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48
The Lee-Yang and Pólya-Schur programs I. Linear operators preserving stability
, 2008
"... 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 an essential tool in this program as well as various contexts in complex analysis, probability theory, combinatorics, matrix t ..."
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Cited by 35 (10 self)
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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 an essential tool in this program as well as various contexts in complex analysis, probability theory, combinatorics, matrix theory. We characterize all linear operators on finite or infinite-dimensional spaces of multivariate polynomials preserving the property of being non-vanishing when the variables are in prescribed open circular domains. In particular, this supersedes [7, 9] and solves the higher dimensional counterpart of a long-standing classification problem originating from classical works of Hermite, Laguerre,
Multivariate stable polynomials: theory and applications
- BULL. AM. MATH. SOC
, 2010
"... Univariate polynomials with only real roots, while special, do occur often enough that their properties can lead to interesting conclusions in diverse areas. Due mainly to the recent work of two young mathematicians, Julius Borcea and Petter Brändén, a very successful multivariate generalization of ..."
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Cited by 29 (1 self)
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Univariate polynomials with only real roots, while special, do occur often enough that their properties can lead to interesting conclusions in diverse areas. Due mainly to the recent work of two young mathematicians, Julius Borcea and Petter Brändén, a very successful multivariate generalization of this method has been developed. The first part of this paper surveys some of the main results of this theory of multivariate stable polynomials—the most central of these results is the characterization of linear transformations preserving stability of polynomials. The second part presents various applications of this theory in complex analysis, matrix theory, probability and statistical mechanics, and combinatorics.
Multivariate Pólya-Schur classification problems in the Weyl algebra
, 2008
"... A multivariate polynomial is stable if it is nonvanishing whenever all variables have positive imaginary parts. We classify all linear partial differential operators in the Weyl algebra An that preserve stability. An important tool that we develop in the process is the higher dimensional generalizat ..."
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Cited by 28 (9 self)
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A multivariate polynomial is stable if it is nonvanishing whenever all variables have positive imaginary parts. We classify all linear partial differential operators in the Weyl algebra An that preserve stability. An important tool that we develop in the process is the higher dimensional generalization of Pólya-Schur’s notion of multiplier sequence. We characterize all multivariate multiplier sequences as well as those of finite order. Next, we establish a multivariate extension of the Cauchy-Poincaré interlacing theorem and prove a natural analog of the Lax conjecture for real stable polynomials in two variables. Using the latter we describe all operators in A1 that preserve univariate hyperbolic polynomials by means of determinants and homogenized symbols. Our methods also yield homotopical properties for symbols of linear stability preservers and a duality theorem showing that an operator in An preserves stability if and only if its Fischer-Fock adjoint does. These are powerful multivariate extensions of the classical Hermite-Poulain-Jensen theorem, Pólya’s curve theorem and Schur-Maló-Szegö composition theorems. Examples, applications to strict stability preservers and further directions are also discussed.
The Lee-Yang and Pólya-Schur programs. II. Theory of stable polynomials and applications
, 2008
"... 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 Pó ..."
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Cited by 19 (5 self)
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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 mechanics, combinatorics, and geometric function theory in one or several variables. in
The Palm measure and the Voronoi tessellation for the Ginibre process
- Ann. Appl. Probab
, 2010
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Proof of the monotone column permanent conjecture, in Notions of positivity and the geometry of polynomials
- Trends in Mathematics, Birkhauser
, 2011
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Distributional Limits for the Symmetric Exclusion Process
, 2008
"... Strong negative dependence properties have recently been proved for the symmetric exclusion process. In this paper, we apply these results to prove convergence to the Poisson and Gaussian distributions for various functionals of the process. 1 ..."
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Cited by 8 (3 self)
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Strong negative dependence properties have recently been proved for the symmetric exclusion process. In this paper, we apply these results to prove convergence to the Poisson and Gaussian distributions for various functionals of the process. 1
Random complexes and ℓ 2 -Betti numbers
, 2005
"... Abstract. Uniform spanning trees on finite graphs and their analogues on infinite graphs are a well-studied area. On a Cayley graph of a group, we show that they are related to the first ℓ 2-Betti number of the group. Our main aim, however, is to present the basic elements of a higher-dimensional an ..."
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Cited by 7 (4 self)
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Abstract. Uniform spanning trees on finite graphs and their analogues on infinite graphs are a well-studied area. On a Cayley graph of a group, we show that they are related to the first ℓ 2-Betti number of the group. Our main aim, however, is to present the basic elements of a higher-dimensional analogue on finite and infinite CW-complexes, which relate to the higher ℓ 2-Betti numbers. One consequence is a uniform isoperimetric inequality extending work of Lyons, Pichot, and Vassout. We also present an enumeration similar to recent work of Duval, Klivans, and Martin. §1. Introduction. Enumeration of spanning trees in graphs began with Kirchhoff (1847). Cayley (1889) evaluated this number in the special case of a complete graph. Cayley’s theorem was
Complete monotonicity for inverse powers of some combinatorially defined polynomials
, 2013
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