Percolation on supercritical causal triangulations

TL;DR

This paper investigates phase transitions in oriented percolation on random causal triangulations derived from infinite trees, establishing critical thresholds, coexistence of infinite clusters, and convergence to continuum limits.

Contribution

It provides the first rigorous analysis of percolation thresholds and cluster behavior on supercritical causal triangulations, linking discrete models to continuum random trees.

Findings

Percolation undergoes a phase transition at a specific threshold p_c(m).

Above p_c(m), multiple infinite clusters coexist.

Large critical clusters converge to the Brownian continuum random tree.

Abstract

We study oriented percolation on random causal triangulations, those are random planar graphs obtained roughly speaking by adding horizontal connections between vertices of an infinite tree. When the underlying tree is a geometric Galton--Watson tree with mean $m>1$, we prove that the oriented percolation undergoes a phase transition at $p_c(m)$, where $p_c(m) = \frac{\eta}{1+\eta}$ with $\eta = \frac{1}{m+1} \sum_{n \geq 0} \frac{m-1}{m^{n+1}-1}$. We establish that strictly above the threshold $p_c(m)$, infinitely many infinite components coexist in the map. This is a typical percolation result for graphs with a hyperbolic flavour. We also demonstrate that large critical oriented percolation clusters converge after rescaling towards the Brownian continuum random tree. The proof is based on a Markovian exploration method, similar in spirit to the peeling process of random planar maps.