Persistence properties of a system of coagulating and annihilating random walkers.

Abstract

We study a d-dimensional system of diffusing particles that on contact either annihilate with probability 1/(q-1) or coagulate with probability (q-2)/(q-1). In one dimension, the system models the zero-temperature Glauber dynamics of domain walls in the q-state Potts model. We calculate P(m,t), the probability that a randomly chosen lattice site contains a particle whose ancestors have undergone exactly (m-1) coagulations. Using perturbative renormalization group analysis for d<2, we show that, if the number of coagulations m is much less than the typical number M(t), then P(m,t) approximately m(zeta/d)t(-theta), with theta=dQ+Q(Q-1/2)epsilon+O(epsilon(2)), zeta=(2Q-1)epsilon+(2Q-1)(Q-1)(1/2+AQ)epsilon(2)+O(epsilon(3)), where Q=(q-1)/q, epsilon=2-d and A=-0.006....M(t) is shown to scale as M(t) approximately t(d/2-delta), where delta=d(1-Q)+(Q-1)(Q-1/2)epsilon+O(epsilon(2)). In two dimensions, we show that P(m,t) approximately ln(t)(Q(3-2Q))ln(m)((2Q-1)(2))t(-2Q) for m<<t(2Q-1). We also derive an exact nonperturbative relation between the exponents: namely delta(Q)=theta(1-Q). The one-dimensional results corresponding to epsilon=1 are compared with results from Monte Carlo simulations.