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arXiv · 2006.07899

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

Assaf Shapira·Kay Jörg Wiese
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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.

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Assaf Shapira, Kay Jörg Wiese. 2020-06-14. An exact mapping between loop-erased random walks and an interacting field theory with two fermions and one boson. https://doi.org/10.21468/scipostphys.9.5.063

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