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Multi-point correlations for two dimensional coalescing random walks
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Lukins, Jamie, Tribe, Roger and Zaboronski, Oleg V. (2018) Multi-point correlations for two dimensional coalescing random walks. Journal of Applied Probability, 55 (4). pp. 1158-1185. doi:10.1017/jpr.2018.77 ISSN 0021-9002.
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Official URL: https://doi.org/10.1017/jpr.2018.77
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
| Item Type: | Journal Article | |||||||||
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| Subjects: | Q Science > QA Mathematics | |||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | |||||||||
| Library of Congress Subject Headings (LCSH): | Random walks (Mathematics) | |||||||||
| Journal or Publication Title: | Journal of Applied Probability | |||||||||
| Publisher: | Applied Probability Trust | |||||||||
| ISSN: | 0021-9002 | |||||||||
| Official Date: | December 2018 | |||||||||
| Dates: |
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| Volume: | 55 | |||||||||
| Number: | 4 | |||||||||
| Page Range: | pp. 1158-1185 | |||||||||
| DOI: | 10.1017/jpr.2018.77 | |||||||||
| Status: | Peer Reviewed | |||||||||
| Publication Status: | Published | |||||||||
| Access rights to Published version: | Restricted or Subscription Access | |||||||||
| Date of first compliant deposit: | 15 November 2018 | |||||||||
| Date of first compliant Open Access: | 16 July 2019 | |||||||||
| RIOXX Funder/Project Grant: |
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