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The Julia sets and complex singularities in hierarchical Ising models

Pavel Bleher·Mikhail Lyubich
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Abstract

We study the analytical continuation in the complex plane of free energy of the Ising model on diamond-like hierarchical lattices. It is known that the singularities of free energy of this model lie on the Julia set of some rational endomorphism $f$ related to the action of the Migdal-Kadanoff renorm-group. We study the asymptotics of free energy when temperature goes along hyperbolic geodesics to the boundary of an attractive basin of $f$. We prove that for almost all (with respect to the harmonic measure) geodesics the complex critical exponent is common, and compute it.

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Pavel Bleher, Mikhail Lyubich. 1990-09-26. The Julia sets and complex singularities in hierarchical Ising models. https://doi.org/10.1007/bf02102810

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