Graduation Year

2018

Document Type

Dissertation

Degree

Ph.D.

Degree Name

Doctor of Philosophy (Ph.D.)

Degree Granting Department

Mathematics and Statistics

Major Professor

Seung-Yeop Lee, Ph.D.

Committee Member

Abey López-García, Ph.D.

Committee Member

Dmitry Khavinson, Ph.D.

Committee Member

Evguenii A. Rakhmanov, Ph.D.

Committee Member

Razvan Teodorescu, Ph.D.

Committee Member

Wen-Xiu Ma, Ph.D.

Keywords

Discontinuity, Multiple orthogonal polynomials, Orthogonal polynomials, Random Matrices, Riemann-Hilbert problem, Skeleton

Abstract

In chapter 1, we present some background knowledge about random matrices, Coulomb gas, orthogonal polynomials, asymptotics of planar orthogonal polynomials and the Riemann-Hilbert problem. In chapter 2, we consider the monic orthogonal polynomials, $\{P_{n,N}(z)\}_{n=0,1,\cdots},$ that satisfy the orthogonality condition,

\begin{equation}\nonumber \int_\mathbb{C}P_{n,N}(z)\overline{P_{m,N}(z)}e^{-N Q(z)}dA(z)=h_{n,N}\delta_{nm} \quad(n,m=0,1,2,\cdots), \end{equation}

where $h_{n,N}$ is a (positive) norming constant and the external potential is given by

$$Q(z)=|z|^2+ \frac{2c}{N}\log \frac{1}{|z-a|},\quad c>-1,\quad a>0.$$

The orthogonal polynomial is related to the interacting Coulomb particles with charge $+1$ for each, in the presence of an extra particle with charge $+c$ at $a.$ For $N$ large and a fixed ``c'' this can be a small perturbation of the Gaussian weight. The polynomial $P_{n,N}(z)$ can be characterized by a matrix Riemann--Hilbert problem \cite{Ba 2015}. We then apply the standard nonlinear steepest descent method \cite{Deift 1999, DKMVZ 1999} to derive the strong asymptotics of $P_{n,N}(z)$ when $n$ and $N$ go to $\infty.$ From the asymptotic behavior of $P_{n,N}(z),$ we find that, as we vary $c,$ the limiting distribution behaves discontinuously at $c=0.$ We observe that the mother body (a kind of potential theoretic skeleton) also behaves discontinuously at $c=0.$ The smooth interpolation of the discontinuity is obtained by further scaling of $c=e^{-\eta N}$ in terms of the parameter $\eta\in[0,\infty).$ To obtain the results for arbitrary values of $c$, we used the ``partial Schlesinger transform'' method developed in \cite{BL 2008} to derive an arbitrary order correction in the Riemann--Hilbert analysis.

In chapter 3, we consider the case of multiple logarithmic singularities. The planar orthogonal polynomials $\{p_n(z)\}_{n=0,1,\cdots}$ with respect to the external potential that is given by $$Q(z)=|z|^2+ 2\sum_{j=1}^lc_j\log \frac{1}{|z-a_j|},$$

where $\{a_1, a_2, \cdots, a_l\}$ is a set of nonzero complex numbers and $\{c_1, c_2, \cdots, c_l\}$ is a set of positive real numbers. We show that the planar orthogonal polynomials $p_{n}(z)$ with $l$ logarithmic singularities in the potential are the multiple orthogonal polynomials $p_{{\bf{n}}}(z)$ (Hermite-Pad\'e polynomials) of Type II with $l$ measures of degree $|{\bf{n}}|=n=\kappa l+r,$ ${\bf{n}}=(n_1,\cdots,n_l)$ satisfying the orthogonality condition,

$$ \frac{1}{2\ii}\int_{\Gamma}p_{{\bf{n}}}(z) z^k\chi_{{\bf{n}}-{\bf{e}}_j}(z)\dd z=0, \quad 0\leq k\leq n_j-1,\quad 1\leq j\leq l,$$

where $\Gamma$ is a certain simple closed curve with counterclockwise orientation and

$$ \chi_{{\bf{n}}-{\bf{e}}_j}(z):= \prod_{i=1}^l(z-a_i)^{c_i }\int_{0}^{\overline{z}\times\infty}\frac{\prod_{i=1}^l(s-\bar{a}_i)^{n_i+c_i}}{(s-\bar{a}_j)\ee^{zs}}\,\dd s. $$

Such equivalence allows us to formulate the $(l+1)\times(l+1)$ Riemann--Hilbert problem for $p_n(z)$. We also find the ratio between the determinant of the moment matrix corresponding to the multiple orthogonal polynomials and the determinant of the moment matrix from the original planar measure.

Included in

Mathematics Commons

Share

COinS