Abstract: We define a graph process 
based on a discrete branching process with deletions and mergers, which is inspired by the 4-cycle structure of both the hypercube 
and the lattice 
for large 
. We prove survival and extinction under certain conditions on 
and 
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 is monotone in or .
In the first half of the talk, we introduce the model and prove that represents an idealized version of a percolation cluster in either or . 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.