Yang–Lee zeros of the Ising model on random graphs of non planar topology

Abstract

We obtain in a closed form the 1/N^2 contribution to the free energy of the two Hermitian N\times N random matrix model with non symmetric quartic potential. From this result, we calculate numerically the Yang-Lee zeros of the 2D Ising model on dynamical random graphs with the topology of a torus up to n=16 vertices. They are found to be located on the unit circle on the complex fugacity plane. In order to include contributions of even higher topologies we calculated analytically the nonperturbative (sum over all genus) partition function of the model Z_n = \sum_{h=0}^{\infty} \frac{Z_n^{(h)}}{N^{2h}} for the special cases of N=1,2 and graphs with n\le 20 vertices. Once again the Yang-Lee zeros are shown numerically to lie on the unit circle on the complex fugacity plane. Our results thus generalize previous numerical results on random graphs by going beyond the planar approximation and strongly indicate that there might be a generalization of the Lee-Yang circle theorem for dynamical random graphs.