Self-interacting walks, random spin systems, and the zero-component limit

Abstract

Emery's analysis of $n$-component "spin" systems is extended and simplified for the limits $n=0,-2,-4,\dots$. Restrictions on the spin-weighting function implied by an interpretation of the $n\to 0$ limit as a random system are pointed out. Fully explicit relations are also derived for the interpretation of the $n\to 0$ correlation function in terms of self-avoiding and, more generally, weighted self-interacting lattice walks.