Browsing by Author "Lyubich, Mikhail"

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    Lee–Yang zeros for the DHL and 2D rational dynamics, I. Foliation of the physical cylinder
    (Elsevier, 2017-05) Bleher, Pavel; Lyubich, Mikhail; Roeder, Roland; Department of Mathematical Sciences, School of Science
    In a classical work of the 1950's, Lee and Yang proved that the zeros of the partition functions of a ferromagnetic Ising model always lie on the unit circle. Distribution of these zeros is physically important as it controls phase transitions in the model. We study this distribution for the Migdal–Kadanoff 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 renormalization transformation). We prove that R is partially hyperbolic on an invariant cylinder C. The Lee–Yang zeros are organized in a transverse measure for the central-stable foliation of R|C. Their distribution is absolutely continuous. Its density is C∞ (and non-vanishing) below the critical temperature. Above the critical temperature, it is C∞ on a open dense subset, but it vanishes on the complementary set of positive measure.
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    Lee–Yang–Fisher Zeros for the DHL and 2D Rational Dynamics, II. Global Pluripotential Interpretation
    (Springer, 2019) Bleher, Pavel; Lyubich, Mikhail; Roeder, Roland; Mathematical Sciences, School of Science
    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.