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

TL;DR

This paper establishes a simplified proof linking loop-erased random walks to a lattice field theory with two fermions and one boson, applicable to arbitrary directed graphs and reducing to a scalar  theory at large scales.

Contribution

It provides a simplified proof of the equivalence between loop-erased random walks and a specific interacting field theory, extending to arbitrary directed graphs and exploring novel bosonic field formulations.

Findings

Equivalent lattice model with two fermions and one boson for loop-erased random walks

Reduction to a scalar  theory in large-scale hypercubic lattices

Introduction of a nilpotent bosonic field formulation with advantages in lattice models

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 $\phi^4$-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 $\phi(x)\in \mathbb C$ (standard formulation) or a nilpotent one satisfying $\phi(x)^2 =0$. We discuss basic properties of the latter formulation, which has distinct advantages in the lattice model.