We derive a lace expansion for the survival probability for critical spread-out oriented percolation above 4+1 dimensions, i.e., the probability ¿n that the origin is connected to the hyperplane at time n, at the critical threshold pc. Our lace expansion leads to a non-linear recursion relation for ¿n, with coefficients that we bound via diagrammatic estimates. This lace expansion is for point-to-plane connections and differs substantially from previous lace expansions for point-to-point connections. In particular, to be able to deduce the asymptotics of ¿n for large n, we need to derive the recursion relation up to quadratic order. The present paper is Part II in a series of two papers. In Part I, we use the recursion relation and the diagrammatic estimates to prove that limn¿8n¿n=1/B(0,8), and also deduce consequences of this asymptotics for the geometry of large critical clusters and for the incipient infinite cluster.
|Annales de l'institut Henri Poincare (B): Probability and Statistics
|Published - 2007