Mathematics – Probability
Scientific paper
2010-11-06
Mathematics
Probability
30 pages
Scientific paper
We consider the discrete time threshold-two contact process on a random r-regular graph on n vertices. In this process, a vertex with at least two occupied neighbors at time t will be occupied at time t+1 with probability p, and vacant otherwise. We use a suitable isoperimetric inequality to show that if r is larger than 3 and p is close enough to 1, then starting from all vertices occupied, there is a positive density of occupied vertices up to time exp(c(p) n) for some positive constant c(p). In the other direction, another appropriate isoperimetric inequality allows us to show that there is a decreasing function epsilon(p) and a constant C_0(p) := 2/log(2/(1+p)) so that if the number of occupied vertices in the initial configuration is at most epsilon(p)n, then with high probability all vertexs are vacant at time C_0(p) log n. These two conclusions imply that the density of occupied vertexs in the quasi-stationary distribution (defined in the paper) is discontinuous at the critical probability p_c, which is in the interval (0,1).
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