Motivated by the longest increasing subsequence problem, we examine sundry topics at the interface of enumerative/algebraic combinatorics and random matrix theory. We begin with an expository account of the increasing subsequence problem, contextualizing it as an ``exactly solvable'' Ramsey-type problem and introducing the RSK correspondence. New proofs and generalizations of some of the key results in increasing subsequence theory are given. These include Regev's single scaling limit, Gessel's Toeplitz determinant identity, and Rains' integral representation. The double scaling limit (Baik-Deift-Johansson theorem) is briefly described, although we have no new results in that direction. Following up on the appearance of determinantal generating functions in increasing subsequence type problems, we are led to a connection between combinatorics and the ensemble of truncated random unitary matrices, which we describe in terms of Fisher's random-turns vicious walker model from statistical mechanics. We prove that the moment generating function of the trace of a truncated random unitary matrix is the grand canonical partition function for Fisher's random-turns model with reunions. Finally, we consider unitary matrix integrals of a very general type, namely the ``correlation functions'' of entries of Haar-distributed random matrices. We show that these expand perturbatively as generating functions for class multiplicities in symmetric functions of Jucys-Murphy elements, thus addressing a problem originally raised by De Wit and t'Hooft and recently resurrected by Collins. We argue that this expansion is the CUE counterpart of genus expansion. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2009-09-27 12:27:21.479

Scattering is a fundamental phenomenon in physics, e.g. large parts of the knowledge about quantum systems stem from scattering experiments. A scattering process can be completely characterized by its K-matrix, also known as the \Wigner reaction matrix" in nuclear scattering or \impedance matrix" in the electromagnetic wave scattering. For chaotic quantum systems it can be modelled within the framework of Random Matrix Theory (RMT), where either the K-matrix itself or its underlying Hamiltonian is taken as a random matrix. These two approaches are believed to lead to the same results due to a universality conjecture by P. Brouwer, which is equivalent to the claim that the probability distribution of K, for a broad class of invariant ensembles of random Hermitian matrices H, converges to a matrix Cauchy distribution in the limit of large matrix dimension of H. For unitarily invariant ensembles, this conjecture will be proved in the thesis by explicit calculation, utilising results about ensemble averages of characteristic polynomials. This thesis furthermore analyses various characteristics of the K-matrix such as the distribution of a diagonal element at the spectral edge or the distribution of an off-diagonal element in the bulk of the spectrum. For the latter it is necessary to know correlation functions involving products and ratios of half-integer powers of characteristic polynomials of random matrices for the Gaussian Orthogonal Ensemble (GOE), which is an interesting and important topic in itself, as they frequently arise in various other applications of RMT to physics of quantum chaotic systems, and beyond. A larger part of the thesis is dedicated to provide an explicit evaluation of the large-N limits of a few non-trivial objects of that sort within a variant of the supersymmetry formalism, and via a related but different method.

We give overcrowding estimates for the Sine(beta) process, the bulk point process limit of the Gaussian beta-ensemble. We show that the probability of having exactly n points in a fixed interval is given by e(-beta/2n2) log(n)+O(n(2)) as n -> infinity. We also identify the next order term in the exponent if the size of the interval goes to zero.

Motivated by dimension reduction in regression analysis and signal detection, we investigate order determination for large dimensional matrices with spiked structures in which the dimensions of the matrices are proportional to the sample sizes. Because the asymptotic behaviors of the estimated eigenvalues differ completely from those in fixed dimension scenarios, we then discuss the largest possible order, say q, we can identify and introduce criteria for different settings of q. When q is assumed to be fixed, we propose a "valley-cliff" criterion with two versions - one based on the original differences of eigenvalues and the other based on the transformed differences - to reduce the effect of ridge selection in the criterion. This generic method is very easy to implement and computationally inexpensive, and it can be applied to various matrices. As examples, we focus on spiked population models, spiked Fisher matrices and factor models with auto-covariance matrices. For the case of divergent q, we propose a scale-adjusted truncated double ridge ratio (STDRR) criterion, where a scale adjustment is implemented to deal with the bias in scale parameter for large q. Again, examples include spiked population models, spiked Fisher matrices. Numerical studies are conducted to examine the finite sample performances of the method and to compare it with existing methods. As for theoretical contributions, we investigate the limiting properties, including convergence in probability and central limit theorems, for spiked eigenvalues of spiked Fisher matrices with divergent q. Keywords: Auto-covariance matrix, factor model, finite-rank perturbation, Fisher matrix, principal component analysis (PCA), phase transition, random matrix theory (RMT), ridge ratio, spiked population model.

