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Zeros, chaotic ratios and the computational complexity of approximating the independence polynomial

Part of: Graph theory Theory of computing Complex dynamical systems Equilibrium statistical mechanics

Published online by Cambridge University Press:  24 November 2023

DAVID DE BOER ,
PJOTR BUYS ,
LORENZO GUERINI ,
HAN PETERS  and
GUUS REGTS
DAVID DE BOER
Affiliation:
Korteweg de Vries Institute for Mathematics, University of Amsterdam, P.O. Box 94248 1090 GE Amsterdam, The Netherlands. e-mails: fdaviddeboer2795@gmail.com, pjotr.buys@gmail.com, lorenzo.guerini92@gmail.com, hanpeters77@gmail.com, guusregtsg@gmail.com
PJOTR BUYS
Affiliation:
Korteweg de Vries Institute for Mathematics, University of Amsterdam, P.O. Box 94248 1090 GE Amsterdam, The Netherlands. e-mails: fdaviddeboer2795@gmail.com, pjotr.buys@gmail.com, lorenzo.guerini92@gmail.com, hanpeters77@gmail.com, guusregtsg@gmail.com
LORENZO GUERINI
Affiliation:
Korteweg de Vries Institute for Mathematics, University of Amsterdam, P.O. Box 94248 1090 GE Amsterdam, The Netherlands. e-mails: fdaviddeboer2795@gmail.com, pjotr.buys@gmail.com, lorenzo.guerini92@gmail.com, hanpeters77@gmail.com, guusregtsg@gmail.com
HAN PETERS
Affiliation:
Korteweg de Vries Institute for Mathematics, University of Amsterdam, P.O. Box 94248 1090 GE Amsterdam, The Netherlands. e-mails: fdaviddeboer2795@gmail.com, pjotr.buys@gmail.com, lorenzo.guerini92@gmail.com, hanpeters77@gmail.com, guusregtsg@gmail.com
GUUS REGTS
Affiliation:
Korteweg de Vries Institute for Mathematics, University of Amsterdam, P.O. Box 94248 1090 GE Amsterdam, The Netherlands. e-mails: fdaviddeboer2795@gmail.com, pjotr.buys@gmail.com, lorenzo.guerini92@gmail.com, hanpeters77@gmail.com, guusregtsg@gmail.com
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Abstract

The independence polynomial originates in statistical physics as the partition function of the hard-core model. The location of the complex zeros of the polynomial is related to phase transitions, and plays an important role in the design of efficient algorithms to approximately compute evaluations of the polynomial.

In this paper we directly relate the location of the complex zeros of the independence polynomial to computational hardness of approximating evaluations of the independence polynomial. We do this by moreover relating the location of zeros to chaotic behaviour of a naturally associated family of rational functions; the occupation ratios.

MSC classification

Primary: 37F10: Polynomials; rational maps; entire and meromorphic functions 05C31: Graph polynomials
Secondary: 68Q17: Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) 82B30: Statistical thermodynamics
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 176 , Issue 2 , March 2024 , pp. 459 - 494
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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