An exact mapping between loop-erased random walks and an interacting field theory with two fermions and one boson

Abstract

We give a simplified proof for the equivalence of loop-erased random walks to a lattice model containing two complex fermions, and one complex boson. This equivalence works on an arbitrary directed graph. Specifying to the $d$-dimensional hypercubic lattice, at large scales this theory reduces to a scalar ${\varphi }^4$-type theory with two complex fermions, and one complex boson. While the path integral for the fermions is the Berezin integral, for the bosonic field we can either use a complex field $\varphi (x)\in \mathbb{C}$ (standard formulation) or a nilpotent one satisfying $\varphi (x{)}^2=0$. We discuss basic properties of the latter formulation, which has distinct advantages in the lattice model.

14 pages