Annealed Ising model on configuration models

Van Hao Can; Cristian Giardinà; Claudio Giberti; Remco van der Hofstad

doi:10.1214/21-AIHP1156

Annealed Ising model on configuration models

Van Hao Can
, Cristian Giardinà
, Claudio Giberti
, Remco van der Hofstad

Research output: Contribution to journal › Article › Academic › peer-review

2 Citations (Scopus)

Abstract

In this paper, we study the annealed ferromagnetic Ising model on the configuration model. In an annealed system, we take the average on both sides of the ratio defining the Boltzmann–Gibbs measure of the Ising model. In the configuration model, the degrees are specified. Remarkably, when the degrees are deterministic, the critical value of the annealed Ising model is the same as that for the quenched Ising model. For independent and identically distributed (i.i.d.) degrees, instead, the annealed critical value is strictly smaller than that of the quenched Ising model. This identifies the degree structure of the underlying graph as the main driver for the critical value. Furthermore, in both contexts (deterministic or random degrees), we provide the variational expression for the annealed pressure. Interestingly, our rigorous results establish that only part of the heuristic conjectures in the physics literature were correct.

Original language	English
Pages (from-to)	134-163
Number of pages	30
Journal	Annales de l'institut Henri Poincare (B): Probability and Statistics
Volume	58
Issue number	1
DOIs	https://doi.org/10.1214/21-AIHP1156
Publication status	Published - Feb 2021

Bibliographical note

Funding Information:
The work of RvdH is supported by the Netherlands Organisation for Scientific Research (NWO) through VICI grant 639.033.806 and the Gravitation NETWORKS grant 024.002.003. C. Giardinà and C. Giberti acknowledge financial supports from Fondo di Ateneo per la Ricerca, Università di Modena e Reggio Emilia. The work of V. H. Can is supported by the fellowship no. 17F17319 of the Japan Society for the Promotion of Science, and by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.03–2019.310.

Funding

The work of RvdH is supported by the Netherlands Organisation for Scientific Research (NWO) through VICI grant 639.033.806 and the Gravitation NETWORKS grant 024.002.003. C. Giardinà and C. Giberti acknowledge financial supports from Fondo di Ateneo per la Ricerca, Università di Modena e Reggio Emilia. The work of V. H. Can is supported by the fellowship no. 17F17319 of the Japan Society for the Promotion of Science, and by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.03–2019.310.

Keywords

Annealed Ising model
Configuration model
Critical temperature
Phase transition

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10.1214/21-AIHP1156

Cite this

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title = "Annealed Ising model on configuration models",

abstract = "In this paper, we study the annealed ferromagnetic Ising model on the configuration model. In an annealed system, we take the average on both sides of the ratio defining the Boltzmann–Gibbs measure of the Ising model. In the configuration model, the degrees are specified. Remarkably, when the degrees are deterministic, the critical value of the annealed Ising model is the same as that for the quenched Ising model. For independent and identically distributed (i.i.d.) degrees, instead, the annealed critical value is strictly smaller than that of the quenched Ising model. This identifies the degree structure of the underlying graph as the main driver for the critical value. Furthermore, in both contexts (deterministic or random degrees), we provide the variational expression for the annealed pressure. Interestingly, our rigorous results establish that only part of the heuristic conjectures in the physics literature were correct.",

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Annealed Ising model on configuration models

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Keywords

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