Random walks in logarithmic and power-law potentials, nonuniversal persistence, and vortex dynamics in the two-dimensional XY model

The Langevin equation for a particle (''random walker") moving in d-dimensi onal space under an attractive central force and driven by a Gaussian white noise is considered for the case of a power-law force, F(r) similar to -r( -sigma). The "persistence probability," P-0(t), that the particle has not v isited the origin up to time t is calculated for a number of cases. For sig ma>1, the force is asymptotically irrelevant (with respect to the noise), a nd the asymptotics of P-0(t) are those of a free random walker. For sigma< 1, the noise is (dangerously) irrelevant and the asymptotics of P-0(f) can be extracted from a weak noise limit within a path-integral formalism emplo ying the Onsager-Machlup functional. The case sigma=1,corresponding to a lo garithmic potential, is most interesting because the noise is exactly margi nal. In this case, P-0(t) decays as a power law, P-0(t)similar to t(-theta) With an exponent theta that depends continuously on the ratio of the stren gth of the potential to the strength of the noise. This case, with d=2, is relevant to the annihilation dynamics of a voaex-antivortex pair in the two -dimensional XY model. Although the noise is multiplicative in the latter c ase, the relevant Langevin equation can be transformed to the standard form discussed in the first part of the paper. The mean annihilation time for a pair initially separated by r is given by t(r)similar to r(2) In(r/a) wher e a is a microscopic cutoff (the vortex core size). Implications for the no nequilibrium critical dynamics of the system are discussed and compared to numerical simulation results.