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Lattice random walks and quantum A-period conjecture
by Li Gan
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: | 2025-02-07 15:45 |
| Submitted by: | Gan, Li |
| 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 CN(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 CN(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 F0 geometry, while triangular lattice walks are associated with local B3.
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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.
1. 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.
2. 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.
3. 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.
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