Abstract:A polynomial f is said to have the half-plane property if there is an open
half-plane H, whose boundary contains the origin, such that f is non-zero
whenever all the variables are in H. This paper answers several open questions
regarding multivariate polynomials with the half-plane property and matroid
theory.
* We prove that the support of a multivariate polynomial with the half-plane
property is a jump system. This answers an open question posed by Choe, Oxley,
Sokal and Wagner and generalizes their recent… Show more
“…To add to this list, in §3 we show that both the inequalities characterizing multi-affine real stable polynomials [8,9,15] and Hadamard-FischerKotelyansky type inequalities in matrix theory [28,40,39] may in fact be viewed as natural manifestations of negative dependence properties.…”
Section: F Dµ Gdµ ≤ F Gdµmentioning
confidence: 99%
“…In particular, this allows us to prove several conjectures made by Liggett [55], Pemantle [65], and Wagner [72], respectively, and to recover and extend Lyons' main results [57] on negative association and stochastic domination for determinantal probability measures induced by positive contractions. Moreover, we define a partial order on the set of strongly Rayleigh measures (by means of the notion of proper position for multivariate stable polynomials studied in [7,8,9,15]) and use it to settle Pemantle's questions and conjectures on stochastic domination for truncations of "negatively dependent" measures [65].…”
Section: (Nlc) µ(S)µ(t ) ≥ µ(S ∪ T )µ(S ∩ T ) For All S T ⊆ [N]mentioning
We introduce the class of strongly Rayleigh probability measures by means of geometric properties of their generating polynomials that amount to the stability of the latter. This class covers important models such as determinantal measures (e.g. product measures and uniform random spanning tree measures) and distributions for symmetric exclusion processes. We show that strongly Rayleigh measures enjoy all virtues of negative dependence, and we also prove a series of conjectures due to Liggett, Pemantle, and Wagner, respectively. Moreover, we extend Lyons’ recent results on determinantal measures, and we construct counterexamples to several conjectures of Pemantle and Wagner on negative dependence and ultra log-concave rank sequences.
“…To add to this list, in §3 we show that both the inequalities characterizing multi-affine real stable polynomials [8,9,15] and Hadamard-FischerKotelyansky type inequalities in matrix theory [28,40,39] may in fact be viewed as natural manifestations of negative dependence properties.…”
Section: F Dµ Gdµ ≤ F Gdµmentioning
confidence: 99%
“…In particular, this allows us to prove several conjectures made by Liggett [55], Pemantle [65], and Wagner [72], respectively, and to recover and extend Lyons' main results [57] on negative association and stochastic domination for determinantal probability measures induced by positive contractions. Moreover, we define a partial order on the set of strongly Rayleigh measures (by means of the notion of proper position for multivariate stable polynomials studied in [7,8,9,15]) and use it to settle Pemantle's questions and conjectures on stochastic domination for truncations of "negatively dependent" measures [65].…”
Section: (Nlc) µ(S)µ(t ) ≥ µ(S ∪ T )µ(S ∩ T ) For All S T ⊆ [N]mentioning
We introduce the class of strongly Rayleigh probability measures by means of geometric properties of their generating polynomials that amount to the stability of the latter. This class covers important models such as determinantal measures (e.g. product measures and uniform random spanning tree measures) and distributions for symmetric exclusion processes. We show that strongly Rayleigh measures enjoy all virtues of negative dependence, and we also prove a series of conjectures due to Liggett, Pemantle, and Wagner, respectively. Moreover, we extend Lyons’ recent results on determinantal measures, and we construct counterexamples to several conjectures of Pemantle and Wagner on negative dependence and ultra log-concave rank sequences.
“…From (4.7) and [13,Corollary 3.7] we deduce that λ(γ) > 0 for all ξ ≤ γ ≤ κ. The proposed formula (4.4) now follows by induction over k := |α| − |ξ|.…”
Section: Hard Pólya-schur Theory: Bounded Degree Multiplier Sequencesmentioning
confidence: 99%
“…Recently, Lee-Yang like problems and techniques have appeared in various mathematical contexts such as combinatorics, complex analysis, matrix theory and probability theory [1,6,7,8,9,10,13,14,16,21,24,25,47,56,59]. The past decade has also been marked by important developments on other aspects of phase transitions, conformal invariance, percolation theory [27,29,55].…”
Abstract. 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 particular, we answer a question of Hinkkanen on multivariate apolarity.
Abstract. The Heine-Stieltjes Theorem describes the polynomial solutions, (v, f ) such that T (f ) = vf , to specific second order differential operators, T , with polynomial coefficients. We extend the theorem to concern all (nondegenerate) differential operators preserving the property of having only real zeros, thus solving a conjecture of B. Shapiro
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