The present work studies some aspects of random matrix theory. Its first part is devoted to the asymptotics of random matrices with infinitely divisible, in particular heavy-tailed, entries. Its second part focuses on relations between limiting law in subsequence problems and spectra of random matrices.

Thesis (Ph. D.)--Rutgers University, 2007. / "Graduate Program in Mathematics." Includes bibliographical references (p. 64-65).

by Lo Tai-yan, Milton. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1995. / Includes bibliographical references (leaves 112-114). / Chapter 1 --- Preliminaries --- p.1 / Chapter 1.1 --- Introduction --- p.1 / Chapter 1.1.1 --- The Noncentral Multivariate F distribution --- p.2 / Chapter 1.1.2 --- The Central Problems and the Approach --- p.4 / Chapter 1.2 --- Concepts and Terminology --- p.7 / Chapter 1.3 --- Choice of Estimates --- p.10 / Chapter 1.4 --- Related Work --- p.11 / Chapter 2 --- Estimation of the noncentrality parameter of a Noncentral Mul- tivariate F distribution --- p.19 / Chapter 2.1 --- Unbiased and Linear Estimators --- p.19 / Chapter 2.1.1 --- The unbiased estimate --- p.20 / Chapter 2.1.2 --- The Class of Linear Estimates --- p.24 / Chapter 2.2 --- Optimal Linear Estimate --- p.32 / Chapter 2.3 --- Nonlinear Estimate --- p.34 / Chapter 2.4 --- Monte Carlo Simulation Study --- p.41 / Chapter 2.5 --- Evaluation and Further Investigation --- p.42 / Chapter 3 --- Estimation of Canonical Correlation Coefficients --- p.73 / Chapter 3.1 --- Preliminary --- p.73 / Chapter 3.2 --- The Estimation Problem --- p.76 / Chapter 3.3 --- Orthogonally Invariant Estimates --- p.77 / Chapter 3.3.1 --- The Unbiased Estimate --- p.78 / Chapter 3.3.2 --- The Class of Linear Estimates --- p.78 / Chapter 3.3.3 --- The Class of Nonlinear Estimates --- p.80 / Chapter 3.4 --- Monte Carlo Simulation Study --- p.87 / Chapter 3.5 --- Evaluation and Further Investigation --- p.89 / Chapter A --- p.104 / Chapter A.1 --- Lemma 3.2 --- p.104 / Chapter A.2 --- Theorem 3.3 Leung(1992) --- p.105 / Chapter A.3 --- The Noncentral F Identity --- p.106 / Chapter B --- Bibliography --- p.111

Leung Kit Ying. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 86-88). / Abstracts in English and Chinese. / Declaration --- p.i / Acknowledgement --- p.ii / Chapter 1 --- INTRODUCTION --- p.1 / Chapter 2 --- IMPROVED ESTIMATION OF THE NORMAL PRECISION MATRIX USING THE L1 AND L2 LOSS FUNCTIONS --- p.7 / Chapter 2.1 --- Previous Work --- p.9 / Chapter 2.2 --- Important Lemmas --- p.13 / Chapter 2.3 --- Improved Estimation of Σ-1 under L1 Loss Function --- p.20 / Chapter 2.4 --- Improved Estimation of Σ-1 under L2 Loss Function --- p.26 / Chapter 2.5 --- Simulation Study --- p.31 / Chapter 2.6 --- Comparison with Krishnammorthy and Gupta's result --- p.38 / Chapter 3 --- IMPROVED ESTIMATION OF THE NORMAL PRECISION MATRIX USING THE L3 AND L4 LOSS FUNCTIONS --- p.43 / Chapter 3.1 --- Justification of the Loss Functions --- p.46 / Chapter 3.2 --- Important Lemmas for Calculating Risks --- p.48 / Chapter 3.3 --- Improved Estimation of Σ-1 under L3 Loss Function --- p.55 / Chapter 3.4 --- Improved Estimation of Σ-1 under L4 Loss Function --- p.62 / Chapter 3.5 --- Simulation Study --- p.69 / Appendix --- p.77 / Reference --- p.35

