# Ising critical exponents on random trees and graphs

S. Dommers, C. Giardina, R.W. Hofstad, van der

### Uittreksel

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 t > 2. We show that the critical inverse 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 critical inverse temperature equals zero when t ¿ (2,3] where this mean equals infinity. We further study the critical exponents d, ß and ¿, 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. [12] and Leone et al. [23]. These values depend on the power-law exponent t taking the mean-field values for t > 5, but different values for t ¿ (3,5).
Originele taal-2 Engels Eindhoven Eurandom 39 Gepubliceerd - 2013

### Publicatie series

Naam Report Eurandom 2013025 1389-2355

### Vingerafdruk

Random Trees
Random Graphs
Ising
Critical Exponents
Degree Distribution
Inverse hyperbolic tangent
Ising Model
Power Law
Exponent
Tail Behavior
Power-law Distribution
Criticality
Critical Behavior
Magnetization
Mean Field
Infinity
Roots
Zero
Graph in graph theory

### Citeer dit

Dommers, S., Giardina, C., & Hofstad, van der, R. W. (2013). Ising critical exponents on random trees and graphs. (Report Eurandom; Vol. 2013025). Eindhoven: Eurandom.
Dommers, S. ; Giardina, C. ; Hofstad, van der, R.W. / Ising critical exponents on random trees and graphs. Eindhoven : Eurandom, 2013. 39 blz. (Report Eurandom).
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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 t > 2. We show that the critical inverse 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 critical inverse temperature equals zero when t ¿ (2,3] where this mean equals infinity. We further study the critical exponents d, {\ss} and ¿, 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. [12] and Leone et al. [23]. These values depend on the power-law exponent t taking the mean-field values for t > 5, but different values for t ¿ (3,5).",
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Dommers, S, Giardina, C & Hofstad, van der, RW 2013, Ising critical exponents on random trees and graphs. Report Eurandom, vol. 2013025, Eurandom, Eindhoven.

Ising critical exponents on random trees and graphs. / Dommers, S.; Giardina, C.; Hofstad, van der, R.W.

Eindhoven : Eurandom, 2013. 39 blz. (Report Eurandom; Vol. 2013025).

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N2 - 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 t > 2. We show that the critical inverse 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 critical inverse temperature equals zero when t ¿ (2,3] where this mean equals infinity. We further study the critical exponents d, ß and ¿, 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. [12] and Leone et al. [23]. These values depend on the power-law exponent t taking the mean-field values for t > 5, but different values for t ¿ (3,5).

AB - 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 t > 2. We show that the critical inverse 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 critical inverse temperature equals zero when t ¿ (2,3] where this mean equals infinity. We further study the critical exponents d, ß and ¿, 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. [12] and Leone et al. [23]. These values depend on the power-law exponent t taking the mean-field values for t > 5, but different values for t ¿ (3,5).

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Dommers S, Giardina C, Hofstad, van der RW. Ising critical exponents on random trees and graphs. Eindhoven: Eurandom, 2013. 39 blz. (Report Eurandom).