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In and Out of Equilibrium 3: Celebrating Vladas SidoraviciusGlauber Dynamics on the Erdős-Rényi Random Graph

In and Out of Equilibrium 3: Celebrating Vladas Sidoravicius: Glauber Dynamics on the... [We investigate the effect of disorder on the Curie-Weiss model with Glauber dynamics. In particular, we study metastability for spin-flip dynamics on the Erdős-Rényi random graph ERn(p) with n vertices and with edge retention probability p ∈ (0, 1). Each vertex carries an Ising spin that can take the values − 1 or + 1. Single spins interact with an external magnetic field h ∈ (0, ∞), while pairs of spins at vertices connected by an edge interact with each other with ferromagnetic interaction strength 1∕n. Spins flip according to a Metropolis dynamics at inverse temperature β. The standard Curie-Weiss model corresponds to the case p = 1, because ERn(1) = Kn is the complete graph on n vertices. For β > βc and h ∈ (0, pχ(βp)) the system exhibits metastable behaviour in the limit as n →∞, where βc = 1∕p is the critical inverse temperature and χ is a certain threshold function satisfying limλ→∞χ(λ) = 1 and limλ↓1χ(λ) = 0. We compute the average crossover time from the metastable set (with magnetization corresponding to the ‘minus-phase’) to the stable set (with magnetization corresponding to the ‘plus-phase’). We show that the average crossover time grows exponentially fast with n, with an exponent that is the same as for the Curie-Weiss model with external magnetic field h and with ferromagnetic interaction strength p∕n. We show that the correction term to the exponential asymptotics is a multiplicative error term that is at most polynomial in n. For the complete graph Kn the correction term is known to be a multiplicative constant. Thus, apparently, ERn(p) is so homogeneous for large n that the effect of the fluctuations in the disorder is small, in the sense that the metastable behaviour is controlled by the average of the disorder. Our model is the first example of a metastable dynamics on a random graph where the correction term is estimated to high precision.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

In and Out of Equilibrium 3: Celebrating Vladas SidoraviciusGlauber Dynamics on the Erdős-Rényi Random Graph

Part of the Progress in Probability Book Series (volume 77)
Editors: Vares, Maria Eulália; Fernández, Roberto; Fontes, Luiz Renato; Newman, Charles M.

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References (16)

Publisher
Springer International Publishing
Copyright
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
ISBN
978-3-030-60753-1
Pages
519 –589
DOI
10.1007/978-3-030-60754-8_24
Publisher site
See Chapter on Publisher Site

Abstract

[We investigate the effect of disorder on the Curie-Weiss model with Glauber dynamics. In particular, we study metastability for spin-flip dynamics on the Erdős-Rényi random graph ERn(p) with n vertices and with edge retention probability p ∈ (0, 1). Each vertex carries an Ising spin that can take the values − 1 or + 1. Single spins interact with an external magnetic field h ∈ (0, ∞), while pairs of spins at vertices connected by an edge interact with each other with ferromagnetic interaction strength 1∕n. Spins flip according to a Metropolis dynamics at inverse temperature β. The standard Curie-Weiss model corresponds to the case p = 1, because ERn(1) = Kn is the complete graph on n vertices. For β > βc and h ∈ (0, pχ(βp)) the system exhibits metastable behaviour in the limit as n →∞, where βc = 1∕p is the critical inverse temperature and χ is a certain threshold function satisfying limλ→∞χ(λ) = 1 and limλ↓1χ(λ) = 0. We compute the average crossover time from the metastable set (with magnetization corresponding to the ‘minus-phase’) to the stable set (with magnetization corresponding to the ‘plus-phase’). We show that the average crossover time grows exponentially fast with n, with an exponent that is the same as for the Curie-Weiss model with external magnetic field h and with ferromagnetic interaction strength p∕n. We show that the correction term to the exponential asymptotics is a multiplicative error term that is at most polynomial in n. For the complete graph Kn the correction term is known to be a multiplicative constant. Thus, apparently, ERn(p) is so homogeneous for large n that the effect of the fluctuations in the disorder is small, in the sense that the metastable behaviour is controlled by the average of the disorder. Our model is the first example of a metastable dynamics on a random graph where the correction term is estimated to high precision.]

Published: Nov 4, 2020

Keywords: Erdős-Rényi random graph; Glauber spin-flip dynamics; Metastability; Crossover time

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