Proof of Triangle Condition: The Infrared Bound
We state one of the key results in high-dimensional percolation, the so-called infrared bound that bounds the Fourier transform of the critical two-point function close to its singularity. This key result was first proved by Hara and Slade. We extend the discussion to a slightly modified percolation model known as spread-out percolation that signals the role of the upper-critical dimension more clearly than the usual nearest-neighbor model. We give an overview of the proof of the infrared bound that relies on the lace expansion. This proof stretches over the next three chapters. Finally, we discuss random walk triangles. The main idea is that if the random walk triangle is sufficiently small, then the infrared bound follows.