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

by Assaf Shapira, Kay Jörg Wiese

Submission summary

As Contributors: Assaf Shapira · Kay Joerg Wiese
Arxiv Link: (pdf)
Date submitted: 2020-06-16 02:00
Submitted by: Shapira, Assaf
Submitted to: SciPost Physics
Discipline: Physics
Subject area: Statistical and Soft Matter Physics
Approach: Theoretical


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.

Current status:
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Submission & Refereeing History

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Submission 2006.07899v1 on 16 June 2020

Reports on this Submission

Report 1 by Ilya Gruzberg on 2020-8-4 Invited Report


1. The paper contains a new exact relation between a probabilistic object, which is intrinsically non-local (loop erased random walk), and a local field theory. The relation holds for any directed graph.
2. The paper is well written and organized, clearly stating its results and outlining the derivations.
3. The paper is sufficiently compact to be read reasonably quickly in order to grasp the main results.
4. The paper provide enough details to reproduce all the intermediate steps.


I liked reading and refereeing the paper, which is written in a very logical way, and includes sufficient amount of detail for readers to either understand the authors' point right away or to be able to reproduce the necessary steps for themselves.
The paper presents a new and exact relation between loop-erased walks on arbitrary directed graphs and discrete field theories of two complex fermions and one complex boson. Such a relation was known before from the work of one of the authors (and collaborators), where the complex boson involved in the construction was nilpotent. The present paper demonstrates that the nilpotetn boson can be replaced a much more familiar canonical complex bosonic field. The applications and extensions of this work are numerous and partially summarized by the authors at the end of the paper.

Requested changes

1. I have never before encountered nilpotent bosons, and would like the authors to provide some clarifications. I had to go to the previously published paper by T. Helmuth and A. Shapira to make sense of the nilpotent bosons. They turned out to be even members of a Grassman algebra made up by to fermionc fields. However, this is not sufficient to understand why the definition of the functional integral for the nilpotent bosons can be taken as the Berezin integral, literally in parallel with the fermionic degrees of freedom. Therefore, I request that the authors provide an explanation of this choice of the functional integral, and its consequences: what is the result for a Gaussian integral of this form, and what is the statement of the Wick's theorem for the nilpotent bosons?

  • validity: top
  • significance: high
  • originality: top
  • clarity: high
  • formatting: excellent
  • grammar: excellent

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