Limiting Measure of Lee–Yang Zeros for the Cayley Tree

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 Müller-Hartmann and Zittartz (Z Phys B 22:59, 1975), Müller-Hartmann (Z Phys B 27:161–168, 1977), Barata and Marchetti (J Stat Phys 88:231–268, 1997) and Barata and Goldbaum (J Stat Phys 103:857–891, 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 Müller-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.