Contact process on regular tree with random vertex weights

Abstract

This paper is concerned with the contact process with random vertex weights on regular trees, and studies the asymptotic behavior of the critical infection rate as the degree of the trees increasing to infinity. In this model, the infection propagates through the edge connecting vertices xand yat rate λρ(x)ρ(y) for some λ>0,where {ρ(x), x∈T^d} are independent and identically distributed (i.i.d.) vertex weights. We show that when dis large enough, there is a phase transition at λ_c(d) ∈ (0,∞) such that for λ<λ_c (d),the contact process dies out, and for λ>λ_c(d),the contact process survives with a positive probability. Moreover, we also show that there is another phase transition at λ_e(d) such that for λ<λ_e(d),the contact process dies out at an exponential rate. Finally, we show that these two critical values have the same asymptotic behavior as dincreases.

Key words： Contact process; random vertex weights; critical value; asymptotic behavior

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