Lee-Yang-Fisher zeros for DHL and 2D rational dynamics, II. Global Pluripotential Interpretation

TL;DR

This paper analyzes the distribution of Lee-Yang-Fisher zeros in the complex plane for the Diamond Hierarchical Lattice, using complex dynamics and pluripotential theory to understand phase transitions.

Contribution

It introduces a global pluripotential framework for describing Lee-Yang-Fisher zeros via rational dynamics and extends equidistribution results to mappings with indeterminate points.

Findings

Zeros are equidistributed with respect to a dynamical current

The free energy is interpreted as a pluripotential of this current

The results apply to various hierarchical lattices with rational maps

Abstract

In a classical work of the 1950's, Lee and Yang proved that for fixed nonnegative temperature, the zeros of the partition functions of a ferromagnetic Ising model always lie on the unit circle in the complex magnetic field. Zeros of the partition function in the complex temperature were then considered by Fisher, when the magnetic field is set to zero. Limiting distributions of Lee-Yang and of Fisher zeros are physically important as they control phase transitions in the model. One can also consider the zeros of the partition function simultaneously in both complex magnetic field and complex temperature. They form an algebraic curve called the Lee-Yang-Fisher (LYF) zeros. In this paper we continue studying their limiting distribution for the Diamond Hierarchical Lattice (DHL). In this case, it can be described in terms of the dynamics of an explicit rational function R in two variables (the Migdal-Kadanoff renormalization transformation). We study properties of the Fatou and Julia sets of this transformation and then we prove that the Lee-Yang-Fisher zeros are equidistributed with respect to a dynamical (1,1)-current in the projective space. The free energy of the lattice gets interpreted as the pluripotential of this current. We also prove a more general equidistribution theorem which applies to rational mappings having indeterminate points, including the Migdal-Kadanoff renormalization transformation of various other hierarchical lattices.