L. Eslava & S. Penington: A branching process with deletions and mergers that matches the threshold for hypercube percolation

Abstract: We define a graph process G(p,q) based on a discrete branching process with deletions and mergers, which is inspired by the 4-cycle structure of both the hypercube Q_d and the lattice \mathbb{Z}^d for large d. We prove survival and extinction under certain conditions on p and q that heuristically match the known expansions of the critical probabilities for bond percolation on these graphs. However, it is left open whether the survival probability of G(p,q) is monotone in p or q.

In the first half of the talk, we introduce the model and prove that G(p,q) represents an idealized version of a percolation cluster in either Q_d or \mathbb{Z}^d. We then provide a heuristic to recover the known expansion of the corresponding critical probabilities. In the second half of the talk, we analyse the graph process survival which is considerably more challenging than branching processes in discrete time, due to the interdependence between the descendants of different individuals in the same generation.

This is joint work with Fiona Skerman.


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