Survival probability of a diffusing test particle in a system of coagulating and annihilating random walkers
Abstract
We calculate the survival probability of a diffusing test particle in an environment of diffusing particles that undergo coagulation at rate lambdac and annihilation at rate lambdaa. The test particle dies at rate lambda' on coming into contact with the other particles. The survival probability decays algebraically with time as t-theta. The exponent theta in d<2 is calculated using the perturbative renormalization group formalism as an expansion in epsilon=2-d. It is shown to be universal, independent of lambda', and to depend only on delta, the ratio of the diffusion constant of test particles to that of the other particles, and on the ratio lambdaa/lambdac. In two dimensions we calculate the logarithmic corrections to the power law decay of the survival probability. Surprisingly, the log corrections are non-universal. The one loop answer for theta in one dimension obtained by setting epsilon=1 is compared with existing exact solutions for special values of delta and lambdaa/lambdac. The analytical results for the logarithmic corrections are verified by Monte Carlo simulations.
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