Contact process on regular tree with random vertex weights

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 x and y at rate lambda rho(x)rho(y) for some lambda > 0, where {rho(x), x is an element of T-d} are independent and identically distributed (i.i.d.) vertex weights. We show that when d is large enough, there is a phase transition at lambda(c)(d) is an element of (0,infinity) such that for lambda < lambda(c)(d), the contact process dies out, and for lambda > lambda(c)(d), the contact process survives with a positive probability. Moreover, we also show that there is another phase transition at lambda(e)(d) such that for lambda < lambda(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 d increases.