SciPost Physics

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

• doi: 10.21468/SciPostPhys.9.5.063
• pdf
• BiBTeX
• RIS
• Submissions/Reports
• 

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.

• Semeikin et al., Roughness and critical force for depinning at 3-loop order
  Phys. Rev. B 109, 134203 (2024) [Crossref]
• Helmuth et al., Loop-Erased Random Walk as a Spin System Observable
  J Stat Phys 181, 1306 (2020) [Crossref]
• Nahum et al., Renormalization group for measurement and entanglement phase transitions
  Phys. Rev. B 108, 104203 (2023) [Crossref]
• Wiese, Theory and experiments for disordered elastic manifolds, depinning, avalanches, and sandpiles
  Rep. Prog. Phys. 85, 086502 (2022) [Crossref]