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 > 2. We show that the critical temperature of the Ising model equals the inverse hyperbolic tangent of the inverse of the mean offspring or mean forward degree distribution. In particular, the inverse critical temperature equals zero when ∈ (2, 3] where this mean equals infinity. We further study the critical exponents , and , describing how the (root) magnetiza- tion behaves close to criticality. We rigorously identify these critical exponents and show that they take the values as predicted by Dorogovstev, et al. [9] and Leone et al. [17]. These values depend on the power-law exponent , taking the mean-field values for > 5, but different values for ∈ (3, 5).

Ising Critical Exponents on Random Trees and Graphs / Sander, Dommers; Giardina', Cristian; Remco van der, Hofstad. - In: COMMUNICATIONS IN MATHEMATICAL PHYSICS. - ISSN 0010-3616. - STAMPA. - 328:1(2014), pp. 355-395. [10.1007/s00220-014-1992-2]

Ising Critical Exponents on Random Trees and Graphs

GIARDINA', Cristian;
2014

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 > 2. We show that the critical temperature of the Ising model equals the inverse hyperbolic tangent of the inverse of the mean offspring or mean forward degree distribution. In particular, the inverse critical temperature equals zero when ∈ (2, 3] where this mean equals infinity. We further study the critical exponents , and , describing how the (root) magnetiza- tion behaves close to criticality. We rigorously identify these critical exponents and show that they take the values as predicted by Dorogovstev, et al. [9] and Leone et al. [17]. These values depend on the power-law exponent , taking the mean-field values for > 5, but different values for ∈ (3, 5).
2014
328
1
355
395
Ising Critical Exponents on Random Trees and Graphs / Sander, Dommers; Giardina', Cristian; Remco van der, Hofstad. - In: COMMUNICATIONS IN MATHEMATICAL PHYSICS. - ISSN 0010-3616. - STAMPA. - 328:1(2014), pp. 355-395. [10.1007/s00220-014-1992-2]
Sander, Dommers; Giardina', Cristian; Remco van der, Hofstad
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11380/1055714
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