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The Interpolation Method for Random Graphs with Prescribed Degrees

Part of: Statistical mechanics, structure of matter Graph theory

Published online by Cambridge University Press:  19 June 2015

JUSTIN SALEZ
JUSTIN SALEZ*
Affiliation:
Université Paris Diderot–LPMA, UFR de Mathématiques, 5 rue Thomas Mann, 75205 Paris CEDEX 13, France (e-mail: [email protected])
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Abstract

We consider large random graphs with prescribed degrees, as generated by the configuration model. In the regime where the empirical degree distribution approaches a limit μ with finite mean, we establish the systematic convergence of a broad class of graph parameters that includes the independence number, the maximum cut size, the logarithm of the Tutte polynomial, and the free energy of the anti-ferromagnetic Ising and Potts models. Contrary to previous works, our results are not a priori limited to the free energy of some prescribed graphical model. They apply more generally to any additive, Lipschitz and concave graph parameter. In addition, the corresponding limits are shown to be Lipschitz and concave in the degree distribution μ. This considerably extends the applicability of the celebrated interpolation method, introduced in the context of spin glasses, and recently related to the challenging question of right-convergence of sparse graphs.

MSC classification

Primary: 05C80: Random graphs
Secondary: 82-08: Computational methods
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 25 , Issue 3 , May 2016 , pp. 436 - 447
Copyright
Copyright © Cambridge University Press 2015 

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