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This paper focuses on different reductions of the two-dimensional (2d)-Toda hierarchy. Symmetric and skew-symmetric moment matrices are first considered, resulting in differential relations between symmetric/skew-symmetric tau functions and 2d-Toda’s tau functions, respectively. Furthermore, motivated by the Cauchy two-matrix model and Bures ensemble from random matrix theory, we study the rank-one shift condition in the symmetric case and rank-two shift condition in the skew-symmetric case, from which the C-Toda and B-Toda hierarchies are determined, together with their special Lax matrices and integrable structures. [ABSTRACT FROM AUTHOR]

The theory of Pólya ensembles of positive definite random matrices provides structural formulas for the corresponding biorthogonal pair, and correlation kernel, which are well suited to computing the hard edge large N asymptotics. Such an analysis is carried out for products of Laguerre ensembles, the Laguerre Muttalib–Borodin ensemble, and products of Laguerre ensembles and their inverses. The latter includes, as a special case, the Jacobi unitary ensemble. In each case, the hard edge scaled kernel permits an expansion in powers of 1/N, with the leading term given in a structured form involving the hard-edge scaling of the biorthogonal pair. The Laguerre and Jacobi ensembles have the special feature that their hard edge scaled kernel – the Bessel kernel – is symmetric and this leads to there being a choice of hard edge scaling variables for which the rate of convergence of the correlation functions is O (1 / N 2). [ABSTRACT FROM AUTHOR]

The eigenvalue probability density function for symplectic invariant random matrix ensembles can be generalized to discrete settings involving either a linear or an exponential lattice. The corresponding correlation functions can be expressed in terms of certain discrete and q skew orthogonal polynomials, respectively. We give a theory of both of these classes of polynomials, and the correlation kernels determining the correlation functions, in the cases in which the weights for the corresponding discrete unitary ensembles are classical. Crucial for this are certain difference operators which relate the relevant symmetric inner products to the skew symmetric ones, and have a tridiagonal action on the corresponding (discrete or q) orthogonal polynomials. [ABSTRACT FROM AUTHOR]

Previous works have considered the leading correction term to the scaled limit of various correlation functions and distributions for classical random matrix ensembles and their β generalisations at the hard and soft edge. It has been found that the functional form of this correction is given by a derivative operation applied to the leading term. In the present work we compute the leading correction term of the correlation kernel at the spectrum singularity for the circular Jacobi ensemble with Dyson indices β = 1 , 2 and 4, and also to the spectral density in the corresponding β -ensemble with β even. The former requires an analysis involving the Routh–Romanovski polynomials, while the latter is based on multidimensional integral formulas for generalised hypergeometric series based on Jack polynomials. In all cases this correction term is found to be related to the leading term by a derivative operation. [ABSTRACT FROM AUTHOR]

There is a two-component log-gas system with Boltzmann factor which provides an interpolation between the eigenvalue probability density function (PDF) for β = 1 and β = 4 invariant random matrix ensembles. Its solvability relies on the construction of particular skew orthogonal polynomials, with the skew inner product a linear combination of the β = 1 and β = 4 inner products, each involving weight functions. For suitably related classical weight functions, we seek to express the skew orthogonal polynomials as linear combinations of the underlying orthogonal polynomials. It is found that in each case (Gaussian, Laguerre, Jacobi and generalised Cauchy) the coefficients can be expressed in terms of hypergeometric polynomials with argument relating to the fugacity. In the Jacobi case, for example, the coefficients are Wilson polynomials. [ABSTRACT FROM AUTHOR]