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Stability on $$\{0, 1, 2, \dots \}^{S}$$: birth-death chains and particle systems. (English) Zbl 1278.60144
Brändén, Petter (ed.) et al., Notions of positivity and the geometry of polynomials. Dedicated to the memory of Julius Borcea. Basel: Birkhäuser (ISBN 978-3-0348-0141-6/hbk; 978-3-0348-0142-3/ebook). Trends in Mathematics, 311-329 (2011).
Authors’ abstract: A strong negative dependence property for measures on $$\{0,1\}^n$$-stability was recently developed in [J. Borcea et al., J. Am. Math. Soc. 22, No. 2, 521–567 (2009; Zbl 1206.62096)], 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.
For the entire collection see [Zbl 1222.00033].
##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 60G50 Sums of independent random variables; random walks 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 62H20 Measures of association (correlation, canonical correlation, etc.)
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