Year: 2012
Type: preprint
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 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 $τ\in(2,3]$ 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. and Leone et al. These values depend on the power-law exponent $τ$, taking the mean-field values for $τ>5$, but different values for $τ\in(3,5)$.
Source: arXiv (Cornell University)
Language: English
Subfield: Condensed Matter Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
Award: G54886314
OpenAlex ID: W2807791793