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Information, Physics, and Computation$
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Marc Mézard and Andrea Montanari

Print publication date: 2009

Print ISBN-13: 9780198570837

Published to Oxford Scholarship Online: September 2009

DOI: 10.1093/acprof:oso/9780198570837.001.0001

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Ising models on random graph

Ising models on random graph

Chapter:
(p.381) 17 Ising models on random graph
Source:
Information, Physics, and Computation
Author(s):

Marc Mézard

Andrea Montanari

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198570837.003.0017

This chapter studies two problems of statistical physics: the ferromagnet and the spin glass, on large random graphs with fixed degree profile. It describes the use of the replica symmetric cavity method in this context, and studies its stability. The analysis relies on physicists methods, without any attempt at being rigorous. It provides a complete solution of the ferromagnetic problem at all temperatures. In the spin glass case, the replica symmetric solution is asymptotically correct in the high temperature ‘paramagnetic’ phase, but it turns out to be wrong in the spin glass phase. The phase transition temperature can be computed exactly.

Keywords:   ferromagnet, spin glass, random graph, replica symmetric, cavity method

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