Multiscaling of correlation functions in single species reaction-diffusion systems

We derive the multi-scaling of probability distributions of multi-particle configurations for the binary reaction-diffusion system in A + A --> O in d < or =2 and for the ternary system in 3A --> O in d=1. For the binary reaction we find that the probability Pt(N, Delta V) of finding N particles in a fixed volume element Delta V at time t decays in the limit of large time as (ln t/t)N(ln t)-N(N-1)/2 for d=2 and t-Nd/2 t-N(N-1)epsilon/4+O(epsilon2) for d<2. Here epsilon=2-d. For the ternary reaction in one dimension we find that Pt(N, delta V) approximately (ln t/t)N/2(ln t)-N(N-1)(N-2)/6 . The principal tool of our study is the dynamical renormalization group. We compare predictions of epsilon expansions for Pt(N, Delta V) for a binary reaction in one dimension against the exact known results. We conclude that the epsilon corrections of order two and higher are absent in the previous answer for Pt(N, Delta V) for N=1, 2, 3, 4. Furthermore, we conjecture the absence of epsilon2 corrections for all values of N.