The algebraic area of closed lattice random walks

Abstract

We propose a formula for the enumeration of closed lattice random walks of length $n$ enclosing a given algebraic area. The information is contained in the Kreft coefficients which encode, in the commensurate case, the Hofstadter secular equation for a quantum particle hopping on a lattice coupled to a perpendicular magnetic field. The algebraic area enumeration is possible because it is split in $2^{n/2-1}$ pieces, each tractable in terms of explicit combinatorial expressions.

20 pages, 1 figure, misprints corrected, one section in the Appendix added

References in corpus (1)

• Hofstadter point spectrum trace and the Almost Mathieu operator

Cited by in corpus (11)

• Exclusion statistics and lattice random walks
• Combinatorics of generalized Dyck and Motzkin paths
• Algebraic area enumeration for open lattice walks
• Exclusion statistics for particles with a discrete spectrum
• Hamiltonian and exclusion statistics approach to discrete forward-moving paths
• Algebraic area enumeration of random walks on the honeycomb lattice
• Combinatorics of Multicompositions
• Length and area generating functions for height-restricted Motzkin meanders
• On the algebraic area of cubic lattice walks
• Lattice random walks and quantum A-period conjecture
• Lattice walk area combinatorics, some remarkable trigonometric sums and Apéry-like numbers