SciPost logo

SciPost Submission Page

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

This Submission thread is now published as

Submission summary

Authors (as registered SciPost users): Assaf Shapira · Kay Joerg Wiese
Submission information
Preprint Link:  (pdf)
Date accepted: 2020-10-29
Date submitted: 2020-10-05 11:18
Submitted by: Shapira, Assaf
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • 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.

Author comments upon resubmission

We would first like to thank both referees for the detailed reviews.

The points raised in the second report concerning specific corrections were addressed as suggested by the referee.

As requested in the first report, we added details on the interpretation of the nilpotent bosons as even members of the Grassmann algebra and how this relates to the Berezin integral. The discussion on the physical and combinatorial meaning of these nilpotent bosons appears in section 5 and is summarized in section 7.

List of changes

1) • p. 3, Example (10): Clarify that the graph is assumed to contain no other directed edges except between x and y (or perhaps a milder but similar assumption). Perhaps the authors in fact thought of the graph consisting only of these two vertices, which should then be stated clearly.

Response: Clarification added.

2) • p. 3, Definition of loop: The text defines loops as paths whose first and last vertices are equal and all other vertices are distinct. For the following combinatorics to work out correctly, this definition should be modified so that any two paths related by a cyclic permutation of vertices are considered the same loop. A possible terminology for this is (oriented) unrooted loops, as opposed to rooted loops.

Response: The definition was modified

3) • p. 3, Definition of Lγ: Sentence: “We define the set Lγ to be the collections of…” -> “We define Lγ to be the set of collections of…” or “We define the set Lγ to consist of the collections of…”.

Response: Changed as suggested by the referee.

4) • p. 4, Paragraph of Eqn. (17)-(18) or of Eqn. (16): It is worth emphasizing that in the prescription, whenever a loop is transferred from the walk to the collection of loops, the disjointness of the loop collection is indeed preserved.

Response: Clarification was added.

5) • p. 5, Paragraph of Eqn. (21): The terminology is otherwise accurate in referring to “pairs of complex conjugate fields”, except at the first mention of them. Change: “one pair (ϕ,ϕ∗) of complex fermionic fields” -> “one pair (ϕ,ϕ∗) of complex conjugate fermionic fields”.

Response: Fixed according to the referee's suggestion.

6) • p. 6, Definition of “graded commutation relations”: The intended meaning of Eqn. (24) and (25) can probably be guessed correctly, but its current statement is not sufficiently clear. Specifically, (24) should be imposed for ϕ(x)ψ(y) for any x,y, whereas (25) should only be imposed for ϕ(x)ϕ(y) when x=y.

Response: We rewrote the relations more explicitly.

7) • p. 6, Eqn (28): One further pair of parentheses would avoid any potential misunderstanding about whether the product ∏i is inside the product ∏x (as is should be), or whether the two are separate.

Response: Added parentheses.

8) • p. 7, Paragraph of Eqn (30): The most interesting observable that the authors calculate in the field theory is U(a,b,c), proportional to the probability that the loop-erased random walk from a to c goes via b. It appears very natural to ask about a generalization: does the field theory at least in principle also allow for the calculation of the probability that the loop-erased random walk from a to c goes via all of b1,…,bn? If the answer is no, the field theory formulation is still quite interesting, but it seems appropriate to briefly admit that there are questions about loop-erased random walks that can not be addressed in the theory. If the answer is yes, I recommend mentioning this interesting generalization and giving a brief hint for how it is done.

Response: We added a paragraph after equation (35), explaining how our technique could be adapted in order to find higher-order observables.

9) • p. 9, Paragraph of Eqn. (42): “standard bosonic fields” -> “standard pair of complex conjugate bosonic fields”.

Response: Fixed as suggested by the referee.

10) • p. 11, Last paragraph before Section 7: “on a 3-regular graph as the honeycomb lattice” -> “on a 3-regular graph such as the honeycomb lattice”.

Response: Fixed as suggested by the referee.

Published as SciPost Phys. 9, 063 (2020)

Reports on this Submission

Report 2 by Ilya Gruzberg on 2020-10-20 (Invited Report)


I am satisfied with the response of the authors to referees. I believe the paper can be published in tis present form

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Anonymous Report 1 on 2020-10-20 (Invited Report)


The authors have addressed all the issues in my earlier report appropriately. I recommend this interesting article for publication in SciPost Physics.

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

Login to report or comment