Ising model with external magnetic field on random planar maps: Critical exponents

TL;DR

This paper analyzes the critical behavior of the Ising model with an external magnetic field on random tetravalent planar maps, deriving explicit formulas and critical exponents through combinatorial methods.

Contribution

It provides the first explicit expressions for magnetization and susceptibility, and determines critical exponents for the model on random planar maps.

Findings

Critical exponents: α=-1, β=1/2, γ=2, δ=5

Explicit formulas for spontaneous magnetization and susceptibility

Asymptotic analysis of the partition function in weak magnetic field

Abstract

We study the Ising model with an external magnetic field on random tetravalent planar maps and investigate its critical behavior. Explicit expressions for spontaneous magnetization and the susceptibility are computed and the critical exponents $\alpha=-1$ (third order phase transition), $\beta=\frac{1}{2}$ (spontaneous magnetization), $\gamma=2$ (susceptibility at zero external magnetic field) and $\delta=5$ (magnetization at critical temperature) are derived. To do so, we study the asymptotic behavior of the partition function of the model in the case of a weak external magnetic field using analytic combinatorics.