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 ᵢ₌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\'{a}s [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=ᵢ₌₁ⁿ K₂ to the n-dimensional Hamming graph ᵢ₌₁ⁿ Kₖ, determining the asymptotic value of pc(ᵢ₌₁ⁿ Kₖ,2), up to multiplicative constants (when n → ∞), for arbitrary k ∈ N satisfying 2 ≤ k≤ 2√n.