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139
Negative dependence and the geometry of polynomials
, 2008
"... 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, uniform random spanning tree measur ..."
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Cited by 48 (13 self)
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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, 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
Transfer matrices and partitionfunction zeros for antiferromagnetic Potts models. I. General theory and squarelattice chromatic polynomial
 J. Stat. Phys
, 2001
"... We study the chromatic polynomials ( = zerotemperature antiferromagnetic Pottsmodel partition functions) PG(q) for m × n rectangular subsets of the square lattice, with m ≤ 8 (free or periodic transverse boundary conditions) and n arbitrary (free longitudinal boundary conditions), using a transfer ..."
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Cited by 41 (6 self)
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We study the chromatic polynomials ( = zerotemperature antiferromagnetic Pottsmodel partition functions) PG(q) for m × n rectangular subsets of the square lattice, with m ≤ 8 (free or periodic transverse boundary conditions) and n arbitrary (free longitudinal boundary conditions), using a transfer matrix in the FortuinKasteleyn representation. In particular, we extract the limiting curves of partitionfunction zeros when n → ∞, which arise from the crossing in modulus of dominant eigenvalues (Beraha–Kahane–Weiss theorem). We also provide evidence that the Beraha numbers B2,B3,B4,B5 are limiting points of partitionfunction zeros as n → ∞ whenever the strip width m is ≥ 7 (periodic transverse b.c.) or ≥ 8 (free transverse b.c.). Along the way, we prove that a noninteger Beraha number (except perhaps B10) cannot be a chromatic root of any graph. Key Words: Chromatic polynomial; chromatic root; antiferromagnetic Potts model; square lattice; transfer matrix; FortuinKasteleyn representation; TemperleyLieb algebra; Beraha–Kahane–Weiss theorem; Beraha numbers.
The LeeYang and PólyaSchur 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 nonvanishing 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 nonvanishing 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 infinitedimensional spaces of multivariate polynomials preserving the property of being nonvanishing when the variables are in prescribed open circular domains. In particular, this supersedes [7, 9] and solves the higher dimensional counterpart of a longstanding classification problem originating from classical works of Hermite, Laguerre,
Generalized duality for graphs on surfaces and the signed BollobasRiordan polynomial
 J. COMBIN. THEORY SER. B
, 2008
"... We generalize the natural duality of graphs embedded into a surface to a duality with respect to a subset of edges. The dual graph might be embedded into a different surface. We prove a relation between the signed BollobásRiordan polynomials of dual graphs. This relation unifies various recent re ..."
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Cited by 31 (2 self)
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We generalize the natural duality of graphs embedded into a surface to a duality with respect to a subset of edges. The dual graph might be embedded into a different surface. We prove a relation between the signed BollobásRiordan polynomials of dual graphs. This relation unifies various recent results expressing the Jones polynomial of links as specializations of the BollobásRiordan polynomials.
COMBINATORICS AND GEOMETRY OF POWER IDEALS
, 2009
"... We investigate ideals in a polynomial ring which are generated by powers of linear forms. Such ideals are closely related to the theories of fat point ideals, Cox rings, and box splines. We pay special attention to a family of power ideals that arises naturally from a hyperplane arrangement A. We pr ..."
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Cited by 28 (2 self)
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We investigate ideals in a polynomial ring which are generated by powers of linear forms. Such ideals are closely related to the theories of fat point ideals, Cox rings, and box splines. We pay special attention to a family of power ideals that arises naturally from a hyperplane arrangement A. We prove that their Hilbert series are determined by the combinatorics of A, and can be computed from its Tutte polynomial. We also obtain formulas for the Hilbert series of certain closely related fat point ideals and zonotopal Cox rings. Our work unifies and generalizes results due to DahmenMicchelli, HoltzRon, PostnikovShapiroShapiro, and SturmfelsXu, among others. It also settles a conjecture of HoltzRon on the spline interpolation of functions on the lattice points of a zonotope.
Negatively correlated random variables and Mason’s conjecture for independent sets in matroids
 Ann. Comb
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Topological Graph Polynomials in Colored Group Field Theory
, 2009
"... In this paper we analyze the open Feynman graphs of the Colored Group Field Theory introduced in [1]. We define the boundary graph G ∂ of an open graph G and prove it is a cellular complex. Using this structure we generalize the topological (BollobásRiordan) Tutte polynomials associated to (ribbon ..."
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Cited by 24 (4 self)
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In this paper we analyze the open Feynman graphs of the Colored Group Field Theory introduced in [1]. We define the boundary graph G ∂ of an open graph G and prove it is a cellular complex. Using this structure we generalize the topological (BollobásRiordan) Tutte polynomials associated to (ribbon) graphs to topological polynomials adapted to Colored Group Field Theory graphs in arbitrary dimension.
Topological graph polynomials and quantum field theory, Part I: Heat kernel theories
, 2008
"... We investigate the relationship between the universal topological polynomials for graphs in mathematics and the parametric representation of Feynman amplitudes in quantum field theory. In this first paper we consider translation invariant theories with the usual heatkernelbased propagator. We show ..."
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Cited by 23 (4 self)
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We investigate the relationship between the universal topological polynomials for graphs in mathematics and the parametric representation of Feynman amplitudes in quantum field theory. In this first paper we consider translation invariant theories with the usual heatkernelbased propagator. We show how the Symanzik polynomials of quantum field theory are particular multivariate versions of the Tutte polynomial, and how the new polynomials of noncommutative quantum field theory are particular versions of the BollobásRiordan polynomials.
Knot Invariants and the BollobásRiordan Polynomial of embedded graphs
, 2006
"... For a graph G embedded in an orientable surface Σ, we consider associated links L(G) in the thickened surface Σ × I. We relate the HOMFLY polynomial of L(G) to the recently defined BollobásRiordan polynomial of a ribbon graph. This generalizes celebrated results of Jaeger and Traldi. We use knot th ..."
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Cited by 21 (6 self)
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For a graph G embedded in an orientable surface Σ, we consider associated links L(G) in the thickened surface Σ × I. We relate the HOMFLY polynomial of L(G) to the recently defined BollobásRiordan polynomial of a ribbon graph. This generalizes celebrated results of Jaeger and Traldi. We use knot theory to prove results about graph polynomials and, after discussing questions of equivalence of the polynomials, we go on to use our formulae to prove a duality relation for the BollobásRiordan polynomial. We then consider the specialization to the Jones polynomial and recent results of Chmutov and Pak to relate the BollobásRiordan polynomials of an embedded graph and its tensor product with a cycle.
From a zoo to a zoology: Towards a general theory of graph polynomials
 Theory of Computing Systems
, 2007
"... Abstract. We outline a general theory of graph polynomials which covers all the examples we found in the vast literature, in particular, the chromatic polynomial, various generalizations of the Tutte polynomial, matching polynomials, interlace polynomials, and the cover polynomial of digraphs. We in ..."
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Cited by 21 (7 self)
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Abstract. We outline a general theory of graph polynomials which covers all the examples we found in the vast literature, in particular, the chromatic polynomial, various generalizations of the Tutte polynomial, matching polynomials, interlace polynomials, and the cover polynomial of digraphs. We introduce two classes of (hyper)graph polynomials definable in second order logic, and outline a research program for their classification in terms of definability and complexity considerations, and various notions of reducibilities. 1