Voronoi percolation on a product of trees

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**This talk will be broadcast at 13:30 BT / 14:30 CET / 15:30 EET, January 28th, 2026 on MS Teams only. **

Meeting ID: 393 527 089 878
Passcode: dsm7py

Abstract

Bernoulli–Voronoi percolation on a graph $G=(V,E)$ is a percolation model with two parameters $\lambda\in(0,1]$ and $p\in(0,1)$ defined as follows. From the set $V$ of vertices, sample a set of {\em nuclei} independently with probability $\lambda$. To each nucleus $x$ associate its {\em Voronoi cell}, which is the set of all vertices $v\in V$ closest to $x$ among the nuclei. Color each Voronoi cell black independently with probability $p$ and consider the union of black cells. When we fix $\lambda$ and vary $p\in(0,1)$, this model typically undergoes a phase transition, similar to that of classical Bernoulli percolation, at two important critical parameters. Namely $p_c(\lambda)$ for the existence of an infinite cluster and $p_u(\lambda)$ for the existence of a {\em unique} infinite cluster. In this talk, I will give an introduction to this model and its continuum analogue known as Poisson–Voronoi percolation. I will then discuss a new phenomenon for the behavior of the uniqueness threshold $p_u(\lambda)$ in the low intensity limit $\lambda\to0$, using the product $G:=\mathbb T_3\times\mathbb T_3$ of two $3$-regular trees as the guiding example. I will also discuss an application of this phenomenon to a question of Hutchcroft and Pete (2020) and Pete and Rokob (2025). Joint work with Matteo d’Achille (Lorraine), Jan Grebík (Leipzig), Ali Khezeli (Tehran) and Amanda Wilkens (Pittsburgh).

About Konstantin

Konstantin currently is a Titchmarsh Research Fellow at the University of Oxford and a Junior Research Fellow at Jesus College. He obtained his PhD in 2025 at the University of Münster under the supervision of Chiranjib Mukherjee. His research interests lie primarily in probability theory with a focus on percolation. He is further interested in connections to functional analysis, group theory and metric geometry.