Averages of Products and Ratios of Characteristic Polynomials in Polynomial Ensembles
- PDF / 548,490 Bytes
- 30 Pages / 439.37 x 666.142 pts Page_size
- 76 Downloads / 189 Views
Annales Henri Poincar´ e
Averages of Products and Ratios of Characteristic Polynomials in Polynomial Ensembles Gernot Akemann , Eugene Strahov and Tim R. W¨ urfel Abstract. Polynomial ensembles are a sub-class of probability measures within determinantal point processes. Examples include products of independent random matrices, with applications to Lyapunov exponents, and random matrices with an external field, that may serve as schematic models of quantum field theories with temperature. We first analyse expectation values of ratios of an equal number of characteristic polynomials in general polynomial ensembles. Using Schur polynomials, we show that polynomial ensembles constitute Giambelli compatible point processes, leading to a determinant formula for such ratios as in classical ensembles of random matrices. In the second part, we introduce invertible polynomial ensembles given, e.g. by random matrices with an external field. Expectation values of arbitrary ratios of characteristic polynomials are expressed in terms of multiple contour integrals. This generalises previous findings by Fyodorov, Grela, and Strahov. for a single ratio in the context of eigenvector statistics in the complex Ginibre ensemble.
1. Introduction In this paper, we study correlation functions of characteristic polynomials in a sub-class of determinantal random point processes. They are called polynomial ensembles [39] and belong to biorthogonal ensembles in the sense of Borodin [10]. Polynomial ensembles are characterised by the fact that one of the two determinants in the joint density of points is given by a Vandermonde determinant, while the other one is kept general. Thus they are generalising the classical ensembles of Gaussian random matrices [41]. Polynomial ensembles appear in various contexts as the joint distribution of eigenvalues (or singular values) of random matrices, see [3,14,19,20,27]. They enjoy many
G. Akemann et al.
Ann. Henri Poincar´e
invariance properties on the level of joint density, kernel and bi-orthogonal functions [35,38] and provide examples for realisations of multiple orthogonal polynomials, see e.g. [8,20,39] and Muttalib–Borodin ensembles [10,42]. Random matrices enjoy many different applications in physics and beyond, see [2] and references therein. Polynomial ensembles in particular are relevant in the following contexts: Ensembles with an external field have been introduced as a tool to count intersection numbers of moduli spaces on Riemann surfaces [16]. In the application to the quantum field theory of the strong interactions, quantum chromodynamics (QCD), they have been used as a schematic model to study the influence of temperature in the chiral phase transition [29]. Detailed computations of Dirac operator eigenvalues [28,45] within this class of models have been restricted to supersymmetric techniques so far, that can now be addressed in the framework of biorthogonal ensembles. Recently, sums and products of random matrices have been shown to lead to polynomial ensembles [3,18,36]—see [4] for a revi
Data Loading...
