An induced real quaternion spherical ensemble of random matrices
We study the induced spherical ensemble of non-Hermitian matrices with real quaternion entries (considering each quaternion as a complex matrix). We define the ensemble by the matrix probability distribution function that is proportional to
These matrices can also be constructed via a procedure called ‘inducing’, using a product of a Wishart matrix (with parameters ) and a rectangular Ginibre matrix of size . The inducing procedure imposes a repulsion of eigenvalues from and in the complex plane, with the effect that in the limit of large matrix dimension, they lie in an annulus whose inner and outer radii depend on the relative size of , and .
By using functional differentiation of a generalized partition function, we make use of skew-orthogonal polynomials to find expressions for the eigenvalue -point correlation functions, and in particular the eigenvalue density (given by ).
We find the scaled limits of the density in the bulk (away from the real line) as well as near the inner and outer annular radii, in the four regimes corresponding to large or small values of and . After a stereographic projection the density is uniform on a spherical annulus, except for a depletion of eigenvalues on a great circle corresponding to the real axis (as expected for a real quaternion ensemble). We also form a conjecture for the behaviour of the density near the real line based on analogous results in the and ensembles; we support our conjecture with data from Monte Carlo simulations of a large number of matrices drawn from the induced spherical ensemble.
1 Introduction and main results
Non-Hermitian random matrices largely began with the pioneering work of Ginibre in 1965 [Gini1965], which discussed three ensembles of matrices having independent real, complex and real quaternion111When we say ‘real quaternion’ we mean a quaternion in the sense of a number , where and obeys the quaternionic multiplication and addition rules. These real quaternions can be represented as complex matrices and it is in this representation that we calculate the (complex) eigenvalues of real quaternionic matrices. We provide an overview of quaternionic definitions and properties in Appendix LABEL:app:quatPf. random entries respectively, in keeping with Dyson’s three-fold way [Dyso1962]. As with Hermitian ensembles, these non-Hermitian ensembles correspond to the indices respectively, which represent the number of independent real components in each matrix entry.
More recently, various other non-Hermitian ensembles have attracted interest (see [Akem2005, ForrNaga2007, KhorSommZycz2010, KhorSomm2011, AkemPhil2014] for a small selection). One particular categorization relevant to the present work is the ‘geometrical triumvirate’ of ensembles described in [Kris2006, HougKrisPereVira2009, Mays2011, Fisc2013], which identifies random matrix ensembles with the three classical surfaces of constant curvature: the plane, the sphere and the pseudo- or anti-sphere. We leave the interested reader to investigate for themselves all the details contained in those works; here we highlight only the spherical ensemble, which is given by the matrix ‘ratio’
where and are independent Gaussian matrices (i.e., they are drawn from the Ginibre ensembles, which correspond to the plane), and is non-singular. By analogy with Cauchy random variables (which can be described as the ratio of two Gaussian random variables) these matrices have been called Cauchy matrices [EdelKostShub1994], and have the matrix Cauchy distribution function [Fein2004, HougKrisPereVira2009, ForrMays2011, Mays2013]
where the ‘dagger’ should be interpreted as ‘transpose’, ‘Hermitian conjugate’ or ‘quaternion dual’ for (real matrices), (complex matrices) and (real quaternion matrices) respectively. Note that for , the determinant is to be understood as a quaternion determinant (see Appendix LABEL:app:quatPf). We note also that the eigenvalues of the matrix defined in (1) are equal to the generalized eigenvalues of the pair , given as the solutions to the equation
which is the viewpoint of [EdelKostShub1994].
As in the case of the Ginibre ensembles, the eigenvalue density has distinctive symmetries depending on the value of :
( and complex): rotational symmetry in the complex plane;
( and real): positive density of eigenvalues along the real axis, with reflective symmetry across the real axis;
( and real quaternion): depletion of eigenvalues near the real axis, with reflective symmetry across the real axis.
