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

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.