Jack Hanson

Chemical distances and k-point functions in high-dimensional percolation

In 1984, Aizenman and Newman conjectured that k-point functions in high-dimensional critical percolation should behave as "simple combinations of the two-point function" governed by tree diagrams resembling those of a phi^3 field theory. We prove this conjecture. We also establish an asymptotic distributional law for the intrinsic or "chemical" distance in large critical clusters.