(The reflective symmetry is a property of all finite-size real and real quaternion matrices, where the non-real eigenvalues come in complex-conjugate pairs.) The reason for the name ‘spherical ensemble’ is that the eigenvalues of have uniform distribution (under stereographic projection) on the unit sphere in the limit of large matrix dimension, which is a consequence of the spherical law [Roge2010, ForrMays2011, Bord2011], a result analogous to the more famous circular law for Ginibre matrices (see for example [Girk1984, Bai1997, GoetTikh2010, TaoVuKris2010]). For details concerning the eigenvalue statistics of these ensembles the interested reader may refer to [HougKrisPereVira2009, Mays2011], in addition to the works listed above. One may also seek a physical interpretation of these processes in terms of minimizing some energy function on a sphere, in which case we refer the reader to [LeCaHo1990, ArmeBeltShub2011, BoroSerf2013].
Each class in the geometrical triumvirate can be generalized by the introduction of parameters whose effect is to restrict the eigenvalue density to an annulus in the complex plane through a procedure called ‘inducing’ (we provide a brief overview of this procedure in Appendix LABEL:s:genrand, but comprehensive descriptions are given in [FBKSZ2012, FiscForr2011, Fisc2013]). However, we take as our definition of the induced spherical ensemble those matrices that are defined by the matrix probability density function (pdf)
where and ; as mentioned above the parameter corresponds to matrices with real (), complex () or real quaternion () entries. The normalization constant is given by
The ensemble corresponding to was the subject of [FiscForr2011], while that corresponding to was discussed in [Fisc2013]; the aim of the work in this paper is to study the eigenvalue statistics of the analogous real quaternion ensemble. First note that with , , (3) reduces to (2) and we are back in the regime of the spherical law, which was mentioned above. The result of the generalization (3) is to keep the eigenvalues away from the origin and , effectively squeezing the support into an annulus. This annulus projects (stereographically) to a belt of eigenvalues centered on the great circle corresponding to the circle . As an aid to visualization in Figures LABEL:f:quatBB–LABEL:f:quatSS of Section LABEL:s:slims we present simulated eigenvalue plots for and their stereographic projections onto the unit sphere. In brief, as in the and cases we find four regimes that correspond to large and small values (compared to ) of and . Although we take the pdf (3) to be our definition of the induced spherical matrices, as alluded to above, it is possible to explicitly construct them from products of Wishart and Ginibre matrices. While the real quaternion construction is a natural modification of the discussions in the above references, there are some subtleties related to numerical Monte Carlo simulations of the real quaternion induced spherical ensemble and so we make some comments on this point in Appendix LABEL:s:genrand.
We note that these matrices are similar to the class of matrices that relate to the Feinberg–Zee single ring theorem, which was discussed in [FeinZee1997] and rigorously proved in [GuioKrisZeit2009]. The theorem states that for complex matrices from a distribution
where is a polynomial with positive leading coefficient, the support of the eigenvalue density tends toward an annulus around the origin, and the density is rotationally symmetric. From the figures in Section LABEL:s:slims we see that the eigenvalue densities certainly have these properties, yet (3) is a special case of (5) only formally (in the sense that we require to be a general analytic function). More work is needed to make this connection precise.
The explicit goal of the present work is to calculate the eigenvalue correlation functions and various scaled limits of the eigenvalue density for the real quaternion matrices drawn from the distribution (3), which therefore generalizes the results in [Mays2013]. As mentioned above, the complex analogue of this work was presented in [FiscForr2011, Fisc2013] while the real case can be found in [Fisc2013]. Since quaternions, quaternion determinants and Pfaffians play a crucial role in this work we provide a review in Appendix LABEL:app:quatPf.
