Benjamini-Schramm conjecture and the loop O(n) model
Vienna Probability Seminar
Date
Monday, April 28, 2025 17:00 - 18:15
Speaker
Alexander Glazman (University of Innsbruck)
Location
Central Bldg / O1 / Mondi 2a (I01.O1.008)
Series
Seminar/Talk
Tags
mathematical_seminar_ics
Host
Laszlo Erdös, Jan Maas
Contact
We witness many phase transitions in everyday life (eg. ice melting to water). The mathematical approach to these phenomena revolves around the percolation model: given a graph, call each vertex open with probability p independently of the others and look at the subgraph induced by open vertices. Benjamini and Schramm conjectured in 1996 that, at p=1/2, on any planar graph, either there is no infinite connected components or infinitely many.
We prove a stronger version of this conjecture and use this to establish fractal macroscopic behaviour in the loop O(n) model. The latter includes a random discrete Lipschitz surface as a particular case.
Joint work with Matan Harel and Nathan Zelesko.