TY - JOUR

T1 - Averages of Products and Ratios of Characteristic Polynomials in Polynomial Ensembles

AU - Akemann, Gernot

AU - Strahov, Eugene

AU - Würfel, Tim R.

N1 - Publisher Copyright:
© 2020, The Author(s).

PY - 2020/12

Y1 - 2020/12

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85092454172&partnerID=8YFLogxK

U2 - 10.1007/s00023-020-00963-9

DO - 10.1007/s00023-020-00963-9

M3 - Article

AN - SCOPUS:85092454172

SN - 1424-0637

VL - 21

SP - 3973

EP - 4002

JO - Annales Henri Poincare

JF - Annales Henri Poincare

IS - 12

ER -