On Random Matrix Averages Involving Half-Integer Powers of GOE Characteristic Polynomials

1.1 Introduction

Although our methods are specifically tailored for dealing with the GOE we expect our results in the bulk scaling limit to be universal and shared by a broad class of invariant measures on real symmetric matrices \(H\) [8] and by so-called Wigner ensembles of random real symmetric matrices with independent, identically distributed entries satisfying relevant moments conditions [9, 10].

1.2 Motivations and Background

To explain the origin of interest in the correlation functions (1) we start with recalling that the phenomenon of Quantum Chaos attracted considerable theoretical and experimental interest for more than three decades and remains one of the areas where applications of Random Matrix Theory are most fruitful and successful [11]. The applications are based on the famous Bohigas–Giannoni–Schmit (BGS) [12] conjecture claiming that in appropriately chosen energy window sequences of highly excited discrete energy levels of generic quantum systems whose classical counterparts are chaotic are statistically indistinguishable from sequences of real eigenvalues of large random matrices of appropriate symmetry. Although not yet fully rigorously proven, this conjecture has an overwhelming support in experimental, numerical and analytical work of the last decades [13]. Inspired by this analogy as well as by the fact of universality of many random matrix properties (i.e. insensitivity to the particular choice of the probability measure on the matrix space), see [9, 10] and references therein, one of the common strategies for predicting universal observables of quantum chaotic systems has been expressing them in terms of resolvents of underlying Hamiltonians, then replacing the actual Hamiltonians by random matrices taken from analytically tractable (usually, Gaussian) ensembles of \(N\times N\) random matrices. The characteristic functions of the probability densities of the observables under consideration can be frequently computed explicitly by appropriate ensemble averages. Note that the eigenvalues of the standard Gaussian Ensembles, Unitary (GUE, \(\beta =2\)), Orthogonal (GOE, \(\beta =1\)) or Symplectic (GSE, \(\beta =4\)) are independent of the eigenvectors, with the matrix of \(N\) orthonormal eigenvectors being uniformly distributed over the Haar’s measure of the Unitary \(U(N)\), Orthogonal \(O(N)\) or Symplectic \(Sp(2N)\) group, correspondingly. To that end it is natural to evaluate the corresponding characteristic functions by performing first the ensemble average over the eigenvectors. For the \(\beta =2\) case the average can be frequently done exactly for any \(N\) by employing the so-called Itzykson–Zuber–Harish–Chandra [14, 15] formula, which is not yet available for \(\beta =1,4\) group averages. Nevertheless, one is able to perform the eigenvector averages in the limit \(N\gg 1\) by using a heuristic idea (going back to [16]) that the set of eigenvectors essentially behaves for \(N\gg 1\) as if their components were independent, identically distributed Gaussian variables with mean zero and variance \(1/N\). One can rigorously justify this procedure if only a number \(n\ll N^{1/2}\) of eigenvectors is involved in the set, see e.g. [17], but in general a rigorous justification of such a step requires some nontrivial estimates on the resolvents. The heuristic procedure is widely employed in Theoretical Physics for RMT applications to Quantum Chaos using the properties of the standard Gaussian integrals over complex or real variables. In this way the analysis of many distributions of practical interest is reduced to correlation functions of products and ratios involving integer (for \(\beta =2,4\)) or half-integer (for \(\beta =1\)) powers of characteristic polynomials of random matrices. Similar averages arise if one is interested in statistics of the matrix elements of the resolvents computed in the basis of random Gaussian vectors, as it is frequently done in applications to scattering systems with Quantum Chaos, see e.g. the recent paper [18] for an example and further references. For those and other reasons averages of products and ratios of powers of characteristic polynomials of random matrices attracted much interest over the years. When only integer powers are involved in the average the corresponding theory was developed for \(\beta =2\) in [2, 3, 4] and extended to \(\beta =1,4\) in [5]. The case of half-integer powers for \(\beta =1\) remains however outstanding, despite the fact that it is most relevant for an overwhelming majority of experiments in Quantum Chaos due to the preserved time-reversal invariance of the underlying Hamiltonians. Additional interest to this type of averages gives the fact that they are closely related to the problem of evaluating averages of quantities involving absolute values of characteristic polynomials due to the relation \(|\det (E-H)| = \lim _{\epsilon \rightarrow 0} \det (E-H+\tfrac{i\epsilon }{N})^{1/2} \det (E-H-\tfrac{i\epsilon }{N})^{1/2}\) valid for matrices \(H\) with real eigenvalues. Such averages emerge, for example, when studying the statistics of the so-called “level curvatures” in quantum chaotic systems [19, 20], see Eq. (5) below, as well as in the problem of counting the number of stationary points of random Gaussian surfaces, see [21, 22].

