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Generalised unitary group integrals of Ingham-Siegel and Fisher-Hartwig type

Gernot Akemann, Noah Aygün, Tim R. Würfel

Abstract

We generalise well-known integrals of Ingham-Siegel and Fisher-Hartwig type over the unitary group $U(N)$ with respect to Haar measure, for finite $N$ and including fixed external matrices. When depending only on the eigenvalues of the unitary matrix, they can be related to a Toeplitz determinant with jump singularities. After introducing fixed deterministic matrices as external sources, the integrals can no longer be solved using Andréiéf's integration formula. Resorting to the character expansion as put forward by Balantekin, we derive explicit determinantal formulae containing Kummer's confluent and Gauss' hypergeometric function. They depend only on the eigenvalues of the deterministic matrices and are analytic in the parameters of the jump singularities. Furthermore, unitary two-matrix integrals of the same type are proposed and solved in the same manner. When making part of the deterministic matrices random and integrating over them, we obtain similar formulae in terms of Pfaffian determinants. This is reminiscent to a unitary group integral found recently by Kanazawa and Kieburg.

Generalised unitary group integrals of Ingham-Siegel and Fisher-Hartwig type

Abstract

We generalise well-known integrals of Ingham-Siegel and Fisher-Hartwig type over the unitary group with respect to Haar measure, for finite and including fixed external matrices. When depending only on the eigenvalues of the unitary matrix, they can be related to a Toeplitz determinant with jump singularities. After introducing fixed deterministic matrices as external sources, the integrals can no longer be solved using Andréiéf's integration formula. Resorting to the character expansion as put forward by Balantekin, we derive explicit determinantal formulae containing Kummer's confluent and Gauss' hypergeometric function. They depend only on the eigenvalues of the deterministic matrices and are analytic in the parameters of the jump singularities. Furthermore, unitary two-matrix integrals of the same type are proposed and solved in the same manner. When making part of the deterministic matrices random and integrating over them, we obtain similar formulae in terms of Pfaffian determinants. This is reminiscent to a unitary group integral found recently by Kanazawa and Kieburg.
Paper Structure (10 sections, 11 theorems, 86 equations)

This paper contains 10 sections, 11 theorems, 86 equations.

Table of Contents

  1. Introduction and Main Results
  2. The Character Expansion Revisited
  3. Proof of Main Results
  4. Proof of Theorem \ref{['Thm-IS']}
  5. Proof of Corollary \ref{['Cor-TW-IS']} and consistency checks
  6. Proof of Theorem \ref{['Thm-GFH']}
  7. Proof of Corollary \ref{['cor:JIS']}
  8. Conclusions and Open Questions
  9. Properties of Binomial Coefficients and Gauß' Hypergeometric Function
  10. Andréiéf Identity and Standard Fisher-Hartwig Integral

Key Result

theorem 1.1

Let $A,B,C,D\in{\rm GL}(N,\mathbb{C})$ be fixed complex, invertible $N\times N$ matrices and denote by $\mu_j^2$ and $\nu_j^2$, $j=1,\ldots,N$, the eigenvalues of $AD$ and $BC$, respectively. Without loss of generality we can assume that $||C||,||D||<1$, and let $\alpha\in\mathbb{C}$ be fixed. Then, and with $g_N(\alpha):=\prod_{j=1}^{N}\frac{(j-1)!^2\Gamma(\alpha+N-j+1)}{\Gamma(\alpha+N)}$.

Theorems & Definitions (16)

  • theorem 1.1: Unitary Ingham-Siegel Integrals
  • corollary 1.2
  • theorem 1.3: Generalised Fisher-Hartwig Integrals
  • corollary 1.4: Integrated Unitary Ingham-Siegel Integral
  • corollary 2.1: Dimension Formulae
  • proof
  • lemma 2.2: Cauchy--Binet Identity
  • proposition 2.3
  • proof
  • proof
  • ...and 6 more