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 $N \geq 2$, as \[ \rho_N (x_1,\ldots,x_N;t) = \frac{c_0(x_1,\ldots,x_N)}{\pi^N} (\log t)^{N-{N \choose 2}} t^{-N} \left(1 + O\left( \frac{1}{\log^{\frac12-\delta}\!t} \right) \right) \] 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 $p_{NC}(t) \sim c_0 (\log t)^{-{N \choose 2}}$ 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.