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144
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 51 (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 42 (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 37 (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 32 (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 29 (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 26 (5 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.
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 23 (8 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
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.
ON THE COMPLEXITY OF THE INTERLACE POLYNOMIAL
, 2008
"... We consider the twovariable interlace polynomial introduced by Arratia, Bollobás and Sorkin (2004). We develop two graph transformations which allow us to derive pointtopoint reductions for the interlace polynomial. Exploiting these reductions we obtain new results concerning the computational co ..."
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Cited by 21 (5 self)
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We consider the twovariable interlace polynomial introduced by Arratia, Bollobás and Sorkin (2004). We develop two graph transformations which allow us to derive pointtopoint reductions for the interlace polynomial. Exploiting these reductions we obtain new results concerning the computational complexity of evaluating the interlace polynomial at a fixed point. Regarding exact evaluation, we prove that the interlace polynomial is #Phard to evaluate at every point of the plane, except at one line, where it is trivially polynomial time computable, and four lines and two points, where the complexity mostly is still open. This solves a problem posed by Arratia, Bollobás and Sorkin (2004). In particular, we observe that three specializations of the twovariable interlace polynomial, the vertexnullity interlace polynomial, the vertexrank interlace polynomial and the independent set polynomial, are almost everywhere #Phard to evaluate, too. For the independent set polynomial, our reductions allow us to prove that it is even hard to approximate at every point except at −1 and 0.