Kotani's Theorem and the Lace Expansion

Gordon Slade

Published 2020-06-11Version 1

In 1991, Shinichi Kotani proved a theorem giving a sufficient condition to conclude that a function $f(x)$ on ${\mathbb Z}^d$ decays like $|x|^{-(d-2)}$ for large $x$, assuming that its Fourier transform $\hat f(k)$ is such that $|k|^{2}\hat f(k)$ is well behaved for $k$ near zero. We prove an extension of Kotani's Theorem and combine it with the lace expansion to give a simple proof that the critical two-point function for weakly self-avoiding walk has decay $|x|^{-(d-2)}$ in dimensions $d>4$.