# Critical percolation and A+B --> 2A dynamics

## TL;DR

This paper investigates a particle activation process linked to critical percolation, proving finite-time explosion and superlinear growth on various lattices, with implications for understanding critical phenomena in interacting systems.

## Contribution

It establishes the occurrence of finite-time explosions and superlinear expansion in a particle system related to critical percolation, confirming a conjecture and extending known results.

## Key findings

- Explosions occur almost surely on regular trees and 2D lattices.
- The process expands superlinearly with one particle per site.
- New results on infinite paths with finite passage time in oriented percolation.

## Abstract

We study an interacting particle system in which moving particles activate dormant particles linked by the components of critical bond percolation. Addressing a conjecture from Beckman, Dinan, Durrett, Huo, and Junge for a continuous variant, we prove that the process can reach infinity in finite time i.e., explode. In particular, we prove that explosions occur almost surely on regular trees as well as oriented and unoriented two-dimensional integer lattices with sufficiently many particles per site. The oriented case requires an additional hypothesis about the existence and value of a certain critical exponent. We further prove that the process with one particle per site expands at a superlinear rate on integer lattices of any dimension. Some arguments use connections to critical first passage percolation, including a new result about the existence of an infinite path with finite passage time on the oriented two-dimensional lattice.