1.3 The Results

• As it has been mentioned above, we were not yet able to reveal nice mathematical structures for (1) at finite values of the matrix size \(N\) beyond the simplest case \(K=1, L=1\), where the methods outlined below yielded a determinantal structure which we give here for completeness:

  $$\begin{aligned} \mathcal {C}_{1,1}(\mu _F;\mu _B)=&\left( \frac{J^2}{2N}\right) ^{N/4} \frac{[-i{{\mathrm{sgn}}}({{\mathrm{Im}}}(\mu _B))]^{N+1}}{ \Gamma (N/2)} \nonumber \\&\times \det {\left( \begin{array}{cc} H_{N-1}\left( \frac{\sqrt{N}}{J}\mu _F\right) &{} F_{N/2-1}\left( \frac{\sqrt{N}}{\sqrt{2}J}\mu _B\right) \\ H_{N}\left( \frac{\sqrt{N}}{J}\mu _F\right) &{} F_{N/2}\left( \frac{\sqrt{N}}{\sqrt{2}J}\mu _B\right) \end{array}\right) } \end{aligned}$$

  (12)

  where \(\Gamma (x)\) is the Euler Gamma-function, \(H_{N}(z)=\frac{i^N}{\sqrt{2\pi }}\int _{-\infty }^{\infty }dt\, t^N \exp [-\tfrac{1}{2}(t+iz)^2]\) is a Hermite polynomial and the function

  $$\begin{aligned} F_{N}(z)=[i {{\mathrm{sgn}}}({{\mathrm{Im}}}(z))]^N\int _0^{\infty }dt\, t^N\exp [-\tfrac{1}{2}(t^2+2i {{\mathrm{sgn}}}({{\mathrm{Im}}}(z))z t) ] \end{aligned}$$

  may be associated with the Cauchy transforms of Hermite polynomials [2].
• The explicit forms for the “bulk” correlation functions \(\mathcal {C}^{(\mathrm{bulk})}_{1,2}(\omega _{F1};\omega _{B1},\omega _{B2})\) (see Eq. (2)) and \(\mathcal {C}^{(\mathrm{bulk})}_{2,2}(\omega _{F1},\omega _{F2};\omega _{B1},\omega _{B2})\) (see Eq. (3)) depend very essentially on the signs of \(\omega _{B1}\) and \(\omega _{B2}\). In particular, if \({{\mathrm{sgn}}}\omega _{B1}={{\mathrm{sgn}}}\omega _{B2}\) the first correlation function is given by

  $$\begin{aligned} \mathcal {C}^{(\mathrm{bulk, } {{\mathrm{sgn}}}\omega _{B1}={{\mathrm{sgn}}}\omega _{B2})}_{1,2}(\omega _{F1};\omega _{B1},\omega _{B2})\approx e^{\frac{2\omega _{F1}-\omega _{B1}-\omega _{B2}}{4J^2}(iE+{{\mathrm{sgn}}}{\omega _B} \sqrt{4J^2-E^2})}, \end{aligned}$$

