Phase transition, scaling of moments, and order-parameter distributions in Brownian particles and branching processes with finite-size effects

Abstract: We revisit the problem of Brownian diffusion with drift in order to study finite-size effects in the geometricGalton-Watson branching process. This is possible because of an exact mapping between one-dimensional randomwalks and geometric branching processes, known as the Harris walk. In this way, first-passage times of Brownianparticles are equivalent to sizes of trees in the branching process (up to a factor of proportionality). Brownianparticles that reach a distant reflecting boundary correspond to percolating trees, and those that do not correspondto nonpercolating trees. In fact, both systems display a second-order phase transition between “conducting” and“insulating” phases, controlled by the drift velocity in the Brownian system. In the limit of large system size,we obtain exact expressions for the Laplace transforms of the probability distributions and their first and secondmoments. These quantities are also shown to obey finite-size scaling laws.