The Lee‐Yang and Pólya‐Schur programs. II. Theory of stable polynomials and applications

Borcea, Julius; Brändén, Petter

doi:10.1002/cpa.20295

Cited by 83 publications

(104 citation statements)

References 66 publications

(195 reference statements)

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“…Many proofs of this result and its equivalent forms exist; see the appendix of [3], or [19, chapter 5]. Definition 3.2.…”

Section: Stable Measures Onmentioning

confidence: 99%

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Stability on {0, 1, 2, …} S : Birth-Death Chains and Particle Systems

Liggett

Vandenberg-Rodes

2011

Notions of Positivity and the Geometry of Polynomials

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Abstract. A strong negative dependence property for measures on {0, 1} nstability -was recently developed in [5], by considering the zero set of the probability generating function. We extend this property to the more general setting of reaction-diffusion processes and collections of independent Markov chains. In one dimension the generalized stability property is now independently interesting, and we characterize the birth-death chains preserving it. Mathematics Subject Classification (2000). Primary 60K35; Secondary 33C45, 60G50, 60J80.

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“…Many proofs of this result and its equivalent forms exist; see the appendix of [3], or [19, chapter 5]. Definition 3.2.…”

Section: Stable Measures Onmentioning

confidence: 99%

“…[2], with the investigation providing a general account of such polynomials and unifying several Lee-Yang-type theorems [3].…”

Section: Introductionmentioning

confidence: 99%

Stability on {0, 1, 2, …} S : Birth-Death Chains and Particle Systems

Liggett

Vandenberg-Rodes

2011

Notions of Positivity and the Geometry of Polynomials

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“…A new development is the emergence of the class of real stable generating functions [COSW04,BB09a,BB09b,BBL09]. A real polynomial in d-variables is said to be stable if it has no zeros all of whose coordinates are strictly in the complex upper half-plane.…”

Section: Introductionmentioning

confidence: 99%

Multivariate CLT follows from strong Rayleigh property

Ghosh¹,

Liggett²,

Pemantle³

2017

2017 Proceedings of the Fourteenth Workshop on Analytic Algorithmics and Combinatorics (ANALCO)

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Let (X1, . . . , X d ) be a random nonnegative integer vector. Many conditions are known to imply a central limit theorem for a sequence of such random vectors, for example, independence and convergence of the normalized covariances, or various combinatorial conditions allowing the application of Stein's method, couplings, etc. Here, we prove a central limit theorem directly from hypotheses on the probability generating function f (z1, . . . , z d ). In particular, we show that the f being real stable (meaning no zeros with all coordinates in the open upper half plane) is enough to imply a CLT under a nondegeneracy condition on the variance. Known classes of distributions with real stable generating polynomials include spanning tree measures, conditioned Bernoullis and counts for determinantal point processes. Soshnikov [Sos02] showed that occupation counts of disjoint sets by a determinantal point process satisfy a multivariate CLT. Our results extend Soshnikov's to the class of real stable laws. The class of real stable laws is much larger than the class of determinantal laws, being defined by inequalities rather than identities. Along the way we investigate the related problem of stable multiplication.

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“…Recently, the theoretical framework underlying these investigations has experienced a renaissance with new developments, multivariate extensions and applications (see especially the work of the Swedish school and notably that of P. Brändén [21], J. Borcea and P. Brändén [16,17,18,19,20] and B. Shapiro [95]. These inequalities were subsequently extended not only to the classical orthogonal polynomials, but also to other families of functions.…”

Section: Introductionmentioning

confidence: 99%