  (13)

  whereas for \({{\mathrm{sgn}}}\omega _{B1}=-{{\mathrm{sgn}}}\omega _{B2}\) the same object takes instead the form

  $$\begin{aligned}&\mathcal {C}^{(\mathrm{bulk, }{{\mathrm{sgn}}}\omega _{B1}=-{{\mathrm{sgn}}}\omega _{B2})}_{1,2}(\omega _{F1};\omega _{B1},\omega _{B2}) \approx \frac{(-i)^N}{\pi \sqrt{2 N \rho }(2J)^{N+1}}\,e^{-\frac{iE}{4J^2}(\omega _{B1}+\omega _{B2}-2\omega _{F1})} \nonumber \\&\times \bigg \{[Ae^{-\pi \rho \omega _{F1}} -(-1)^N A^*e^{+\pi \rho \omega _{F1}} ](\omega _{B1}+\omega _{B2}-2\omega _{F1})K_0\left( \tfrac{\pi \rho }{2}|\omega _{B1}-\omega _{B2}|\right) \nonumber \\&+[Ae^{-\pi \rho \omega _{F1}}+(-1)^N A^*e^{+\pi \rho \omega _{F1}}]|\omega _{B1}-\omega _{B2}|K_1\left( \tfrac{\pi \rho }{2}|\omega _{B1}-\omega _{B2}|\right) \bigg \} \end{aligned}$$

  (14)

  with

  $$\begin{aligned} A(E,N)=(2\pi J^2 \rho +iE)^{N-1/2}\ e^{\frac{i\pi N}{2}\rho E}, \end{aligned}$$

  (15)

  where we introduced \(\rho =\frac{1}{2\pi J^2}\sqrt{4J^2-E^2}\) for the mean eigenvalue density of large GOE matrices in the bulk of the spectrum and used the standard notation \(K_m(z)\) for the modified Bessel (Macdonald) functions of second kind and index \(m\). Note that the asymptotic expression (14) shows an interesting “parity effect”: it behaves differently depending on whether \(N\) is even or odd for arbitrary large values of \(N\).
  Similarly the second correlation function for \({{\mathrm{sgn}}}\omega _{B1}={{\mathrm{sgn}}}\omega _{B2}\) is given by

  $$\begin{aligned}&\mathcal {C}^{(\mathrm{bulk, }{{\mathrm{sgn}}}\omega _{B1}={{\mathrm{sgn}}}\omega _{B2})}_{2,2}(\omega _{F1},\omega _{F2};\omega _{B1},\omega _{B2})\approx \nonumber \\&\left( \frac{J}{\sqrt{N}}\right) ^N \frac{3\tilde{H}_N\left( \frac{\sqrt{N}E}{J}\right) }{[\pi \rho (\omega _{F1}-\omega _{F2})]^3} e^{\frac{iE(\omega _{F1}+\omega _{F2})}{2J^2}}\,e^{-\frac{iE(\omega _{B1}+\omega _{B2})}{4J^2}} e^{-\frac{\pi \rho (|\omega _{B1}|+|\omega _{B2}|)}{2}} \nonumber \\&\times \left[ \pi \rho (\omega _{F1}-\omega _{F2})\cosh \left( \pi \rho (\omega _{F1}-\omega _{F2})\right) -\sinh \left( \pi \rho (\omega _{F1}-\omega _{F2})\right) \right] \!, \end{aligned}$$

