SciPost Phys. 9, 063 (2020) ·
published 4 November 2020
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.