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Lee-Yang zeros of the antiferromagnetic Ising model

Part of: Complex dynamical systems Graph theory

Published online by Cambridge University Press:  08 April 2021

FERENC BENCS ,
PJOTR BUYS ,
LORENZO GUERINI  and
HAN PETERS
FERENC BENCS
Affiliation:
Alfréd Rényi Institute of Mathematics, Reáltanoda u. 13–15, 1053Budapest, Hungary Department of Mathematics, Central European University, Nádor u. 9, 1051Budapest, Hungary Korteweg de Vries Institute for Mathematics, University of Amsterdam, Science Park 107, 1090GEAmsterdam, The Netherlands (e-mail: ferenc.bencs@gmail.com, pjotr.buys@gmail.com, hanpeters77@gmail.com)
PJOTR BUYS
Affiliation:
Korteweg de Vries Institute for Mathematics, University of Amsterdam, Science Park 107, 1090GEAmsterdam, The Netherlands (e-mail: ferenc.bencs@gmail.com, pjotr.buys@gmail.com, hanpeters77@gmail.com)
LORENZO GUERINI*
Affiliation:
Korteweg de Vries Institute for Mathematics, University of Amsterdam, Science Park 107, 1090GEAmsterdam, The Netherlands (e-mail: ferenc.bencs@gmail.com, pjotr.buys@gmail.com, hanpeters77@gmail.com)
HAN PETERS
Affiliation:
Korteweg de Vries Institute for Mathematics, University of Amsterdam, Science Park 107, 1090GEAmsterdam, The Netherlands (e-mail: ferenc.bencs@gmail.com, pjotr.buys@gmail.com, hanpeters77@gmail.com)
*
e-mail: lorenzo.guerini92@gmail.com
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Abstract

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We investigate the location of zeros for the partition function of the anti-ferromagnetic Ising model, focusing on the zeros lying on the unit circle. We give a precise characterization for the class of rooted Cayley trees, showing that the zeros are nowhere dense on the most interesting circular arcs. In contrast, we prove that when considering all graphs with a given degree bound, the zeros are dense in a circular sub-arc, implying that Cayley trees are in this sense not extremal. The proofs rely on describing the rational dynamical systems arising when considering ratios of partition functions on recursively defined trees.

Keywords

Ising modelLee–Yang zeroscomplex dynamics

MSC classification

Primary: 05C31: Graph polynomials 37F10: Polynomials; rational maps; entire and meromorphic functions
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 7 , July 2022 , pp. 2172 - 2206
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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