Lattice random walks and quantum A-period conjecture
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866908499676495872 |
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| author | Gan, Li |
| author_facet | Gan, Li |
| contents | 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 $\mathcal{B}_3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_21128 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lattice random walks and quantum A-period conjecture Gan, Li Mathematical Physics Statistical Mechanics High Energy Physics - Theory Combinatorics Quantum Physics 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 $\mathcal{B}_3$. |
| title | Lattice random walks and quantum A-period conjecture |
| topic | Mathematical Physics Statistical Mechanics High Energy Physics - Theory Combinatorics Quantum Physics |
| url | https://arxiv.org/abs/2412.21128 |