterisation of Liouville Brownian motion, for which the notion of Revuz measure of a Markov process plays a crucial role.