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We give a simplified proof for the equivalence of loop-erased random walks toa lattice model containing two complex fermions, and one complex boson. Thisequivalence works on an arbitrary directed graph. Specifying to the$d$-dimensional hypercubic lattice, at large scales this theory reduces to ascalar $\phi^4$-type theory with two complex fermions, and one complex boson.While the path integral for the fermions is the Berezin integral, for thebosonic 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 discussbasic properties of the latter formulation, which has distinct advantages inthe lattice model.

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