Lee-Yang Problems and The Geometry of Multivariate Polynomials

The Lee-Yang and Pólya-Schur programs. III. Zero-preservers on Bargmann-Fock spaces

We characterize linear operators preserving zero-restrictions on entire functions in weighted Bargmann-Fock spaces. The characterization extends previous results of J. Borcea and the author to the realm of entire functions, and translates into an optimal, albeit formal, Lee-Yang theorem.

The Lee-Yang and Pólya-Schur Programs. II. Theory of Stable Polynomials and Applications

In the first part of this series we characterized all linear operators on spaces of multivariate polynomials preserving the property of being non-vanishing in products of open circular domains. For such sets this completes the multivariate generalization of the classification program initiated by Pólya-Schur for univariate real polynomials. We build on these classification theorems to develop here a theory of multivariate stable polynomials. Applications and examples show that this theory provides a natural framework for dealing in a uniform way with Lee-Yang type problems in statistical mecha…

The Lee-Yang and Pólya-Schur Programs. I. Linear Operators Preserving Stability

In 1952 Lee and Yang proposed the program of analyzing phase transitions in terms of zeros of partition functions. Linear operators preserving non-vanishing properties are essential in this program and various contexts in complex analysis, probability theory, combinatorics, and matrix theory. We characterize all linear operators on finite or infinite-dimensional spaces of multivariate polynomials preserving the property of being non-vanishing whenever the variables are in prescribed open circular domains. In particular, this solves the higher dimensional counterpart of a long-standing classifi…

Lee-Yang 张量的几何

The Geometry of Lee-Yang Tensors

本文利用复 Perron-Frobenius 理论给出 Lee-Yang 张量的几何描述，推广关键引理以界定非正规算子谱比，并证明 Suzuki-Fisher 哈密顿量是唯一在所有逆温度下吉布斯态具有 Lee-Yang 性质的哈密顿量。

高维傅里叶准晶体从李-杨品种

Higher Dimensional Fourier Quasicrystals from Lee-Yang Varieties

本文提出了一种在任意维度中构造具有单位质量的傅里叶准晶体的方法，利用复代数簇推广了李-扬多项式定义的超曲面，并证明其具有有限交集的性质。

Lee-Yang zeros for DHL and 2D rational dynamics, I. Foliation of the physical cylinder

In a classical work of the 1950's, Lee and Yang proved that the zeros of the partition functions of a ferromagnetic Ising models always lie on the unit circle. Distribution of these zeros is physically important as it controls phase transitions in the model. We study this distribution for the Migdal-Kadanoff Diamond Hierarchical Lattice (DHL). In this case, it can be described in terms of the dynamics of an explicit rational function $\RR$ in two variables (the renormalization transformation). We prove that $\RR$ is partially hyperbolic on an invariant cylinder $\CC$. The Lee-Yang zeros are or…