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\documentclass{article}
\usepackage[margin=1in]{geometry}
\usepackage{parskip}
\usepackage{float}
\usepackage{xcolor}
\usepackage{graphicx}
\graphicspath{{./figures/}}
\usepackage{amsmath}
\usepackage{booktabs}
\usepackage{longtable}
\usepackage{hyperref}
\hypersetup{colorlinks=true,
linkcolor=black,
urlcolor=blue,
citecolor=black,
pdftitle={Reconstructing the Tracy-Widom Distribution},
pdfpagemode=FullScreen}
\usepackage{listings}
\usepackage{algpseudocode}
\title{Reconstructing the Tracy-Widom Distribution\vspace{-0.75em}}
\author{Connor Moore, 100826701. \today{}}
\date{}
\begin{document}
\maketitle
\section{Introduction}
Random numbers form the basis of various studies in mathematics and the sciences. One strong example of this is \emph{Random Matrix Theory}, or the study of the construction and properties of random matrices. One of the first applications of random matrix theory was proposed by Eugene Wigner (1902-1995) in his application to model the nuclei and spectrum of various heavy atom \cite{anderson2009An_Int, wigner1955Charac}. It has since found different applications including work in computational mechanics, control theory, systems engineering, and others.
Random matrix theory borrows some distributions from statistics to construct matrices. This includes the Gaussian (normal) distribution, which is used to make various matrices depending on which number system is used. Wigner's work included Orthogonal (real symmetric), Unitary (Complex Hermitian), and Symplectic (Quaternion Hermitian) matrices and their eigenvalues.
This work considers Gaussian orthogonal ensembles (GOEs) specifically and attempts to reconstruct the Tracy-Widom distribution directly by computing eigenvalues. An overview of relevant theory, the necessary methods, and implementation details is provided before results are presented. The distribution is examined as a function of matrix dimension and sample size, and statistical testing is performed to support the notion of a Tracy-Widom being obtained.
\section{Relevant Theory}
\subsection{Random Ensembles}
The GOE-based random matrix has some properties that go beyond simply being normally distributed. Namely, the matrix is square, symmetric, and the standard deviation $\sigma$ is unity except for elements on the diagonal where $\sigma=2$. The mean $\mu$ is considered as zero for all elements. Consider the normal distribution with notation $N(\mu, \sigma$). Then, an example is provided below for a $5\times5$ matrix:
\begin{equation*}
\mathbf{A} =
\begin{pmatrix}
N(0,2) & N(0,1) & N(0,1) & N(0,1) & N(0,1) \\
N(0,1) & N(0,2) & N(0,1) & N(0,1) & N(0,1) \\
N(0,1) & N(0,1) & N(0,2) & N(0,1) & N(0,1) \\
N(0,1) & N(0,1) & N(0,1) & N(0,2) & N(0,1) \\
N(0,1) & N(0,1) & N(0,1) & N(0,1) & N(0,2) \\
\end{pmatrix}
\end{equation*}
The diagonal standard deviation of 2 is what makes this ensemble orthogonal. Note that the representation above does not explicitly state the symmetry, but $\mathbf{A}(i,j) = \mathbf{A}(j,i)\, \forall(i,j)$.
\subsection{The Tracy-Widom Distributions}
Various distributions can be
\subsection{Statistical Differentiation}
The Tracy-Widom and normal distributions are hard to differentiate based on shape alone. Because of this, it is necessary to consider quantifying the differences in some meaningful way. One way to do this is to consider the important differences between the two distributions. For example, while the normal distribution is symmetric, the Tray-Widom is not. The symmetry of a distribution is something that can be quantified
\section{Numerical Methods}
\subsection{The Power Method}
One relatively cheap way to find the maximum eigenvalue is through the so-called power method. The power method is an iterative approach to finding the maximum eigenvalue of a matrix that exploits the spectral gap ($\lambda_1 - \lambda_2$). Consider an initial guess in the form of a vector of length $N$ given as $\mathbf{x_0}$. Then, perform the multiplication:
\begin{equation}
\mathbf{x}_{k+1} = \mathbf{A}\mathbf{x}_k
\end{equation}
After multiplying, then normalize the vector such that $\left| \mathbf{x}_{k+1} \right|$ is unity:
\begin{equation}
\mathbf{x}_{k+1} = \frac{\mathbf{x}_{k+1}}{\left| \mathbf{x}_{k+1} \right|}
\end{equation}
Or, more compactly written as one step:
\begin{equation}
\mathbf{x}_{k+1} = \frac{\mathbf{A}\mathbf{x}_{k}}{\left|\mathbf{A}\mathbf{x}_{k}\right|}
\end{equation}
Repeating this will converge to the dominant eigenvalue $\lambda_1$ so long as $\mathbf{x}_0$ is not orthogonal to the dominant eigenvector \cite{novak2022}. The convergence of the method is $\mathcal{O}\left( \left| \lambda_1 / \lambda_2 \right|^k \right)$ and is largely dependent on the spectral gap. The main advantage of this method is that computationally it consists of only a matrix multiplication ($\mathcal{O}(n^2)$) and normalization ($\mathcal{O}(n)$) which is rather inexpensive compared with other methods per iteration \cite{watkins2002}. Worth noting, however, is that in cases where $\mathbf{A}$ has a small spectral gap it is not guaranteed to converge in an acceptable number of iterations. This poses an issue as it would bias the calculations for reconstructing the Tracy-Widom by not including random ensembles specifically with lower spectral gaps. To ratify this, more expensive methods exist which guarantee a solution that are used as a fallback.
\subsection{Schur Decomposition}
One method that guarantees a solution $\forall \mathbf{A}$ is Schur Decomposition. Named after the late Russian mathematician Issai Schur (1875-1941), Schur Decomposition involves decomposing the matrix $\mathbf{A}$ into a combination of some unitary matrix $\mathbf{U}$ and upper-triangular matrix $\mathbf{T}$ \cite{watkins2002}:
\begin{equation}
\mathbf{A} = \mathbf{U} \mathbf{T} \mathbf{U}^\mathrm{T}
\end{equation}
In this case, the diagonal entries in $\mathbf{T}$ are then the eigenvalues of the original matrix $\mathbf{A}$. The proof in Schur's theorem is non-constructive and does not provide a solution without knowing the eigenvectors of $\mathbf{A}$, but it can be still be used in combination with the so-called QR algorithm developed independently by John Francis (1934-present) and Vera Kublanovskaya (1920-2012) starting from 1959 \cite{golub2009The_QR}. Essentially, the QR algorithm is the computational scheme that provides the matrices $\mathbf{U}$ and $\mathbf{T}$ from the input $\mathbf{A}$. It does so with a complexity of $\mathcal{O}(n^3)$, which for larger values of $N$ deviates significantly from the $\mathcal{O}(n^2)$ of the power method. For this reason it is used only when the power method fails.
\section{Implementation and MPI Parallelization}
\subsection{Worker-Manager Structure}
\subsection{The Manager Process}
\subsection{The Calculation (Worker) Process}
\section{Results}
%\section{Conclusions}
\bibliographystyle{ieeetr}
\bibliography{refs.bib}
\end{document}
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