Electronic Journal of Probability

Ensemble equivalence for dense graphs

F. den Hollander, M. Mandjes, A. Roccaverde, and N.J. Starreveld

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Abstract

In this paper we consider a random graph on which topological restrictions are imposed, such as constraints on the total number of edges, wedges, and triangles. We work in the dense regime, in which the number of edges per vertex scales proportionally to the number of vertices $n$. Our goal is to compare the micro-canonical ensemble (in which the constraints are satisfied for every realisation of the graph) with the canonical ensemble (in which the constraints are satisfied on average), both subject to maximal entropy. We compute the relative entropy of the two ensembles in the limit as $n$ grows large, where two ensembles are said to be equivalent in the dense regime if this relative entropy divided by $n^2$ tends to zero. Our main result, whose proof relies on large deviation theory for graphons, is that breaking of ensemble equivalence occurs when the constraints are frustrated. Examples are provided for three different choices of constraints.

Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 12, 26 pp.

Dates
Received: 27 March 2017
Accepted: 3 January 2018
First available in Project Euclid: 12 February 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1518426060

Digital Object Identifier
doi:10.1214/18-EJP135

Mathematical Reviews number (MathSciNet)
MR3771749

Zentralblatt MATH identifier
1387.05240

Subjects
Primary: 05C80: Random graphs [See also 60B20] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs

Keywords
random graph canonical ensemble microcanonical ensemble constraint ensemble equivalence relative entropy graphon variational representation

Rights
Creative Commons Attribution 4.0 International License.

Citation

den Hollander, F.; Mandjes, M.; Roccaverde, A.; Starreveld, N.J. Ensemble equivalence for dense graphs. Electron. J. Probab. 23 (2018), paper no. 12, 26 pp. doi:10.1214/18-EJP135. https://projecteuclid.org/euclid.ejp/1518426060


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