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Register a new userSkew-orthogonal Laguerre polynomials for chiral real asymmetric random matrices
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We apply the method of skew-orthogonal polynomials (SOP) in the complex plane to asymmetric random matrices with real elements, belonging to two different classes. Explicit integral representations valid for arbitrary weight functions are derived for the SOP and for their Cauchy transforms, given as expectation values of traces and determinants or their inverses, respectively. Our proof uses the fact that the joint probability distribution function for all combinations of real eigenvalues and complex conjugate eigenvalue pairs can be written as a product. Examples for the SOP are given in terms of Laguerre polynomials for the chiral ensemble (also called the non-Hermitian real Wishart-Laguerre ensemble), both without and with the insertion of characteristic polynomials. Such characteristic polynomials play the role of mass terms in applications to complex Dirac spectra in field theory. In addition, for the elliptic real Ginibre ensemble we recover the SOP of Forrester and Nagao in terms of Hermite polynomials.
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Within the context of Supersymmetric Quantum Mechanics and its related hierarchies of integrable quantum Hamiltonians and potentials, a general programme is outlined and applied to its first two simplest illustrations. Going beyond the usual restriction of shape invariance for intertwined potentials, it is suggested to require a similar relation for Hamiltonians in the hierarchy separated by an arbitrary number of levels, N. By requiring further that these two Hamiltonians be in fact identical up to an overall shift in energy, a periodic structure is installed in the hierarchy of quantum systems which should allow for its solution. Specific classes of orthogonal polynomials characteristic of such periodic hierarchies are thereby generated, while the methods of Supersymmetric Quantum Mechanics then lead to generalised Rodrigues formulae and recursion relations for such polynomials. The approach also offers the practical prospect of quantum modelling through the engineering of quantum potentials from experimental energy spectra. In this paper these ideas are presented and solved explicitly for the cases N=1 and N=2. The latter case is related to the generalised Laguerre polynomials, for which indeed new results are thereby obtained. At the same time new classes of integrable quantum potentials which generalise that of the harmonic oscillator and which are characterised by two arbitrary energy gaps are identified, for which a complete solution is achieved algebraically.
In this paper we study the distribution of level crossings for the spectra of linear families A+lambda B, where A and B are square matrices independently chosen from some given Gaussian ensemble and lambda is a complex-valued parameter. We formulate a number of theoretical and numerical results for the classical Gaussian ensembles and some generalisations. Besides, we present intriguing numerical information about the distribution of monodromy in case of linear families for the classical Gaussian ensembles of 3 * 3 matrices.
For a given polynomial $V(x)in mathbb C[x]$, a random matrix eigenvalues measure is a measure $prod_{1leq i<jleq N}(x_i-x_j)^2 prod_{i=1}^N e^{-V(x_i)}dx_i$ on $gamma^N$. Hermitian matrices have real eigenvalues $gamma=mathbb R$, which generalize to $gamma$ a complex Jordan arc, or actually a linear combination of homotopy classes of Jordan arcs, chosen such that integrals are absolutely convergent. Polynomial moments of such measure satisfy a set of linear equations called loop equations. We prove that every solution of loop equations are necessarily polynomial moments of some random matrix measure for some choice of arcs. There is an isomorphism between the homology space of integrable arcs and the set of solutions of loop equations. We also generalize this to a 2-matrix model and to the chain of matrices, and to cases where $V$ is not a polynomial but $V(x)in mathbb C(x)$.
Discrete spectral transformations of skew orthogonal polynomials are presented. From these spectral transformations, it is shown that the corresponding discrete integrable systems are derived both in 1+1 dimension and in 2+1 dimension. Especially in the (2+1)-dimensional case, the corresponding system can be extended to 2x2 matrix form. The factorization theorem of the Christoffel kernel for skew orthogonal polynomials in random matrix theory is presented as a by-product of these transformations.
Skew-orthogonal polynomials (SOPs) arise in the study of the n-point distribution function for orthogonal and symplectic random matrix ensembles. Motivated by the average of characteristic polynomials of the Bures random matrix ensemble studied in [22], we propose the concept of partial-skew-orthogonal polynomials (PSOPs) as a modification of the SOPs, and then the PSOPs with a variety of special skew-symmetric kernels and weight functions are addressed. By considering appropriate deformations of the weight functions, we derive nine integrable lattices in different dimensions. As a consequence, the tau-functions for these systems are shown to be expressed in terms of Pfaffians and the wave vectors PSOPs. In fact, the tau-functions also admit the representations of multiple integrals. Among these integrable lattices, some of them are known, while the others are novel to the best of our knowledge. In particular, one integrable lattice is related to the partition function of the Bures random matrix ensemble. Besides, we derive a discrete integrable lattice, which can be used to compute certain vector Pade approximants. This yields the first example regarding the connection between integrable lattices and vector Pade approximants, for which Hietarinta, Joshi and Nijhoff pointed out that This field remains largely to be explored. in the recent monograph [27, Section 4.4] .
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