### Percolation on trees as a Brownian excursion

### Nicholas Moloney

Resident Fellow, LML

We calculate the distribution of the size of the percolating cluster on a tree in the subcritical, critical and supercritical phase. We do this by exploiting a mapping between continuum trees and Brownian excursions, and arrive at a diffusion equation with suitable boundary conditions. The exact solution to this equation can be conveniently represented as a characteristic function, from which the following distributions are clearly visible: Gaussian (subcritical), Kolmogorov-Smirnov (critical) and exponential (supercritical). In this way we provide an intuitive explanation for the result reported in R. Botet and M. Ploszajczak, Phys. Rev. Lett 95, 185702 (2005) for critical percolation.

(joint collaboration with Francesc Font-Clos,arXiv:1606.03764)

### Location

Department of Mathematical Sciences, University of Essex, Wivenhoe Park, Colchester CO4 3SQ

### Date & Time

Thursday 24^{th} November 2016 at 14.00hrs

### Information

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