Orthogonal polynomials, perturbed Hankel determinants and random matrix models
Creators: name: Haq, Nazmus Saqeeb
Publisher: name: Imperial College London
Year: 2013

description: In this thesis, for a given weight function w(x), supported on [A,B]\subseteq\mathbb{R}, we consider the sequence of monic polynomials orthogonal with respect to w(x), and the Hankel determinant D_n=\det(\mu_{j+k})_{j,k=0}^{n-1}, generated from the moments \mu_j of w(x). A motivating factor for studying such objects is that by observing the Andreief-Heine identity, these determinants represent the partition function of a Hermitian random matrix ensemble. It is well known that the Hankel determinant can be computed via the product of L^2 norms over [A,B]\subseteq\mathbb{R} of the orthogonal polynomials associated with w(x). Since such polynomials satisfy a three-term recurrence relation, we also study the behaviour of the recurrence coefficients, denoted by \alpha_n and \beta_n, as these are intimately related to the behaviour of D_n. We consider orthogonal polynomials and Hankel determinants associated with the following two weight functions: First, we consider a deformation of the Jacobi weight, given by w(x)=(1-x^2)^{\apha} (1-k^2x^2)^{\beta}, x\in[-1,1], \alpha&gt;-1, \beta\in\mathbb{R}, k^2\in(0,1). This is a generalization of a system of orthogonal polynomials studied by C. J. Rees in 1945. Such orthogonal polynomials are of great interest because the corresponding Hankel determinant is related to the \tau-function of a Painlevé VI differential equation, the special cases of which are related to enumerative problems arising from String theory. For finite n---we employ the orthogonal polynomial ladder operators (formulae that raise or lower the index of the polynomial) to find equations for auxiliary quantities defined by the corresponding orthogonal polynomials, from which we derive differential identities satisfied by the Hankel determinant, and differential-difference identities for the recurrence coefficients \alpha_n and \beta_n. Making use of the ladder operators, we find that the recurrence coefficient \beta_n(k^2), n=1,2,...; and P_1(n,k^2), the coefficient of x^{n-2} of the corresponding monic orthogonal polynomials, satisfy second order non-linear difference equations. The large n expansion based on the difference equations when combined Toda-type differential relations satisfied by the associated Hankel determinant yields a complete asymptotic expansion of D_n. The finite n representation of D_n in terms of a particular Painlevé VI equation is also discussed as well as the generalization of the linear second order differential equation found by Rees. Second, we consider the deformed Laguerre weight: w(x)=x^{\alpha} e^{-x} (t+x)^{N_s} (T+x)^{-N_s}, x\in[0,\infty), \alpha&gt;-1, T, t, N_s&gt;0. This weight is of interest since it appears in the study of a multiple-antenna wireless communication scenario. The key quantity determining system performance is the statistical properties of the signal-to-noise ratio (SNR) \gamma, which recent work has characterized through its moment generating function, in terms of the Hankel determinant generated via our deformed Laguerre weight. We make use of the ladder operators to give an exact finite n characterization of the Hankel determinant in terms of a two-variable generalization of a Painlevé V differential equation, which reduces to Painlevé V under certain limits. We also employ Dyson's Coulomb fluid theory to derive an approximation for D_n---in the limit where n is large. The finite and large n characterizations are then used to compute closed-form (non-determinantal) expressions for the cumulants of the distribution of \gamma, and to compute wireless communication performance quantities of engineering interest.