Algebraic area enumeration for open lattice walks

We calculate the number of open walks of fixed length and algebraic area on a square planar lattice by an extension of the operator method used for the enumeration of closed walks. The open walk area is defined by closing the walks with a straight line across their endpoints and can assume half-inte...

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Bibliographic Details
Published in:Journal of physics. A, Mathematical and theoretical Mathematical and theoretical, 2022-12, Vol.55 (48), p.485005
Main Authors: Ouvry, Stéphane, Polychronakos, Alexios P
Format: Article
Language:English
Subjects:
algebraic area enumeration
Condensed Matter
Dyck paths
High Energy Physics - Theory
Hofstadter model
Mathematical Physics
Physics
quantum torus
random walks
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Summary:We calculate the number of open walks of fixed length and algebraic area on a square planar lattice by an extension of the operator method used for the enumeration of closed walks. The open walk area is defined by closing the walks with a straight line across their endpoints and can assume half-integer values in lattice cell units. We also derive the length and area counting of walks with endpoints on specific straight lines and outline an approach for dealing with walks with fully fixed endpoints.
ISSN:1751-8113
1751-8121
DOI:10.1088/1751-8121/aca573