A first order phase transition in the threshold theta %26gt;= 2 contact process on random r-regular graphs and r-trees

作者：Chatterjee Shirshendu; Durrett Rick^*

来源：Stochastic Processes and Their Applications, 2013, 123(2): 561-578.

DOI：10.1016/j.spa.2012.10.001

• Threshold contact process
• Random regular graphs
• Isoperimetric inequality
• First order phase transition
• Binomial large deviations

We consider the discrete time threshold-theta contact process on a random r-regular graph. We show that if theta %26gt;= 2, r %26gt;= theta + 2, epsilon(1) is small and p %26gt;= p(1) (epsilon(1)), then starting from all vertices occupied the fraction of occupied vertices is %26gt;= 1 - 2 epsilon(1) up to time exp(gamma 1(r)n) with high probability. We also show that for p(2) %26lt; 1 there is an epsilon(2)(p(2)) %26gt; 0 so that if p %26lt;= p(2) and the initial density is %26lt;=epsilon(2)(p(2)), then the process dies out in time O (log n). These results imply that the process on the r-tree has a first-order phase transition. C) 2012 Elsevier B.V.

• 出版日期2013-2