Yang-Lee Zeros of the Ising model on Random Graphs of Non Planar Topology

TL;DR

This paper analytically and numerically investigates the distribution of Yang-Lee zeros in the Ising model on non-planar random graphs, revealing they lie on the unit circle and suggesting a possible generalization of the Lee-Yang circle theorem.

Contribution

It provides the first closed-form calculation of the 1/N^2 correction to the free energy for non-symmetric quartic matrix models and extends the analysis of Yang-Lee zeros beyond planar graphs.

Findings

Yang-Lee zeros lie on the unit circle in the complex fugacity plane.

Numerical results extend to non-planar topologies, including torus and higher genus.

Analytical calculations include nonperturbative contributions for small N and graphs with up to 20 vertices.

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.