Off-giant Criticality in the Mean-field Arboreal Gas

Tyler Helmuth (Durham U)

Sep 21. 2026, 14:15 — 15:05

The arboreal gas is Bernoulli percolation conditioned on the event that each connected component is a tree. On the complete graph K_N, there is a phase transition: a subcritical phase in which all components (trees) are small, and a supercritical phase in which a giant tree exists. Unlike percolation, the supercritical phase has many mesoscopic trees of size N^{2/3} in addition to the unique giant. I will discuss a new proof of this result by comparatively soft means, as well as some conjectures that this proof suggests.

Based on joint work with Louigi Addario-Berry and Gourab Ray.

Further Information
Venue:
ESI Boltzmann Lecture Hall
Associated Event:
Statistical Mechanics and Combinatorics of Discrete Planar Structures (Thematic Programme)
Organizer(s):
Nathanael Berestycki (U of Vienna)
Michael Drmota (TU Wien)
Ilse Fischer (U of Vienna)
Mihyun Kang (TU Graz)
Astrid Kollros (U of Vienna)
Christian Krattenthaler (U of Vienna)
Marcin Lis (TU Wien)
Benedikt Stufler (TU Wien)
Fabio Toninelli (TU Wien)