To obtain our results we will make use of a generalized partition function, which, for a general joint probability density function (jpdf) in variables , is defined by the average
where are some well-behaved functions in the variables . In [Sinc2007] it was shown that (6) can be written in a convenient Pfaffian form for various eigenvalue jpdfs, of which the one considered in this work is an example. This allows us to follow [ForrNaga2007, BoroSinc2009] and use (6) to calculate the eigenvalue correlation functions. For a general jpdf the -point correlation function is defined by
in terms of which the eigenvalue density is given by , with the normalization
Equivalently we can obtain the correlation functions via functional differentiation of the generalized partition function
Our method here falls into the category of (skew-)orthogonal polynomial methods and, as such, we will have need of the polynomials corresponding to the generalized partition functions (6). It is not yet known how to complete a calculation analogous to that in [Fisc2013] for , where the skew-orthogonal polynomials are deduced directly from an average over characteristic polynomials, however, a result from [Forr2013] furnishes us with the necessary expressions. Armed with these polynomials we establish the eigenvalue correlation functions in Propositions 4.2. From these correlation functions we find (with ) that the eigenvalue density (normalized according to (7)) is
Having established the correlation functions for finite matrix sizes , we then analyze various scaled limits of the eigenvalue density in Section LABEL:s:slims. As discussed above, it is known from the spherical law that for spherical matrices (2) the eigenvalue density is uniform (under stereographic projection) on the unit sphere. The figures in Section LABEL:s:slims suggest the eigenvalue support is restricted to an annulus in the complex plane for large matrix size, the inner and outer radii of which depend on the relative sizes of , and . Indeed, as in [Fisc2013], we can identify four regimes of interest as : (i) , ; (ii) , ; (iii) , ; and (iv) , . Broadly speaking, for large the eigenvalues are repulsed from the origin (which corresponds to the south pole), and for large the eigenvalues are repulsed from infinity (the north pole). While we find that we can calculate the limiting bulk and annular edge densities in these four regimes, we are not yet able to derive the density near the real line. We present a conjecture for this in Section LABEL:s:gencorrelns4 along with some simulated data to support the claim. Further, we discuss a differential equation, which, if it was to be solved, should also yield the asymptotics for the full eigenvalue correlation function in this and similar ensembles — however, based upon the structure of the equation (and similar difficulties in related studies, eg [Ipse2015]) this appears a remote possibility.
1.1 Some notational conventions
To avoid confusion we state here some of the notation commonly used in this paper. We will usually use upper-case bold letters (e.g. ) to denote matrices, often with an accompanying subscript to denote the matrix dimension. We use the symbol to refer to ‘transpose’, ‘Hermitian conjugate’ or ‘quaternion dual’ for real, complex and real quaternion matrices, respectively; occasionally, in order to be clear on the matrix type, we will use the superscripts and to denote them explicitly. A detailed description of the properties of the relevant quaternion properties is contained in Appendix LABEL:app:quatPf.
Lower case bold letters are lists (they need not be ordered), e.g. , where the subscript denotes the length. In particular, the bold will always denote the list of eigenvalues of a system of size . Generally these eigenvalues will either be real or live in the upper half complex plane, that is .
We will denote the wedge product of complex and real quaternion quantities respectively by for and for . The wedge product of the differentials of the independent real entries of an object (a matrix or list) are then given by
where the indices run over all values corresponding to independent elements.
We make use of the (half-max) Heaviside step function
2 Normalization of the matrix pdf
The normalization in the complex case was presented in [FiscForr2011] and the real case in [Fisc2013]; by performing a similar procedure the normalization can also be calculated explicitly.
We search for such that
using the representation of the quaternion, according to the notation in Appendix LABEL:app:quatPf (where the superscript is the quaternion dual operation). Let , for which we have the Jacobian [Olki2002]
where is independent of , and (10) becomes
where is a unitary eigendecomposition of the block representation of , and similarly, is such that . For the second equality we have made use of the well-known Jacobian for changing variables from the matrix entries to the matrix eigenvalues (see for example [Forr2010, Chapter 1.3]). Now replace , giving and , which leads to the Selberg integral [Selb1944]
An evaluation of the integral over can be found in [Nach1965], however it won’t be necessary for our purposes. Using
we can calculate as in [Mays2013] and find
Substituting this into (11) we have
3 Eigenvalue jpdf
Here we change variables in the matrix pdf (3) to the eigenvalues for the real quaternion ensemble following the methods of [Mays2013] (which deals with the specified ensemble , ). The idea is to use a Schur decomposition
where is a symplectic matrix (i.e., a unitary real quaternion matrix) and is a (block) upper triangular matrix, whose diagonal blocks correspond to the eigenvalues of . We have the relation
between the matrix pdf and the eigenvalue jpdf , where the integral is understood to be over the variables relating to the eigenvectors. Performing the integral involves iteratively integrating column-by-column over the blocks in the strict upper triangle of (a technique introduced to this topic in [HougKrisPereVira2009]) as well as an integral over [Nach1965]. Except for the factors of coming from the numerator of (3) the procedure here is identical and so we will not include it in full; the interested reader is referred to [Mays2013, Fisc2013].
