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

Random walks provide precise mathematical models for diffusion processes, while rooted trees offer a geometric representation of branching processes. Both are of broad importance in probability theory and statistical physics, and some important mathematical results establish links between the two. For example, Theodore Harris in 1952 demonstrated a purely geometric mapping between walks and rooted trees. This mapping, known as the Harris walk, implies that a realization of a finite-size geometric Galton-Watson branching process with no more than L generations corresponds (exactly) to a random walk excursion that reaches height L.
In a new paper, LML Fellow Nicholas Moloney and colleagues employ the Harris mapping to calculate the size distribution of both percolating and non-percolating clusters in a geometric Galton-Watson process with a finite number of generations. They build on other recent work of Font-Clos and Moloney, who applied this mapping to derive the distribution of the size of the percolating clusters in a finite Bethe lattice, by using the first-passage time to the origin of a Brownian particle conditioned to first reach L. In the new work, the authors focus on the size distribution of all clusters, whether they percolate or not. They first solve the corresponding diffusion problem and derive analytical expressions and scaling laws, then translate the results to the random-walk picture, and finally, by means of the mapping from trees to walks, obtain the properties of the associated branching process.
The paper is available at or as a preprint:

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