SciPost Submission Page
Bethe $M$-layer construction for the percolation problem
by Maria Chiara Angelini, Saverio Palazzi, Tommaso Rizzo, Marco Tarzia
Submission summary
| Authors (as registered SciPost users): | Saverio Palazzi · Tommaso Rizzo |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202408_00017v2 (pdf) |
| Date accepted: | 2024-12-13 |
| Date submitted: | 2024-12-02 12:54 |
| Submitted by: | Palazzi, Saverio |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
One way to perform field theory computations for the bond percolation problem is through the Kasteleyn and Fortuin mapping to the $n+1$ states Potts model in the limit of $n \to 0$. In this paper, we show that it is possible to recover the $\epsilon$-expansion for critical exponents in finite dimension directly using the $M$-layer expansion, without the need to perform any analytical continuation. Moreover, we also show explicitly that the critical exponents for site and bond percolation are the same. This computation provides a reference for applications of the $M$-layer method to systems where the underlying field theory is unknown or disputed.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
List of changes
The authors, thanks to the suggestions of the reports, decided to make the following changes:
1) As suggested by the reports, we deleted the Section on the bond percolation case (Section 6 of the version uploaded on 17 August 2024) and we incorporated the discussion of this case in the previous Section, in order to treat site and bond percolation in parallel, since there are few differences between the two cases.
2) Thanks to report #1 we separated Section 5 of the version uploaded on 17 August 2024 (the old one) in two: the new Section 5 is about the results of the $M$-layer construction while the new Section 6 is about a standard RG method used to compute estimates of the critical exponents. Following the suggestion of report #2, at the end of the new Section 5 we also added a paragraph in order to explicitly show how to find that the upper critical dimension for percolation is $D_U=6$, following the same steps done in the same $M$-layer setting, but for a different problem (ref: Phys. Rev. Lett. 128, 075702).
3) We revised Section 3, about the percolation problem on the Bethe lattice, thanks to both the useful comments of the reports. We added few remarks on the cavity method and how it is used for this problem. We added a figure in order to explain the cavity equation written (Eq. (19) of the old version) and the related Eq. for $g(s,p)$ (Eq. (20) of the old version). Moreover, we made some minor corrections, in particular, we corrected the definition of $g(s,p)$ and we added some comments on the divergence of the two-point susceptibility on the Bethe lattice (lines 167 to 180 of the new version).
We also made the following minor corrections:
- corrected the subscript of $C$ in Eq. 5 of the old version;
- added a reference for the symmetry term at line 262 of the new version;
- added the corrections term in Eq. (11) in order to remark the perturbative nature of this relation between $\lambda$ and $u$;
- we defined the notions of "topological loop"
and "topological diagram" at lines 222 and 242 of the new version respectively;
- we corrected the statement done at the end of the conclusions Section (at line 490 of the old version): "To proceed further and obtain the values of the critical exponents in the non-percolating phase" in "To proceed further and obtain the values of the critical exponents in the percolating phase";
- in this new version we always refer to the upper critical dimension with $D_U$, we corrected at line 35 $D_c$ into $D_U$.
Current status:
Editorial decision:
For Journal SciPost Physics: Publish
(status: Editorial decision fixed and (if required) accepted by authors)
Reports on this Submission
Report
I already gave a positive appreciation of the paper in my previous report. The authors made all the changes that I suggested. I recommend publication in the present form.
Recommendation
Publish (meets expectations and criteria for this Journal)