Relations between generalised Wishart matrices, the Muttalib-Borodin model and matrix spherical functions (Q6631546)

Generalised uncorrelated Wishart matrices are formed out of rectangular standard Gaussian data matrices with a certain pattern of zero entries. The development of the theory in the real and complex cases has been done separately. In the real case the contributions have emphasized on the Bellman and Riesz distributions, while the complex case they are closely related to the Muttalib-Borodin model.\N\NIn the first situation, let consider the probability density function (PDF) on the space of real positive definite matrices\N\[\N\frac{1}{\mathcal{N}^{(1)}_{N}(\alpha)} q^{(1)}_{\alpha}(W^{(1)}) \exp (- \mathrm{Trace} (W^{(1)}) /2) \chi_{W^{(1)}>0} \tag{1}.\N\]\NHere \(\mathcal{N}^{(1)}_{N}(\alpha)\) and \(q^{(1)}_{\alpha} \) are real numbers associated with \(W^{(1)}\) and \(\alpha=(\alpha_{1}, \dots, \alpha_{N}),\alpha_{i} >-1, i=1, 2, \dots, N.\) For notational purposes \(\chi_{A} =1\) for \(A\) true and \(\chi_{A}=0,\) otherwise,\N\NLet \(Y^{(1)}\) be a real upper triangular \(N \times N\) random matrix with diagonal entries distributed by the requirement that its square has distribution \(\Gamma[1/2 (\alpha_{k} +1), 1/2], \alpha_{k} >-1,\) while its strictly upper triangular entries are distributed as standard real Gaussians. The joint element PDF of \(W^{(1)}=Y^{{(1)}^T}Y^{(1)}\) is then as in (1). Alternatively, with \(n, n > N, \) a positive integer, let assume the nonnegative integer numbers \(\alpha_{i}\) satisfy \(\alpha_{1} +1 \leq \alpha_{2} +2 \leq \alpha_{N}+N <n, ~~(2), \) and introduce the \(n \times N\) rectangular random matrices \(\tilde{Y}^{(1)}\) with the requirement that \(\tilde{Y}^{(1)}_{j,k}=0, j> k + \alpha_{k},\) as well as the other entries not prescribed to equal zero be independent standard real Gaussians. The joint element PDF of the positive definite random matrix \(\tilde{W}^{(1)}=\tilde{Y}^{{(1)}^T}\tilde{Y}^{(1)}\) is then as in (1), see [\textit{E. Veleva}, ``Stochastic representations of the Bellman gamma distribution'', in: Proceedings of the International Conference on Theory and Applications of Mathematics and Informatics, ICTAMI'09. 463--474 (2009)].\N\NIn the complex case, let us remind that this model of random matrix was first introduced in the study of quantum transport in [\textit{K. A. Muttalib}, J. Phys. A 28, No. 5, L159--L164 (1995; \url{doi:10.1088/0305-4470/28/5/003})], and subsequently it yields the exact evaluation of its limiting hard edge \(k\)-point correlation in [\textit{A. Borodin}, Nucl. Phys., B 536, No. 3, 704--732 (1999; Zbl 0948.82018)], where a biorthogonal structure is isolated.\N\NIn such a case, let consider the class of eigenvalue PDF on the space of complex positive definite matrices given by\N\[\N\frac{1}{ \prod_{l=1}^{N} l! \Gamma( \theta(l-1) + c+1)}\prod x_{l} ^{c} e^{-x_{l}} \chi_{x_{l}>0} \prod_{1\leq j <k \leq N}(x_{j} -x_{k})(x^{\theta}_{j} -x^{\theta}_{k})\tag{3}.\N\]\NLet \(Y^{(2)}\) be a complex upper triangular \(N \times N\) random matrix with diagonal entries distributed by the requirement that the square of its absolute value has distribution \(\Gamma[1/2 (\alpha_{k} +1), 1/2]\) with strictly upper triangular entries distributed as standard complex Gaussians. Introduce the positive definite random matrices \(W^{(2)}=Y^{{(2)}^{\dag}}Y^{(2)}\) and let denote by \(x_{j}, j=1, 2, \dots, N,\) the corresponding eigenvalues. For the choice \(\alpha_{k} = \theta (k-1) + c, \theta \geq0, c\geq -1,\) the eigenvalue PDF of \(W^{(2)}\) is given by (3), see [\textit{D. Cheliotis}, Stat. Probab. Lett. 134, 36--44 (2018; Zbl 1390.60029)].\N\NOn the other hand, for \(\alpha_{k} >-1\), if the parameters \(\alpha_{k}\) are nonnegative integers and satisfy the inequalities (2),then for the matrix \(W^{(2)}\) defined as above the eigenvalue PDF is given by\N\[\N\frac{1}{N!}\frac{1}{\prod_{l=1}^{N} \Gamma(\alpha_{l} +1)}\frac{1}{\prod_{1\leq j <k \leq N}(\alpha_{j} -\alpha_{k})}\N\prod_{l=1}^{N} e^{-x_{l}} \chi_{x_{l}>0} \prod_{1\leq j<k \leq N} (x_{j} -x_{k})\det [x_{j}^{\alpha_{k}}]_{j,k=1}^{N}.\N\]\NIn a similar way as in the real case, let consider \(n\times N\) complex rectangular random matrices \(Y^{(2)},\) and let assume that \(\tilde{Y}^{(2)}_{j,k}=0, j> k + \alpha_{k}\) as well as the other entries not prescribed to equal zero be independent standard complex Gaussians. Then the positive definite random matrix \(\tilde{W}^{(2)}=\tilde{Y}^{{(2)}^{\dag}}\tilde{Y}^{(2)}\) has the same joint element PDF as \(W^{(2)}\) and, in particular, the eigenvalue PDF of \(\tilde{W}^{(2)}\) is the same as the \(W^{(2)},\) see [\textit{P. J. Forrester} and \textit{D. Wang}, Electron. J. Probab. 22, Paper No. 54, 43 p. (2017; Zbl 1366.60009)].\N\NIn the contribution under review, the functional form of the joint element PDF of \(W^{(2)}\) and \(\tilde{W}^{2)}\) as well as the functional form of the joint element PDF of the correlation matrix corresponding to \(W^{(2)}\) and \(\tilde{W}^{2)}\) are deduced. In a next step, the eigenvalue PDF implied by the joint element PDF for \(W^{1)}\) is obtained. Finally, the evaluation of the characteristic polynomial and the large \(N\) limit of the normalised eigenvalue density for \(\tilde{W}^{1)}\) is given.\N\NThus, uniting the lines of development in the real and complex case, a tie-in with matrix spherical functions is identified in the context of deducing the eigenvalue probability density function from the joint element probability density function.