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Breaking of ensemble equivalence for dense random graphs under a single constraint

Part of: Combinatorial probability Equilibrium statistical mechanics Limit theorems Graph theory

Published online by Cambridge University Press:  11 April 2023

Frank Den Hollander  and
Maarten Markering Open the ORCID record for Maarten Markering [Opens in a new window]
Frank Den Hollander*
Affiliation:
Leiden University
Maarten Markering*
Affiliation:
Leiden University
*
*Postal address: Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, The Netherlands. Email: denholla@math.leidenuniv.nl
**Postal address: Pembroke College, Cambridge, CB2 1RF, United Kingdom. Email: mjrm2@cam.ac.uk
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Abstract

Two ensembles are frequently used to model random graphs subject to constraints: the microcanonical ensemble (= hard constraint) and the canonical ensemble (= soft constraint). It is said that breaking of ensemble equivalence (BEE) occurs when the specific relative entropy of the two ensembles does not vanish as the size of the graph tends to infinity. Various examples have been analysed in the literature. It was found that BEE is the rule rather than the exception for two classes of constraints: sparse random graphs when the number of constraints is of the order of the number of vertices, and dense random graphs when there are two or more constraints that are frustrated. We establish BEE for a third class: dense random graphs with a single constraint on the density of a given simple graph. We show that BEE occurs in a certain range of choices for the density and the number of edges of the simple graph, which we refer to as the BEE-phase. We also show that, in part of the BEE-phase, there is a gap between the scaling limits of the averages of the maximal eigenvalue of the adjacency matrix of the random graph under the two ensembles, a property that is referred to as the spectral signature of BEE. We further show that in the replica symmetric region of the BEE-phase, BEE is due to the coexistence of two densities in the canonical ensemble.

Keywords

Constrained random graphGibbs ensemblerelative entropybreaking of ensemble equivalencegraphonvariational representationmaximal eigenvaluesreplica symmetry

MSC classification

Primary: 05C80: Random graphs
Secondary: 60C05: Combinatorial probability 60F10: Large deviations 82B80: Numerical methods (Monte Carlo, series resummation, etc.)
Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 4 , December 2023 , pp. 1181 - 1200
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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