SciPost Submission Page
Lattice random walks and quantum A-period conjecture
by Li Gan
Submission summary
| Authors (as registered SciPost users): | Li Gan |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2412.21128v2 (pdf) |
| Date accepted: | June 12, 2025 |
| Date submitted: | May 21, 2025, 10:53 a.m. |
| Submitted by: | Li Gan |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| 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
Author comments upon resubmission
We hereby resubmit our manuscript [https://arxiv.org/abs/2412.21128v2], incorporating revisions that address the Referees' comments as well as the editorial suggestions. We are grateful for the Referees' overall positive assessment of our work, and we appreciate their constructive feedback and valuable suggestions. Below, we provide a brief response to each Referee. A detailed list of changes made to the revised manuscript is included under the "List of changes" section in the submission system.
Response to Referee \#1
We thank the Referee for the helpful comments and suggestions. The Referee correctly pointed out that the parameters introduced to enumerate closed paths with different step counts in each direction are redundant and can be reduced. However, we have chosen to retain all variables to make the step counts in each direction explicit. This choice facilitates further discussion, particularly in Section 3, where we connect these parameters to complex moduli in the mirror curve. We have clarified this decision with an explanatory footnote added on Page 4 of the revised manuscript.
The Referee also noted ambiguity in the phrase ``ignoring $g_q$''. We clarify that the condition $e^{i k_x}=-b/c' Q^{-1/2}$ does not eliminate $g_q$, and to disregard the ``spurious'' term, we simply treat $g_q$ as zero. We have revised the manuscript accordingly to reflect this explanation more clearly. Additionally, all scientific typos noted by the Referee have been corrected.
In response to the Referee's suggestion to add a comment on the proof of the conjecture in Section 3, we have revised the Conclusion section. There, we suggest that a quantum analog of the classical Picard–Fuchs differential equation may be key to a potential proof. However, to the best of our knowledge, such a difference equation has not yet been studied and remains an open problem.
Response to Referee \#2
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.
Thank you for your kind attention.
Best regards,
Li
List of changes
- Textual revisions
Page 4: Added a footnote addressing Referee #1's suggestion regarding the redundancy of parameters introduced to enumerate closed paths.
Pages 6, 8: Replaced ignore $g_q$'' withneglect $g_q$ by treating it as zero''.
Page 14: Revised the explanation regarding the quantum counterpart of the classical Picard–Fuchs differential equation; added acknowledgments to the anonymous Referees for their insightful comments and suggestions.
- Corrections of misprints
Page 4: this Hamiltonian $H$ describes the Hofstadter model on a triangular lattice'' $\rightarrow$the Hamiltonian $H_{\text{tri}}$ describes the Hofstadter model on a triangular lattice''.
Pages 6, 8: $e^{i k_x}=b/c' Q^{-1/2}$ $\rightarrow$ $e^{i k_x}=-b/c' Q^{-1/2}$
Page 6: $Z_0=0$ $\rightarrow$ $Z_0=1$
Published as SciPost Phys. 19, 053 (2025)
Reports on this Submission
Report #2 by Anonymous (Referee 1) on 2025-5-27 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2412.21128v2, delivered 2025-05-27, doi: 10.21468/SciPost.Report.11283
Report
Recommendation
Publish (meets expectations and criteria for this Journal)
Author: Li Gan on 2025-05-26 [id 5522]
(in reply to Report 1 on 2025-05-23)We appreciate the Referee's feedback and comments. It is reasonable to treat $k_x$ as a free parameter and leave it as it is, since for low enough $N$, the matrix trace $\mathrm{tr}\,H_{\text{sq}}^N$ does not depend on $k_x$ and reproduces the full trace $\mathbf{Tr}\,H_{\text{sq}}^N$. This means that the several lowest cluster coefficients calculated via the $k_x$-dependent partition function are $k_x$-independent. However, the $k_x$-dependent partition function is complicated by the $k_x$-dependent Wilson loop term appearing in the secular determinant $\det(I-zH_{\text{sq}})$.
In our paper, we simplify the secular determinant by eliminating the Wilson loop term. We choose a $k_x$ such that $e^{ik_x} = − b/c' Q^{−1/2}$, ensuring that $f_q=0$, and we also set $k_y=0$ and $g_q = 0$. With these choices, the Wilson loop term vanishes. Note that, in general, the value of $k_x$ that makes $f_q=0$ does not necessarily make $g_q=0$, so we treat $g_q$ as zero, as stated in the paper. Put simply, we not only remove the two corner elements of $H_{\text{sq}}$, but also fix $k_x$ and $k_y$ to eliminate the Wilson loop term, thereby facilitating the subsequent calculation.
The approach we present in the paper is an interesting one, as it allows the calculation of the secular determinant $\det(I-zH_{\text{sq}})$ in the standard exclusion partition function way. Other approaches to computing $\mathrm{tr}\,H_{\text{sq}}^N$ are certainly possible but would require more effort.