Supercritical behaviors in first-passage percolation
The paper obtains some nice results on the existence of routes in standard first passage percolation on \(Z^d\) where the nonnegative random variables \(X(e)\) \((e\) an edge of \(Z^d)\) are i.i.d. with distribution function \(F\). Let \(a_{0,n}\) denote the first passage time from the origin to \((n,0, \dots,0)\), and denote by \(b_{0,n}\) the first passage time from the origin to the set \(\{(n, k_2, \dots, k_d) \in Z^d : k_2, \dots, k_d \in Z\}\). A path \(\gamma\) is called a route for \(a_{0,n}\) \((b_{0,n})\) if the passage time of \(\gamma\) equals \(a_{0,n}\) \((b_{0, n})\). It is shown in the case \(F(0) > p_c(d)\) (where \(p_c (d)\) denotes the critical probability for Bernoulli (bond) percolation on \(Z^d)\): There exist routes for both \(a_{0,n}\) and \(b_{0,n}\). On the other hand there exists a random variable \(\rho (F) \geq 0\) such that \(E [(\rho (F))^k] < \infty\) if \(E [(X(e))^k] < \infty\) for some \(k \geq 1\), and, as \(n \to \infty\), \(E [a_{0,n}] \to 2E [\rho (F)]\), \(E [b_{0,n}] \to E [\rho(F)]\).
scientific article; zbMATH DE number 828753