Self-avoiding walks and polygons on hyperbolic graphs

Christoforos Panagiotis

doi:10.1002/jgt.23087

Self-avoiding walks and polygons on hyperbolic graphs

Christoforos Panagiotis

Department of Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

Abstract

We prove that for the (Formula presented.) -regular tessellations of the hyperbolic plane by (Formula presented.) -gons, there are exponentially more self-avoiding walks of length (Formula presented.) than there are self-avoiding polygons of length (Formula presented.). We then prove that this property implies that the self-avoiding walk is ballistic, even on an arbitrary vertex-transitive graph. Moreover, for every fixed (Formula presented.), we show that the connective constant for self-avoiding walks satisfies the asymptotic expansion (Formula presented.) as (Formula presented.); on the other hand, the connective constant for self-avoiding polygons remains bounded. Finally, we show for all but two tessellations that the number of self-avoiding walks of length (Formula presented.) is comparable to the (Formula presented.) th power of their connective constant. Some of these results were previously obtained by Madras and Wu for all but finitely many regular tessellations of the hyperbolic plane.

Original language	English
Pages (from-to)	435-473
Number of pages	39
Journal	Journal of Graph Theory
Volume	106
Issue number	3
Early online date	19 Feb 2024
DOIs	https://doi.org/10.1002/jgt.23087
Publication status	Published - 31 Jul 2024

Data Availability Statement

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Funding

I would like to thank John Haslegrave and Agelos Georgakopoulos for their comments on a preliminary version of the current paper. I would also like to thank Tom Hutchcroft for useful discussions. This research is supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 639046).

Funders	Funder number
European Research Council
Horizon 2020	639046
Horizon 2020

Keywords

connective constant
hyperbolic graph
planar graph
self-avoiding polygon
self-avoiding walk

ASJC Scopus subject areas

Discrete Mathematics and Combinatorics
Geometry and Topology

Access to Document

10.1002/jgt.23087Licence: CC BY

Cite this

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Self-avoiding walks and polygons on hyperbolic graphs

Abstract

Data Availability Statement

Funding

Keywords

ASJC Scopus subject areas

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