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Multi-point correlations for two-dimensional coalescing or annihilating random walks

Part of: Time-dependent statistical mechanics (dynamic and nonequilibrium) Special processes

Published online by Cambridge University Press:  16 January 2019

James Lukins ,
Roger Tribe  and
Oleg Zaboronski
James Lukins*
Affiliation:
University of Warwick
Roger Tribe*
Affiliation:
University of Warwick
Oleg Zaboronski*
Affiliation:
University of Warwick
*
* Postal address: Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK.
* Postal address: Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK.
* Postal address: Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK.
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Abstract

In this paper we consider an infinite system of instantaneously coalescing rate 1 simple symmetric random walks on ℤ2, started from the initial condition with all sites in ℤ2 occupied. Two-dimensional coalescing random walks are a `critical' model of interacting particle systems: unlike coalescence models in dimension three or higher, the fluctuation effects are important for the description of large-time statistics in two dimensions, manifesting themselves through the logarithmic corrections to the `mean field' answers. Yet the fluctuation effects are not as strong as for the one-dimensional coalescence, in which case the fluctuation effects modify the large time statistics at the leading order. Unfortunately, unlike its one-dimensional counterpart, the two-dimensional model is not exactly solvable, which explains a relative scarcity of rigorous analytic answers for the statistics of fluctuations at large times. Our contribution is to find, for any N≥2, the leading asymptotics for the correlation functions ρN(x1,…,xN) as t→∞. This generalises the results for N=1 due to Bramson and Griffeath (1980) and confirms a prediction in the physics literature for N>1. An analogous statement holds for instantaneously annihilating random walks. The key tools are the known asymptotic ρ1(t)∼logt∕πt due to Bramson and Griffeath (1980), and the noncollision probability 𝒑NC(t), that no pair of a finite collection of N two-dimensional simple random walks meets by time t, whose asymptotic 𝒑NC(t)∼c0(logt)-(N2) was found by Cox et al. (2010). We re-derive the asymptotics, and establish new error bounds, both for ρ1(t) and 𝒑NC(t) by proving that these quantities satisfy effective rate equations; that is, approximate differential equations at large times. This approach can be regarded as a generalisation of the Smoluchowski theory of renormalised rate equations to multi-point statistics.

Keywords

Coalescing random walkannihilating random walkcorrelation functionnoncollision probability

MSC classification

Primary: 82C22: Interacting particle systems
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 4 , December 2018 , pp. 1158 - 1185
Copyright
Copyright © Applied Probability Trust 2018 

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