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Lattice random walks and quantum A-period conjecture
by Li Gan
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Li Gan |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2412.21128v1 (pdf) |
| Date submitted: | Feb. 7, 2025, 3:45 p.m. |
| Submitted by: | Li Gan |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We derive explicit closed-form expressions for the generating function $C_N(A)$, which enumerates classical closed random walks on square and triangular lattices with $N$ steps and a signed area $A$, characterized by the number of moves in each hopping direction. This enumeration problem is mapped to the trace of powers of anisotropic Hofstadter-like Hamiltonian and is connected to the cluster coefficients of exclusion particles: exclusion strength parameter $g = 2$ for square lattice walks, and a mixture of $g = 1$ and $g = 2$ for triangular lattice walks. By leveraging the intrinsic link between the Hofstadter model and high energy physics, we propose a conjecture connecting the above signed area enumeration $C_N(A)$ in statistical mechanics to the quantum A-period of associated toric Calabi-Yau threefold in topological string theory: square lattice walks correspond to local $\mathbb{F}_0$ geometry, while triangular lattice walks are associated with local $\mathcal{B}_3$.
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
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2025-4-28 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2412.21128v1, delivered 2025-04-28, doi: 10.21468/SciPost.Report.11096
Strengths
Report
Furthermore, the author proposed a very intriguing connection between the generating function of closed random walks and the quantum A-periods of toric Calabi-Yau threefolds, which is definitely worth further looking into.
I recommend publication of these quite novel results. There are some scientific typos that should be corrected though.
Requested changes
1- On page 6 on the second line above the second equation, $e^{i k_x} = b/c' Q^{-1/2}$ should be corrected to $e^{i k_x} = - b/c' Q^{-1/2}$. 2- On the same page, in the fourth equation, $Z_0$ should be 1 instead of 0.
Recommendation
Ask for minor revision
Report #1 by Anonymous (Referee 1) on 2025-3-24 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2412.21128v1, delivered 2025-03-24, doi: 10.21468/SciPost.Report.10891
Report
This papers derives expressions for the number of closed walks on a triangular lattice with given area and number of steps in each direction, and presents a conjecture on the relation of the quantum A-period of Calabi-Yau threefolds and lattice walks. The paper is interesting and it should be published. I have, nevertheless, a few remarks.
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The parameters a, a',b,b',c,c' introduced to enumerate closed paths of various numbers of steps in each direction seem redundant: since only closed walks are considered, the number of steps in each direction are restricted. This is born out by the fact that in all formulas for the square lattice only the products aa' and bb' appear, and in all formulas for the triangular lattice only aa', bb', cc' and abc appear. It would be simpler to use the minimal number of parameters by taking, say, a=1 and b=1.
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In p.6, to eliminate the corner elements of the matrix it is stated that "we set (...) and ignore g_q". I do not understand what "ignoring g_q" means: it either vanishes or it does not, and here it does not, so the second term in the parenthesis of the "spurious" term does not vanish. Also, I believe the relation (...) for the vanishing of f_q is missing a minus sign. Similar remarks apply to the beginning of section 2.3.
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The conjecture of section 3, eq. (11), is definitely interesting. Given the close connection of the Hofstadter-like Hamiltonian and the one for the mirror curve, it should be possible to actually prove the conjecture, and I am a bit surprised that the author did not push the analysis to provide such a proof. A comment on this in the paper would be useful.
The paper can be published after the author had a chance to consider the above points and make any related modifications.
Recommendation
Ask for minor revision
We thank the Referee for the helpful comments and suggestions.
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We agree that the introduced parameters are redundant and can be reduced, for example, by setting $a=b=1$ for both closed square and triangular lattice walks. However, we prefer to retain them to make the step counts in each direction explicit. This facilitates later discussion, particularly in Section 3, where we connect them to complex moduli in the mirror curve, e.g., $(b,b',c,c')=(1,1,R^2,R^2)$ on Page 13. A clarifying footnote has been added on Page 4.
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We agree the phrase "ignoring $g_q$" was unclear, and the condition $e^{i k_x}=-b/c' Q^{-1/2}$ does not eliminate $g_q$. Here, we simply set $g_q=0$ by hand so that the spurious term disappears. We have revised the wording accordingly. The missing minus sign and other typos have also been corrected.
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In response to the Referee's suggestion, we have added a comment in the Conclusion noting that a quantum version of the Picard-Fuchs equation may be key to a proof of the conjecture, though to our knowledge this difference equation has not yet been explored.
Author: Li Gan on 2025-05-21 [id 5499]
(in reply to Report 2 on 2025-04-28)We are grateful to the Referee for the kind remarks regarding the content of our paper. We have carefully reviewed the manuscript and corrected all misprints identified.