The One-Dimensional Restriction of the Critical Ising Model under Rescaling and Its Stochastic Integral

Abstract

This paper explores the behavior of the one-dimensional restriction of the critical Ising model under rescaling. In particular, using a “locality” lemma, the rescaled sum PN i=1 σi/N7/8 is proven to converge in distribution. A Monte Carlo experiment is conducted to illustrate what the limiting distribution possibly is. The paper also attempts to construct a stochastic integral based on the model considered above. Finally, the paper provides an exposition of a proof of the “locality” lemma.