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

dc.contributor.authorBleher, Pavel
dc.contributor.authorLyubich, Mikhail
dc.contributor.authorRoeder, Roland
dc.contributor.departmentMathematical Sciences, School of Scienceen_US
dc.date.accessioned2019-12-12T20:51:01Z
dc.date.available2019-12-12T20:51:01Z
dc.date.issued2019
dc.description.abstractIn 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.en_US
dc.eprint.versionAuthor's manuscripten_US
dc.identifier.citationBleher, P., Lyubich, M., & Roeder, R. (2019). Lee–Yang–Fisher Zeros for the DHL and 2D Rational Dynamics, II. Global Pluripotential Interpretation. The Journal of Geometric Analysis. https://doi.org/10.1007/s12220-019-00167-6en_US
dc.identifier.urihttps://hdl.handle.net/1805/21476
dc.language.isoenen_US
dc.publisherSpringeren_US
dc.relation.isversionof10.1007/s12220-019-00167-6en_US
dc.relation.journalThe Journal of Geometric Analysisen_US
dc.rightsPublisher Policyen_US
dc.sourceArXiven_US
dc.subjectcomplex dynamics in higher dimensionsen_US
dc.subjectinvariant currentsen_US
dc.subjectequidistribution problemsen_US
dc.titleLee–Yang–Fisher Zeros for the DHL and 2D Rational Dynamics, II. Global Pluripotential Interpretationen_US
dc.typeArticleen_US
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