With , the eigenvalue jpdf for the real quaternion induced spherical ensemble is
In the definition of above we have kept the factor of separate from the powers of for clarity; this factor comes from splitting the factors into .
4 Eigenvalue correlation functions
As mentioned in the introduction, to find the eigenvalue correlation functions we will first find the generalized partition function (6) and then use the functional differentiation formula (8) to obtain the correlation functions. We know from [DeBr1955, Meht2004, Sinc2007] that pdfs of the form (12) can be transformed to a more convenient Pfaffian or quaternion determinant form using the method of integration over alternate variables via the Vandermonde identity. We state only the results here; the interested reader is referred to [Forr2010, Mays2011, Fisc2013] (in addition to those references mentioned above) for explicit details. (For the real quaternion ensemble we take in (6).)
The generalized partition function for the real quaternion induced spherical ensemble with eigenvalue jpdf (12) is
and the are monic polynomials of degree .
Note that the choice of the polynomials is not unique; indeed, following through the construction of the Pfaffian generalized partition function we find that we may choose any polynomials that satisfy the criteria of being monic and of degree . So, the task of obtaining the correlation functions will be greatly simplified if the polynomials can be chosen such that they skew-orthogonalize the matrix in (13), that is they reduce it to the form of (LABEL:e:skew_diag_mat), where the diagonal blocks are the matrices
with . Specifically, we define the skew-symmetric inner product
and look for polynomials to satisfy the skew-orthogonality conditions
We call these skew-orthogonal polynomials. Assuming the existence of polynomials satisfying (15) (these polynomials do indeed exist, see (LABEL:e:SOPS4)) then we can follow [ForrNaga2007, BoroSinc2009, Forr2010] to calculate the eigenvalue correlation functions from the generalized partition function above. We use the identity for general linear operators, or a Pfaffian or quaternion determinant analogue, to write the generalized partition function (13) as a Fredholm Pfaffian or quaternion determinant (see Appendix LABEL:app:quatPf),
which can then be substituted into (8) to immediately yield the correlation functions, with Pfaffian kernels
The details of the calculation are by now well established, and somewhat involved, so we refer the reader to the papers mentioned above, as well as to [Forr2010, Mays2011].
With polynomials skew-orthogonal with respect to the inner product of (14) the -point correlation function for the real quaternion induced spherical ensemble is
5 Skew-orthogonal polynomials
The expressions for the correlation kernel elements , , given in Proposition 4.2 depend on the skew-orthogonal polynomials (that is, polynomials satisfying (15)) — once we have these polynomials, then we have full knowledge of the correlation functions. In previous studies good use has been made of averages over characteristic polynomials to access the skew-orthogonal polynomials, or to avoid them entirely (see for example [BaikDeifStra2003, FyodSomm2003, Akem2005, FyodKhor2007a, AkemPhilSomm2009, KhorSommZycz2010, KhorSomm2011, Forr2010a]). In particular, for the real analogue of the real quaternion ensemble of this paper, [Fisc2013] uses exactly this method to find the skew-orthogonal polynomials and the eigenvalue density corresponding to (9).
However, the situation is somewhat different in the real quaternion case that we consider here: while we are able to write down an expression for the average over the characteristic polynomial in terms of the skew-orthogonal polynomials analogous to [Fisc2013, Corollary 4.1.11], it is not known how to evaluate it. In the following proposition we state this expression using a method of proof similar to that in [ForrMays2011].
With the characteristic polynomial for a real quaternion matrix,
where the average is over the jpdf (12) with eigenvalues.