Apr 28, 2025

Benjamini-Schramm conjecture and the loop O(n) model

Vienna Probability Seminar

Date: April 28, 2025 | 5:00 pm – 6:15 pm
Speaker: Alexander Glazman, University of Innsbruck
Location: Central Bldg / O1 / Mondi 2a (I01.O1.008)
Language: English

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.