Limiting Measure of Lee--Yang Zeros for the Cayley Tree

TL;DR

This paper analyzes the limiting distribution of Lee--Yang zeros for the Ising Model on Cayley Trees, revealing their support, singularity, and fractal properties, and connecting these to phase transition phenomena.

Contribution

It determines the support and singularity of the limiting measure of Lee--Yang zeros for the Ising model on Cayley trees, and relates measure properties to phase transition exponents.

Findings

The limiting measure is supported on a subset of the unit circle.

The measure is not absolutely continuous with respect to Lebesgue measure.

The pointwise dimension of the measure relates to critical exponents.

Abstract

This paper is devoted to an in-depth study of the limiting measure of Lee--Yang zeroes for the Ising Model on the Cayley Tree. We build on previous works of M\"uller-Hartmann-Zittartz (1974 and 1977), Barata--Marchetti (1997), and Barata--Goldbaum (2001), to determine the support of the limiting measure, prove that the limiting measure is not absolutely continuous with respect to Lebesgue measure, and determine the pointwise dimension of the measure at Lebesgue a.e. point on the unit circle and every temperature. The latter is related to the critical exponents for the phase transitions in the model as one crosses the unit circle at Lebesgue a.e. point, providing a global version of the "phase transition of continuous order" discovered by M\"uller-Hartmann-Zittartz. The key techniques are from dynamical systems because there is an explicit formula for the Lee-Yang zeros of the finite Cayley Tree of level $n$ in terms of the $n$-th iterate of an expanding Blaschke Product. A subtlety arises because the conjugacies between Blaschke Products at different parameter values are not absolutely continuous.