arXiv subjects
Publications and source records attributed to K. T-R McLaughlin.
We characterize the biorthogonal polynomials that appear in the theory of coupled random matrices via a Riemann-Hilbert problem. Our Riemann-Hilbert problem is different from the ones that were proposed recently by Ercolani and McLaughlin, Kapaev, and Bertola et al. We believe that our formulation may be tractable to asymptotic analysis.
We consider polynomials that are orthogonal on $[-1,1]$ with respect to a modified Jacobi weight $(1-x)^α(1+x)^βh(x)$, with $α,β>-1$ and $h$ real analytic and stricly positive on $[-1,1]$. We obtain full asymptotic expansions for the monic and orthonormal polynomials outside the interval $[-1,1]$, for the recurrence coefficients and for the leading coefficients of the orthonormal polynomials. We also deduce asymptotic behavior for the Hankel determinants. For the asymptotic analysis we use the steepest descent technique for Riemann--Hilbert problems developed by Deift and Zhou, and applied to orthogonal polynomials on the real line by Deift, Kriecherbauer, McLaughlin, Venakides, and Zhou. In the steepest descent method we will use the Szegő function associated with the weight and for the local analysis around the endpoints $\pm 1$ we use Bessel functions of appropriate order, whereas Deift et al. use Airy functions.
We consider Laguerre polynomials $L_n^{(α_n)}(nz)$ with varying negative parameters $α_n$, such that the limit $A = -\lim_n α_n/n$ exists and belongs to $(0,1)$. For $A > 1$, it is known that the zeros accumulate along an open contour in the complex plane. For every $A \in (0,1)$, we describe a one-parameter family of possible limit sets of the zeros. Under the condition that the limit $r= - \lim_n \frac{1}{n} \log \dist(α_n, \mathbb Z)$ exists, we show that the zeros accumulate on $Γ_r \cup [β_1,β_2]$ with $β_1$ and $β_2$ only depending on $A$. For $r \in [0,\infty)$, $Γ_r$ is a closed loop encircling the origin, which for $r = +\infty$, reduces to the origin. This shows a great sensitivity of the zeros to $α_n$'s proximity to the integers. We use a Riemann-Hilbert formulation for the Laguerre polynomials, together with the steepest descent method of Deift and Zhou to obtain asymptotics for the polynomials, from which the zero behavior follows.
We study the asymptotic behavior of Laguerre polynomials $L_n^{(α_n)}(nz)$ as $n \to \infty$, where $α_n$ is a sequence of negative parameters such that $-α_n/n$ tends to a limit $A > 1$ as $n \to \infty$. These polynomials satisfy a non-hermitian orthogonality on certain contours in the complex plane. This fact allows the formulation of a Riemann--Hilbert problem whose solution is given in terms of these Laguerre polynomials. The asymptotic analysis of the Riemann--Hilbert problem is carried out by the steepest descent method of Deift and Zhou, in the same spirit as done by Deift et al. for the case of orthogonal polynomials on the real line. A main feature of the present paper is the choice of the correct contour.
We give an explicit formula for the solution to the initial value problem of the full symmetric Toda hierarchy. The formula is obtained by the orthogonalization procedure of Szegö, and is also interpreted as a consequence of the QR factorization method of Symes \cite{symes}. The sorting property of the dynamics is also proved for the case of a generic symmetric matrix in the sense described in the text, and generalizations of tridiagonal formulae are given for the case of matrices with $2M+1$ nonzero diagonals.