Lattice random walks and quantum A-period conjecture

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 ${\mathcal{B}}_3$.

21 pages, 2 figures, 2 tables; minor revisions and typo corrections; submitted to SciPost Physics

References in corpus (34)

• Topological Strings and Integrable Hierarchies
• Spectral signatures of many-body localization with interacting photons
• Chern-Simons Theory and Topological Strings
• Les Houches lectures on matrix models and topological strings
• Quantum Geometry of Refined Topological Strings
• Hofstadter butterfly and integer quantum Hall effect in three dimensions
• Topological Strings and Quantum Spectral Problems
• Universality of Hofstadter butterflies on hyperbolic lattices
• Exact quantization conditions for the relativistic Toda lattice
• Hofstadter's Butterfly in Quantum Geometry
• The Colored Hofstadter Butterfly for the Honeycomb Lattice
• Landau levels in the case of two degenerate coupled bands: kagome lattice tight-binding spectrum
• Duality and integer quantum Hall effect in isotropic 3D crystals
• Exact Random Walk Distributions using Noncommutative Geometry
• On the resurgent structure of quantum periods
• Instantons, Topological Strings and Enumerative Geometry
• Calabi-Yau geometry and electrons on 2d lattices
• A Tale of Two Fractals: The Hofstadter Butterfly and The Integral Apollonian Gaskets
• WKB Method and Quantum Periods beyond Genus One
• Exclusion statistics and lattice random walks
• Spectral equations for the modular oscillator
• Instantons in the Hofstadter butterfly: difference equation, resurgence and quantum mirror curves
• The algebraic area of closed lattice random walks
• Linking topological features of the Hofstadter model to optical diffraction figures
• Nests and Chains of Hofstadter Butterflies
• A numerical study of the window condition for Chern numbers of Hofstadter Butterflies
• On the algebraic area of lattice walks and the Hofstadter model
• Combinatorics of generalized Dyck and Motzkin paths
• Bloch electrons on honeycomb lattice and toric Calabi-Yau geometry
• Algebraic area enumeration of random walks on the honeycomb lattice
• Hofstadter-Toda spectral duality and quantum groups
• On the algebraic area of cubic lattice walks
• Hofstadter point spectrum trace and the Almost Mathieu operator
• Recent Developments in Topological String Theory

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