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Ising Critical Exponents on Random Trees and Graphs

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Abstract

We study the critical behavior of the ferromagnetic Ising model on random trees as well as so-called locally tree-like random graphs. We pay special attention to trees and graphs with a power-law offspring or degree distribution whose tail behavior is characterized by its power-law exponent τ > 2. We show that the critical inverse temperature of the Ising model equals the hyperbolic arctangent of the reciprocal of the mean offspring or mean forward degree distribution. In particular, the critical inverse temperature equals zero when \({\tau \in (2,3]}\) where this mean equals infinity.

We further study the critical exponents δ, β and γ, describing how the (root) magnetization behaves close to criticality. We rigorously identify these critical exponents and show that they take the values as predicted by Dorogovstev et al. (Phys Rev E 66:016104, 2002) and Leone et al. (Eur Phys J B 28:191–197, 2002). These values depend on the power-law exponent τ, taking the same values as the mean-field Curie-Weiss model (Exactly solved models in statistical mechanics, Academic Press, London, 1982) for τ > 5, but different values for \({\tau \in (3,5)}\) .

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands

    Sander Dommers & Remco van der Hofstad

  2. Department of Mathematics, Physics and Computer Science, Modena and Reggio Emilia University, via Campi 231/b, 41125, Modena, Italy

    Cristian Giardinà

Authors
  1. Sander Dommers
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  2. Cristian Giardinà
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  3. Remco van der Hofstad
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Correspondence to Cristian Giardinà.

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Communicated by H. Spohn

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Dommers, S., Giardinà, C. & van der Hofstad, R. Ising Critical Exponents on Random Trees and Graphs. Commun. Math. Phys. 328, 355–395 (2014). https://doi.org/10.1007/s00220-014-1992-2

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