Triangle condition for oriented percolation in high dimensions
By verifying the triangle condition, \textit{T. Hara} and \textit{G. Slade} [Commun. Math. Phys. 128, No. 2, 333-391 (1990; Zbl 0698.60100)] proved mean-field behaviour of the percolation model both for the spread-out model in seven or more dimensions, and for the usual nearest-neighbour model in sufficiently many dimensions. The corresponding program is reported in the current paper, in the case of oriented percolation. This is achieved by a modification of the arguments of Hara and Slade. In the case of oriented percolation, the upper critical dimension is thought to be \(4 + 1\), and this is supported by the proof of mean-field behaviour for the spread-out model on \(\mathbb{Z}^ 5 \times \mathbb{Z}\).
scientific article; zbMATH DE number 528622