[Paper Review] Scaling of a random walk on a supercritical contact process

Frank den Hollander, Renato dos Santos|arXiv (Cornell University)|Sep 7, 2012

Stochastic processes and statistical mechanics13 references4 citations

TL;DR

This paper establishes a strong law of large numbers, functional central limit theorem, and large deviation principle for a one-dimensional random walk evolving in a dynamic random environment driven by a supercritical contact process in equilibrium. Using a coupling argument based on space-time cones of infection clusters, the authors show the walk eventually gets trapped within a single cone when its drift is slower than the infection front speed, enabling regeneration times that facilitate the derivation of scaling limits.

ABSTRACT

A proof is provided of a strong law of large numbers for a one-dimensional random walk in a dynamic random environment given by a supercritical contact process in equilibrium. The proof is based on a coupling argument that traces the space-time cones containing the infection clusters generated by single infections and uses that the random walk eventually gets trapped inside the union of these cones. For the case where the local drifts of the random walk are smaller than the speed at which infection clusters grow, the random walk eventually gets trapped inside a single cone. This in turn leads to the existence of regeneration times at which the random walk forgets its past. The latter are used to prove a functional central limit theorem and a large deviation principle. The qualitative dependence of the speed, the volatility and the rate function on the infection parameter is investigated, and some open problems are mentioned.

Motivation & Objective

• To establish the strong law of large numbers (SLLN) for a random walk in a dynamic random environment generated by a supercritical contact process in equilibrium.
• To derive a functional central limit theorem (FCLT) and a large deviation principle (LDP) for the same process.
• To analyze the qualitative dependence of the walk's speed and volatility on the infection parameter of the contact process.
• To identify regeneration times in the walk's path by exploiting trapping within space-time cones of infection clusters.

Proposed method

• Use of a graphical representation to couple all realizations of the contact process starting from different initial configurations.
• Application of a subadditivity argument to prove the SLLN when the initial configuration is fully infected.
• Construction of space-time cones containing infection clusters generated by single infections to show the random walk eventually gets trapped inside their union.
• Demonstration that when the walk's local drift is smaller than the infection front speed, it gets trapped within a single cone, enabling regeneration times.
• Use of regeneration times to prove the FCLT and LDP via Markov renewal theory and convergence of functionals in the graphical representation.
• Analysis of convergence of key functionals (e.g., hitting times, exit times) in bounded space-time boxes using the graphical representation’s almost sure eventual constancy.

Experimental results

Research questions

• RQ1 Under what conditions does a random walk in a supercritical contact process environment satisfy a strong law of large numbers?
• RQ2 How do the scaling limits—functional central limit theorem and large deviation principle—emerge in the absence of cone-mixing properties?
• RQ3 What is the qualitative dependence of the walk’s speed and volatility on the infection parameter of the contact process?
• RQ4 Can regeneration times be constructed in non-cone-mixing environments like the contact process?
• RQ5 How does the coupling via the graphical representation enable the analysis of long-time behavior in dynamic random environments?

Key findings

• The random walk satisfies a strong law of large numbers almost surely, with the speed determined by the environment's dynamics.
• When the walk's drift is less than the infection front speed, it eventually gets trapped within a single space-time cone, enabling the construction of regeneration times.
• The existence of regeneration times allows the proof of a functional central limit theorem and a large deviation principle for the walk.
• The speed of the walk and the volatility in the FCLT exhibit non-trivial qualitative dependence on the infection parameter of the contact process.
• Convergence of key functionals (e.g., hitting times, exit times) is established via the graphical representation and bounded space-time box analysis.
• The limit of the regeneration time parameter κ^n as n→∞ exists and equals κ^*, ensuring convergence in distribution of the number of regeneration cycles.