The central limit theorem for supercritical oriented percolation in two dimensions

TL;DR

This paper proves that the size of a supercritical oriented percolation process in two dimensions, when conditioned on percolation from the origin, converges to a normal distribution, solving a longstanding open problem.

Contribution

It establishes a central limit theorem for supercritical oriented percolation in two dimensions, including the continuous-time contact process, and introduces general CLTs for related random variables.

Findings

Normalized process converges to the standard normal law

Results apply to both discrete and continuous-time models

Provides new CLTs for associated random variables

Abstract

We consider the cardinality of supercritical oriented bond percolation in two dimensions. We show that, whenever the origin is conditioned to percolate, the process appropriately normalized converges asymptotically in distribution to the standard normal law. This resolves a longstanding open problem pointed out to in several instances in the literature. The result applies also to the continuous-time analog of the process, viz. the basic one-dimensional contact process. We also derive general random-indices central limit theorems for associated random variables as byproducts of our proof.