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Universality of closed nested paths in two-dimensional percolation

by Yu-Feng Song, Jesper Lykke Jacobsen, Bernard Nienhuis, Andrea Sportiello, Youjin Deng

Submission summary

Authors (as registered SciPost users): Jesper Lykke Jacobsen · Bernard Nienhuis
Submission information
Preprint Link: scipost_202312_00045v1  (pdf)
Date submitted: 2023-12-26 10:30
Submitted by: Nienhuis, Bernard
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Statistical and Soft Matter Physics
Approaches: Theoretical, Computational


Recent work on percolation in $d=2$~[J.~Phys.~A {\bf 55} 204002] introduced an operator that gives a weight $k^{\ell}$ to configurations with $\ell$ `nested paths' (NP), i.e.\ disjoint cycles surrounding the origin, if there exists a cluster that percolates to the boundary of a disc of radius $L$, and weight zero otherwise. It was found that $\mathbb{E}(k^{\ell}) \sim L^{-\X{np}(k)}$, and a formula for $\X{np}(k)$ was conjectured. Here we derive an exact result for $\X{np}(k)$, valid for $k \ge -1$, replacing the previous conjecture. We find that the probability distribution $\mathbb{P}_\ell (L)$ scales as $ L^{-1/4} (\ln L)^\ell [(1/\ell!) \Lambda^\ell]$ when $\ell \geq 0$ and $L \gg 1$, with $\Lambda = 1/\sqrt{3} \pi$. Extensive simulations for various critical percolation models confirm our theoretical predictions and support the universality of the NP observables.

Current status:
In refereeing

Reports on this Submission

Anonymous Report 1 on 2024-5-18 (Invited Report)


1) The present work studies a natural operator for 2D percolation, defined in terms of nested paths. The questions studied are both original and well motivated.
2) Several interesting properties are established or observed. This should stimulate further investigation (and it already has, see my comment below).
3) The numerical work is performed very carefully, providing remarkably accurate results for various questions.




This paper builds on earlier work [34] to study a natural operator for critical percolation in dimension two, defined in terms of nested paths. The authors both develop a better theoretical understanding, and perform thorough simulations.

In particular:
1) They check universality of the critical behavior for the nested-path operator.
2) They point out an interesting (and somewhat unexpected) behavior around the special value $k_c = -1$, which seems to display properties which are reminiscent of a BKT transition. However, there does not seem to be any theoretical reason for this at the moment, and it would certainly be interesting to examine this regime further.
3) Finally, the individual probabilities $\mathbb{P}_\ell$ that there exist $\ell$ disjoint closed nested paths, $\ell \geq 0$, are analyzed.

The paper is written very carefully, and raises intriguing questions. I strongly recommend it for publication.

Requested changes

I do not have any substantial request, since as I said, the paper is well written. I would only suggest to mention the recent preprint "Boundary touching probability and nested-path exponent for non-simple CLE" by Morris Ang, Xin Sun, Pu Yu, Zijie Zhuang (, which was posted after the present paper was submitted (and was actually inspired in part by it, see its abstract).

l.395: For clarity, I suggest to add "exactly" $l$ closed...

I also noted a few minor typos: "mostly probably" (l. 482), "enter" (l. 507), "respective" (l. 557).


Publish (surpasses expectations and criteria for this Journal; among top 10%)

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

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