Mandelbrot-like and Julia Sets Associated with Yang-Lee and Fisher Zeros of Ising Models

Abstract

A method for numerical investigation of the Yang-Lee and Fisher zeros of models defined on Cayley-type lattices is given. The method is based on testing the convergence of critical orbits of the corresponding 1D rational mapping to any attracting periodic point. The Yang-Lee and Fisher zeros of multisite interaction Ising model are investigated on Cayley-type lattices. A corresponding C++ program is given in the appendix. Our numerical experiments show that copies of the Mandelbrot set of the quadratic mapping z → z² + c arise in the pictures of Yang-Lee and Fisher zeros. This phenomenon is known as the universality of the Mandelbrot set.