The contact process on a finite set
Let \((\zeta_ t^ N)\) denote the basic contact process on \(\{\) 1,...,N\(\}\) where, at time \(t=0\), all sites are occupied. More precisely, \((\zeta_ t^ N)\) is a Markov process with state space equal to the family of subsets of \(\{1,...,N\}\), having a q-matrix given by \[ q(A, A\setminus \{x\})=1,\quad x\in A;\quad q(A, A\cup \{x\})=\lambda | A\cap \{x-1,x+1\}|,\quad x\not\in A, \] (\(| B|\) denoting the cardinality of a set B). Let \(\sigma_ N=\inf \{t:\zeta^ N_ t=\emptyset \}\) (note that \(\emptyset\) is an absorbing state). The authors prove e.g. the somewhat surprising result that there occurs a ``phase transition'' when \(N\to \infty\). More precisely, there exists a critical value \(0<\lambda_ c<\infty\) for \(\lambda\) such that, as \(N\to \infty\), \[ \sigma_ N/\log N\to 1/\gamma_ 1(\lambda)\quad in\quad probability, \] if \(\lambda <\lambda_ c\) and, for certain constants \(0<\alpha_ 2(\lambda)\leq \beta_ 2(\lambda)<\infty\) and all \(\epsilon >0\), \[ P(\alpha (\lambda)/2-\epsilon \leq (\log \sigma_ N)/N\leq \beta (\lambda)+\epsilon)\to 1,\;if\;\lambda >\lambda_ c. \] The authors show that a similar phase transition also occurs for the biased voter model on \(\{1,...,N\}\).
scientific article; zbMATH DE number 4056747