Il est bien connu que divers solides amorphes ont de nombreuses propriétés universelles. L'une d'entre elles est la variation de la conductivité thermique en fonction de la température. Cependant, le mécanisme microscopique du transfert de chaleur dans le domaine de température supérieure à 20 K est encore mal compris. Simulations numériques récentes du silicium et de la silice amorphes montrent que les modes de vibration dans la gamme de fréquences correspondante (au-dessus de plusieurs THz) sont délocalisés. En même temps ils sont complètement différents des phonons acoustiques de basse fréquence, dits « diffusions ».Dans ce travail, nous présentons un modèle stable de matrice aléatoire d'un solide amorphe. Dans ce modèle, on peut faire varier le degré de désordre allant du cristal parfait jusqu'au milieu mou extrêmement désordonné sans rigidité macroscopique. Nous montrons que les solides amorphes réels sont proches du deuxième cas limite, et que les diffusions occupent la partie dominante du spectre de vibration. La fréquence de transition entre les phonons acoustiques et diffusons est déterminée par le critère Ioffe-Regel. Fait intéressant, cette fréquence de transition coïncide pratiquement avec la position du pic Boson. Nous montrons également que la diffusivité et la densité d'états de vibration de diffusons sont pratiquement constantes en fonction de la fréquence. Par conséquent, la conductivité thermique est une fonction linéaire de la température dans le domaine allant à des températures relativement élevées, puis elle sature. Cette conclusion est en accord avec de nombreuses données expérimentales. En outre, nous considérons un modèle numérique de matériaux de type de silicium amorphe et étudions le rôle du désordre pour les vibrations longitudinales et transverses. Nous montrons aussi que la théorie des matrices aléatoires peut être appliquée avec succès pour estimer la densité d'états vibrationnels des systèmes granulaires bloqués. / It is well known that various amorphous solids have many universal properties. One of them is the temperature dependence of the thermal conductivity. However, the microscopic mechanism of the heat transfer above 20 K is still poorly understood. Recent numerical simulations of amorphous silicon and silica show that vibrational modes in the corresponding frequency range (above several THz) are delocalized, however they are completely different from low-frequency acoustic phonons, called “diffusons”.In this work we present a stable random matrix model of an amorphous solid. In this model one can vary the strength of disorder going from a perfect crystal to extremely disordered soft medium without macroscopic rigidity. We show that real amorphous solids are close to the second limiting case, and that diffusons occupy the dominant part of the vibrational spectrum. The crossover frequency between acoustic phonons and diffusons is determined by the Ioffe-Regel criterion. Interestingly, this crossover frequency practically coincides with the Boson peak position. We also show that, as a function of frequency, the diffusivity and the vibrational density of states of diffusons are practically constant. As a result, the thermal conductivity is a linear function of temperature up to rather high temperatures and then saturates. This conclusion is in agreement with numerous experimental data.Further, we consider a numerical model of amorphous silicon-like materials and investigate the role of disorder for longitudinal and transverse vibrations. We also show that the random matrix theory can be successfully applied to estimate the vibrational density of states of granular jammed systems.

This thesis contains two main chapters. The first chapter focuses on the highdimensional asymptotic regimes of correlated Wishart matrices d−1YY^T , where Y is a n×d Gaussian random matrix with correlated and non-stationary entries. We provide quantitative bounds in the Wasserstein distance for the cases of central convergence and non-central convergence, verify such convergences hold in the weak topology of C([a; b]; M_n(R)), and show that our result can be used to prove convergence in expectation of the empirical spectral distributions of the Wishart matrices to the semicircular law. The second chapter develops a version of the Stein-Malliavin method in an infinite-dimensional and non-diffusive Poissonian setting. In particular, we provide quantitative central limit theorems for approximations by non-degenerate Hilbert-valued Gaussian random elements, as well as fourth moment bounds for approximating sequences with finite chaos expansion. We apply our results to the Brownian approximation of Poisson processes in Besov-Liouville spaces and also derive a functional limit theorem for an edge-counting statistic of a random geometric graph.