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

We revisit the problem of Brownian diffusion with drift in order to study finite-size effects in the geometric Galton-Watson branching process. This is possible because of an exact mapping between one-dimensional random walks and geometric branching processes, known as the Harris walk. In this way, first-passage times of Brownian particles are equivalent to sizes of trees in the branching process (up to a factor of proportionality). Brownian particles that reach a distant reflecting boundary correspond to percolating trees, and those that do not correspond to 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 second moments. These quantities are also shown to obey finite-size scaling laws.

Figure 1

Relations between different physical processes and models, with abbreviations RW (random walk), BP (Galton-Watson branching process), and SOC (self-organized critical phenomena) [11, 34, 35]. BP with geometric and binomial offspring distribution share the same critical properties due to universality.

Figure 2

Distribution of sizes in the geometric Galton-Watson process in the three regimes: [(a) and (d)] subcritical, with $P_e=-2.5$; [(b) and (e)] critical, with $P_e=0$; [(c) and (f)] supercritical, with $P_e=2.5$; together with theoretical predictions from the first-passage times of a diffusion process (numerical inversion of the Laplace transform is done by the Talbot method using the routine in Ref. [36]). Panels (a), (b), and (c) show the full distribution. Panels (d), (e), and (f) show the distribution decomposed into its percolating and nonpercolating contributions. In the critical regime, the complete distribution $f\left(S\right)$ shows a small bump for large sizes (b), which is much more pronounced in the supercritical regime (c). The subcritical distribution, having no bump, can be visually confused with a critical distribution (a). System size is $L=1000$ and total number of realizations is ${10}^7$. Logarithmic binning has been applied [11, 37].

Figure 3

Rescaled mean sizes (simulations) and rescaled times (theory) versus Péclet number for different systems sizes $L$. The number of realizations in the simulations is ${10}^5$ for each value of $P_e$ and $L$.

Figure 4

Correspondence between trees and positive walks (excursions), known as a Harris walk [1]. A walker starts at the root of the tree (the vertex at the bottom) and follows each branch in turn to its end, starting with the leftmost branch. If the tree is generated from a geometric offspring distribution, then the resulting path is a simple random walk.