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arXiv 1806.00403math-phcond-mat.stat-mechmath.DSmath.MP
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Cayley树的Lee-Yang零点的极限测度

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

Ivan Chio, Caleb He, Anthony L. Ji, Roland K. W. Roeder

备注：
Final Version accepted to appear in Communications in Mathematical Physics (some easy proofs that were cut from the published version are still presented here)
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摘要：

本文深入研究了Cayley树上伊辛模型的Lee-Yang零点的极限测度。本文在Müller-Hartmann-Zittartz（1974和1977），Barata-Marchetti（1997）和Barata-Goldbaum（2001）的工作基础上，确定了极限测度的支集，证明了极限测度关于Lebesgue测度不是绝对连续的，并确定了测度在Lebesgue a.e.单位圆上的点和每个温度。后者与模型中的相变的临界指数有关，因为在Lebesgue a.e.穿过单位圆。点，提供了一个全球版本的“连续顺序的相变”发现的穆勒-哈特曼-齐塔尔茨。关键技术来自动力系统，因为有一个关于$n$层的有限Cayley树的Lee-Yang零点的显式公式，该公式表示为扩展Blaschke积的$n$阶.一个微妙之处在于不同参数值下的Blaschke乘积之间的共轭不是绝对连续的。

英文摘要：

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-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 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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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