Multi-point correlations for two dimensional coalescing random walks

Abstract

This paper considers an infinite system of instantaneously coalescing rate one simple random walks on ${\mathbb{Z}}^2$, started from the initial condition with all sites in ${\mathbb{Z}}^2$ occupied. We show that the correlation functions of the model decay, for any $N\ge 2$, as \[ ρ_N (x_1,\ldots,x_N;t) = \frac{c_0(x_1,\ldots,x_N)}{π^N} (\log t)^{N-{N \choose 2}} t^{-N} \left(1 + O\left( \frac{1}{\log^{\frac12-δ}\!t} \right) \right) \] as $t\to \infty$. This generalises the results for $N=1$ due to Bramson and Griffeath 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 ${\rho }_1(t)\sim \log t/\pi t$ due to Bramson and Griffeath, and the non-collision probability $p_{NC}(t)$, that no pair of a finite collection of $N$ two dimensional simple random walks meets by time $t$, whose asymptotic $p_{NC}(t)\sim c_0(\log t{)}^{-\left(\genfrac{}{}{0ex}{}{N}{2}\right)}$ was found by Cox, Merle and Perkins. This paper re-derives the asymptotics both for ${\rho }_1(t)$ and $p_{NC}(t)$ by proving that these quantities satisfy {\it 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.

26 pages