Critical behavior of interacting Brownian motions and L´evy flights
This dissertation studies the late-time critical behavior of interacting many- particle systems. Two examples of such systems considered here are Brownian vicious walks with long-range interactions and vicious L´evy flights. In the first example we consider distinct groups of independent diffusive particles interacting by means of a long ranged potential that decays in d dimensions with distance r as r−d−s such that the process is terminated upon the intersection of any two trajectories of particles in the spatio-temporal plane. The main characteristics of this system are the survival and reunion probabilities defined, respectively, as the probability that no trajectories intersect up to time t and the probability that all trajectories meet each other exactly at time t. We employ methods of renormalized field theory to show that these quantities decay as t−® and t(N−1)d/2−2®, respectively, where N is total number of particles. We calculate, for the first time, the exponent ® for all values of parameters s and d to first order in the double expansion in " = 2−d and ± = 2 − d − s. We show that there are several regions in the s − d plane cor- responding to different scalings for survival and reunion probabilities. Furthermore, we calculate the leading logarithmic corrections, for the first time. In the second example we study the statistics of encounters of L´evy flights by introducing the concept of vicious L´evy flights - distinct groups of walkers performing independent L´evy flights with the process terminating upon the first encounter between walkers of different groups. We show that the probability that the process survives up to time t decays as t−® at late times. We compute ® up to the second order in "-expansion, where " = ¾ − d, ¾ is the L´evy exponent and d is the spatial dimension. For d = ¾, we find the exponent of the logarithmic decay exactly. Theoretical values of the exponents are confirmed by numerical simulations.