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SUMMARY:Nathanaël Berestycki (University of Vienna)
DTSTART:20200330T113000Z
DTEND:20200330T123000Z
DTSTAMP:20260423T041353Z
UID:HSPETDS/11
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/HSPETDS/11/"
 >Random walks on random planar maps and Liouville Brownian motion</a>\nby 
 Nathanaël Berestycki (University of Vienna) as part of Horowitz seminar o
 n probability\, ergodic theory and dynamical systems\n\n\nAbstract\nThe st
 udy of random walks on random planar maps was initiated in a series of sem
 inal papers of Benjamini and Schramm at the end of the 90s\, motivated by 
 contemporary (nonrigourous) works in the study of Liouville Quantum Gravit
 y (LQG). Both topics have been the subject of intense research following r
 emarkable breakthroughs in the last few years.\n\nAfter reviewing some of 
 the recent developments in these fields - including Liouville Brownian mot
 ion\, a canonical notion of diffusion on LQG surfaces - I will describe so
 me joint work with Ewain Gwynne. In this work we show that random walks on
  certain models of random planar maps (known as mated-CRT planar maps) hav
 e a scaling limit given by Liouville Brownian motion. This is true whether
  the maps are embedded using SLE/LQG theory or more intrinsically using th
 e Tutte embedding. This is the first result confirming that Liouville Brow
 nian motion is the scaling limit of random walks on random planar maps.\n\
 nThe proof relies on some earlier work of Gwynne\, Miller and Sheffield wh
 ich proves convergence to Brownian motion\, modulo time-parametrisation. A
 s an intermediate result of independent interest\, we derive an axiomatic 
 characterisation of Liouville Brownian motion\, for which the notion of Re
 vuz measure of a Markov process plays a crucial role.\n
LOCATION:https://researchseminars.org/talk/HSPETDS/11/
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