Aizenman and Newman introduced an unverified condition, the triangle condition, which has been shown to imply that a number of percolation critical exponents take their mean field values, and which is expected to hold above six dimensions for nearest neighbour percolation. We prove that the triangle condition is satisfied in sufficiently high dimensions for the nearest neighbour model, and above six dimensions for a class of “spread-out” models. The proof uses an expansion which is related to the lace expansion for self-avoiding walk.
All Science Journal Classification (ASJC) codes
- Applied Mathematics