Contact processes with random connection weights on regular graphs

In this paper we study the asymptotic critical value of contact processes with random connection weights, sitting on a degree-increasing sequence of r-regular graph Gr. We propose a method to generalize the asymptotics results for λc(Zd) and λc(Td) of classical contact processes as well as of recent work for contact processes on complete graphs with random connection weights. Only the lower bound is rigorously proved; it is conjectured, however, that the lower bound gives the right asymptotic behavior. For comparison purposes we also introduce binary contact path processes with random connection weights, whose asymptotic behavior of the critical value is obtained.