Lee–Yang–Fisher Zeros for the DHL and 2D Rational Dynamics, II. Global Pluripotential Interpretation

• Published in: The Journal of geometric analysis 2020, Vol.30 (1), p.777-833
• Main Authors: Bleher, Pavel, Lyubich, Mikhail, Roeder, Roland
• Format: Article
• Language: English
• Subjects: Abstract Harmonic Analysis; Constraining; Convex and Discrete Geometry; Diamonds; Differential Geometry; Dynamical Systems and Ergodic Theory; Ferromagnetism; Fourier Analysis; Free energy; Geometry; Global Analysis and Analysis on Manifolds; Ising model; Lattices (mathematics); Magnetic fields; Mathematics; Mathematics and Statistics; Partitions (mathematics); Phase transitions; Rational functions; Transformations (mathematics)
• ISSN: 1050-6926; 1559-002X
• DOI: 10.1007/s12220-019-00167-6

In a classical work of the 1950s, 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 LYF 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.