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

In this paper we consider an infinite system of instantaneously coalescing rate 1 simple symmetric random walks on Z(2), started from the initial condition with all sites in Z(2) 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 rho(N) (x(1), ..., x(N)) as t -> infinity. 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 rho(1) (t) similar to log t/pi t 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 p(NC)(t) similar to c(0)(log t) - ((N)(2)) was found by Cox et al. (2010). We re-derive the asymptotics, and establish new error bounds, both for rho(1)(t) and p(NC)(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.