  (16)

  where \(\tilde{H}_N\left( \frac{\sqrt{N}E}{J}\right) =\sqrt{2}\left( \frac{iN}{2J}\right) ^N e^{-N/2}\, e^{\frac{N}{4J^2}E^2} [(-1)^N A(E,N)+ A^*(E,N)]\) is the appropriate large-\(N\) asymptotic of the \(N\)-th Hermite polynomial, with \(A(E,N)\) defined in Eq. (15). In the case \({{\mathrm{sgn}}}\omega _{B1}=-{{\mathrm{sgn}}}\omega _{B2}\) we get instead

  $$\begin{aligned}&\mathcal {C}^{(\mathrm{bulk, }{{\mathrm{sgn}}}\omega _{B1}=-{{\mathrm{sgn}}}\omega _{B2} )}_{2,2}(\omega _{F1},\omega _{F2};\omega _{B1},\omega _{B2}) \approx \nonumber \\&\sqrt{\frac{2N}{\pi }} \frac{J^{N+1} e^{-N/2}}{(\omega _{F1}-\omega _{F2})^3} e^{\frac{N}{4J^2}E^2} e^{\frac{iE(\omega _{F1}+\omega _{F2})}{2J^2}}\,e^{-\frac{iE(\omega _{B1}+\omega _{B2})}{4J^2}} \nonumber \\&\bigg \{ [(\omega _{F1}+\omega _{F2})(\omega _{B1}+\omega _{B2})-2\omega _{F1}\omega _{F2}-2\omega _{B1}\omega _{B2}] K_0 \left( \tfrac{\pi \rho }{2}|\omega _{B1}-\omega _{B2}| \right) \nonumber \\&\qquad \times \left[ \pi \rho (\omega _{F1}-\omega _{F2})\cosh \left( \pi \rho (\omega _{F1}-\omega _{F2})\right) -\sinh \left( \pi \rho (\omega _{F1}-\omega _{F2})\right) \right] \nonumber \\&\qquad + \pi \rho (\omega _{F1}-\omega _{F2})^2 |\omega _{B1}-\omega _{B2}| \sinh \left( \pi \rho (\omega _{F1}-\omega _{F2})\right) K_1 \left( \tfrac{\pi \rho }{2}|\omega _{B1}-\omega _{B2}| \right) \bigg \}. \end{aligned}$$

  (17)

  Note that the parity of \(N\) plays no role for the large-\(N\) behaviour of this correlation function.

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2.1 Evaluation of the Correlation Functions Eqs. (2) and (3)

So far our manipulations were exact and did not use any approximation. As was explained in the introduction we are mainly interested in extracting the “bulk” large-\(N\) asymptotic of these correlation functions. The most natural way to proceed from here is by performing a saddle-point analysis. We believe with due effort such analysis can be done with full mathematical rigor, see e.g. a recent paper [49], but we do not attempt it here concentrating on explaining the gross structures of the saddle-point analysis which yield the correct results.

2.2 Distribution of \(K_{ab}\) via Eq.(8)

For the correlation function (8) associated with the distribution of an individual off-diagonal \(K\)-matrix element we consider for simplicity only the perfect coupling case \(E=0\) and \(\gamma _a=\gamma _b=1\), see Eq. (22)). For evaluating the ensemble average we first tried to follow the same method as described in the previous section. In this way we started with writing \(\det (H^2+\frac{x^2}{N^2})^{1/2}=\det (H+\frac{ix}{N})^{1/2}\det (H-\frac{ix}{N})^{1/2}\) and \(|\det H|=(\det H)^2 /|\det H| = \lim _{\epsilon \rightarrow 0} (\det H)^2 \det (H+\frac{i\epsilon }{N})^{-1/2}\det (H-\frac{i\epsilon }{N})^{-1/2}\) and then replaced the square roots of characteristic polynomials in the denominator by four Gaussian integrals over real commuting vectors and those in the numerator by Gaussian integrals over two vectors with anticommuting components. The ensemble averaging then yields a \(4 \times 4\) \(Q_B\)-matrix, but we found no efficient ways of evaluating the ensuing group integral over the diagonalizing matrices. We also attempted a direct saddle-point analysis for large \(N\) along the same lines as before, and found it to become very tedious as not only the zero-th and first, but also the second order of the integrand expansion in fluctuations around the relevant saddle points turned out to be vanishing at the saddle points. Expanding to an even higher order with the group integrals still present did not seem to us as a viable option.