Laura Eslava

Branching processes with merges and locality of hypercube’s critical percolation

We define a branching process to understand the locality of the critical percolation in the hypercube; that is, whether the local structure of the hypercube has an effect on the critical percolation as a function of the dimension of the hypercube.

The branching process mimics the local behavior of an exploration of a percolated hypercube; it is defined recursively as follows. Start with a single individual in generation 0. On an first stage, each individual has independent Poisson offspring with mean (1+p)(1-q)^k where k depends on the ancestry of the individual; on the merger stage, each pair of cousins merges with probability q.

We exhibit evidence of a critical merger probability q_c=q_c(p) for extinction of the branching process. When p is sufficiently small, the first order terms of q_c coincide with those of the critical percolation for the hypercube, suggesting that percolation in the hypercube is dictated by its local structure. This is work in progress with Sarah Penington and Fiona Skerman.