Random-walk models in the absence of magnetic field

Abstract

Spin systems (such as the Ising model) and polymer models (such as the self-avoiding walk) have long played an important role in the theory of critical phenomena. The analogy between spin systems and random-walk models has intrigued physicists since the 1950’s [498, 208, 206, 197, 135], but the precise relations between these two types of models have emerged only gradually [493, 494, 120, 126, 92, 28]. In recent years, several “artificial” random-walk representations have been introduced as tools with which to study spin systems [92, 5, 8]. The purpose of this chapter (and the next one) is to exhibit an underlying mathematical structure which is common to all random-walk models.