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Polynomial-time approximation algorithms for the antiferromagnetic Ising model on line graphs

Published online by Cambridge University Press:  12 April 2021

Martin Dyer
Affiliation:
School of Computing, University of Leeds, Leeds LS2 9JT, UK
Marc Heinrich
Affiliation:
School of Computing, University of Leeds, Leeds LS2 9JT, UK
Mark Jerrum*
Affiliation:
School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, LondonE1 4NS, UK
Haiko Müller
Affiliation:
School of Computing, University of Leeds, Leeds LS2 9JT, UK
*
*Corressponding author. Email: m.jerrum@qmul.ac.uk

Abstract

We present a polynomial-time Markov chain Monte Carlo algorithm for estimating the partition function of the antiferromagnetic Ising model on any line graph. The analysis of the algorithm exploits the ‘winding’ technology devised by McQuillan [CoRR abs/1301.2880 (2013)] and developed by Huang, Lu and Zhang [Proc. 27th Symp. on Disc. Algorithms (SODA16), 514–527]. We show that exact computation of the partition function is #P-hard, even for line graphs, indicating that an approximation algorithm is the best that can be expected. We also show that Glauber dynamics for the Ising model is rapidly mixing on line graphs, an example being the kagome lattice.

Type
Paper
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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Footnotes

All authors are supported by EPSRC grants EP/S016562/1 and EP/S016694/1, ‘Sampling in hereditary classes’.

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