Bootstrap percolation on the high-dimensional Hamming graph

In the random r-neighbour bootstrap percolation process on a graph G, a set of initially infected vertices is chosen at random by retaining each vertex of G independently with probability p (0, 1), and "healthy" vertices get infected in subsequent rounds if they have at least r infected neighbours. A graph G percolates if every vertex becomes eventually infected. A central problem in this process is to determine the critical probability pc (G, r), at which the probability that G percolates passes through one half. In this paper, we study random 2-neighbour bootstrap percolation on the n-dimensional Hamming graph ₈=₁ⁿ Kₖ, which is the graph obtained by taking the Cartesian product of n copies of the complete graph Kₖ on k vertices. We extend a result of Balogh and Bollob\'as Bootstrap percolation on the hypercube, Probab. Theory Related Fields. 134 (2006), no. 4, 624-648. MR2214907 about the asymptotic value of the critical probability pc (Qⁿ, 2) for random 2-neighbour bootstrap percolation on the n-dimensional hypercube Qⁿ=₈=₁ⁿ K₂ to the n-dimensional Hamming graph ₈=₁ⁿ Kₖ, determining the asymptotic value of pc (₈=₁ⁿ Kₖ, 2), up to multiplicative constants (when n), for arbitrary k N satisfying 2 k 2^n.