[Paper Review] Sharpness of the Percolation Phase Transition for the Contact Process on $\mathbb{Z}^d$

Thomas Beekenkamp|arXiv (Cornell University)|Jul 15, 2018

Stochastic processes and statistical mechanics4 citations

TL;DR

This paper establishes a sharp percolation phase transition for the contact process on $ℤ^d$ by proving exponentially small cluster sizes in the subcritical regime and a mean-field lower bound for the infinite cluster density in the supercritical regime. The proof leverages the OSSS inequality for Boolean functions, generalizing and simplifying Van den Berg's earlier result on $ℤ^2$.

ABSTRACT

We study percolation properties of the upper invariant measure of the contact process on $\mathbb{Z}^d$. Our main result is a sharp percolation phase transition with exponentially small clusters throughout the subcritical regime and a mean-field lower bound for the infinite cluster density in the supercritical regime. This generalizes and simplifies an earlier result of Van den Berg [Ann. App. Prob., 2011], who proved a sharp percolation phase transition on $\mathbb{Z}^2$. Our proof relies on the OSSS inequality for Boolean functions and is inspired by a series of papers by Duminil-Copin, Raoufi and Tassion in which they prove similar sharpness results for a variety of models.

Motivation & Objective

• To establish a sharp percolation phase transition for the upper invariant measure of the contact process on $ℤ^d$.
• To generalize Van den Berg's result on $ℤ^2$ to higher dimensions $d \geq 2$.
• To simplify the proof of sharpness using the OSSS inequality for Boolean functions.
• To provide exponential tail bounds on cluster sizes in the subcritical regime.
• To derive a mean-field lower bound for the density of the infinite cluster in the supercritical regime.

Proposed method

• Utilizes the OSSS inequality for Boolean functions to control noise sensitivity and derive sharp threshold results.
• Applies techniques inspired by Duminil-Copin, Raoufi, and Tassion on sharp thresholds in dependent models.
• Analyzes the upper invariant measure of the contact process to study percolation properties.
• Establishes exponential decay of cluster sizes in the subcritical phase using noise sensitivity arguments.
• Derives a lower bound on the density of the infinite cluster in the supercritical phase via mean-field estimates.
• Relies on the structure of the contact process on $ℤ^d$ and its correlation decay properties.

Experimental results

Research questions

• RQ1 Does the contact process on $ℤ^d$ exhibit a sharp percolation phase transition?
• RQ2 What is the decay rate of cluster sizes in the subcritical regime of the contact process?
• RQ3 Can the mean-field lower bound for the infinite cluster density be rigorously established in the supercritical phase?
• RQ4 To what extent can the OSSS inequality be applied to dependent models like the contact process?
• RQ5 How does the sharpness result generalize from $ℤ^2$ to higher-dimensional lattices?

Key findings

• The subcritical regime exhibits exponentially small cluster sizes, confirming a sharp transition.
• The infinite cluster density in the supercritical regime satisfies a mean-field lower bound.
• The proof technique via the OSSS inequality provides a unified and simplified approach compared to prior methods.
• The sharp phase transition is established for all $d \geq 2$, generalizing Van den Berg's result.
• The method applies to the upper invariant measure of the contact process, not just the process dynamics.
• The results confirm that the contact process on $ℤ^d$ undergoes a sharp transition with no intermediate regime.