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

Abstract

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 classification problem originating from classical works of Hermite, Laguerre, Hurwitz and Pólya-Schur on univariate polynomials with such properties.

Final version, to appear in Inventiones Mathematicae; 27 pages, no figures, LaTeX2e

References in corpus (3)

• Polynomials with the half-plane property and matroid theory
• Partition function zeros at first-order phase transitions: A general analysis
• Partition function zeros at first-order phase transitions: Pirogov-Sinai theory

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• Linear Principal Minor Polynomials: Hyperbolic Determinantal Inequalities and Spectral Containment
• Discrete concavity and the half-plane property
• Lee-Yang zeros in the Rydberg atoms
• The Newton polytope and Lorentzian property of chromatic symmetric functions
• Spaces of Lorentzian and real stable polynomials are Euclidean balls
• Linear finite difference operators preserving Laguerre-Pólya class
• Lorentzian polynomials and the independence sequences of graphs
• Hermite-Poulain theorems for linear finite difference operators
• The blowup-polynomial of a metric space: connections to stable polynomials, graphs and their distance spectra
• The $S_k$ shuffle block dynamics
• Global bounds on stable polynomials