Abstract: A classic result of Erdős and Rényi describes the phase transition that the component structure of the binomial random graph G(n,p) undergoes when p is around 1/n. Below this point, the graph typically contains only small components, of logarithmic order, whereas above this point many of these component coalesce to a unique `giant' component of linear order, and all other components are of logarithmic order. It has been observed that quantitatively similar phase transitions occur in many other percolation models, and, in particular, work of Ajtai, Komlós and Szemerédi and of Bollobás, Kohayakawa and Łuczak shows that such a phenomena occurs in the percolated hypercube. We consider this phase transition in percolation on graphs arising from the cartesian product of many graphs and show that, under some mild conditions on the factor graphs, this phenomena is universal.
Joint work with Sahar Diskin, Mihyun Kang and Michael Krivelevich
