Reunion and survival of interacting walkers

Abstract

The reunion and survival probabilities of p random walkers in d dimensions with a mutual repulsive interaction are formulated via appropriate partition functions of directed polymers. The exponents that describe the decay of these probabilities with length are obtained through renormalization-group theory to O(${\mathrm{\epsilon }}^2$), where ε=2-d. The distribution function and the probability of n out of p walkers meeting are also discussed. To first order, the distribution function is a Gaussian one modified by an anomalous exponent of the length of the polymer, N. The procedure is generalized to multicritical many-body interactions. For these multicritical cases, the exponents are obtained to second order in the relevant εs. At the upper critical dimension of the interaction, there is a logarithmic correction other than the Gaussian exponent. An interesting consequence is a logarithmic correction for one-dimensional walkers with a three-body repulsive interaction.