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Article Dans Une Revue SciPost Physics Année : 2020

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

Résumé

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 ddd-dimensional hypercubic lattice, at large scales this theory reduces to a scalar ϕ4ϕ4\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 ϕ(x)∈Cϕ(x)∈C\phi(x)\in \mathbb C (standard formulation) or a nilpotent one satisfying ϕ(x)2=0ϕ(x)2=0\phi(x)^2 =0. We discuss basic properties of the latter formulation, which has distinct advantages in the lattice model.

Dates et versions

hal-02988816 , version 1 (01-07-2020)
hal-02988816 , version 2 (04-11-2020)

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Assaf Shapira, Kay Joerg Wiese. An exact mapping between loop-erased random walks and an interacting field theory with two fermions and one boson. SciPost Physics, 2020, 9 (5), pp.063. ⟨10.21468/SciPostPhys.9.5.063⟩. ⟨hal-02988816v2⟩
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