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We consider critical spread-out oriented percolation above 4 + 1 dimensions. Our main result is that the extinction probability at time n (i.e., the probability for the origin to be connected to the hyperplane at time n but not to the hyperplane at time n + 1) decays like 1/Bn 2 as $$n\to\infty$$, where B is a finite positive constant. This in turn implies that the survival probability at time n (i.e., the probability that the origin is connected to the hyperplane at time n) decays like 1/Bn as $$n\to\infty$$. The latter has been shown in an earlier paper to have consequences for the geometry of large critical clusters and for the incipient infinite cluster. The present paper is Part I in a series of two papers. In Part II, we derive a lace expansion for the survival probability, adapted so as to deal with point-to-plane connections. This lace expansion leads to a nonlinear recursion relation for the survival probability. In Part I, we use this recursion relation to deduce the asymptotics via induction.

The survival probability for critical spread-out oriented percolation above 4 + 1 dimensions. I. Induction