Ising critical exponents on random trees and graphs

TL;DR

This paper analyzes the critical behavior and exponents of the ferromagnetic Ising model on random trees and graphs with power-law degree distributions, revealing how these exponents depend on the degree distribution tail.

Contribution

It provides rigorous identification of critical exponents for the Ising model on random trees and graphs with power-law degree distributions, confirming predictions and detailing their dependence on the tail exponent.

Findings

Critical temperature equals inverse hyperbolic tangent of inverse mean degree.

Critical exponents depend on the power-law exponent τ, with mean-field values for τ>5.

Inverse critical temperature is zero for τ in (2,3].

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 $\tau>2$. 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 $\tau\in(2,3]$ where this mean equals infinity. We further study the critical exponents $\delta, \beta$ and $\gamma$, 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 $\tau$, taking the mean-field values for $\tau>5$, but different values for $\tau\in(3,5)$.