Related papers: Polynuclear growth model, GOE$^2$ and random matri…
Probabilistic circuits (PCs) have emerged as a powerful framework to compactly represent probability distributions for efficient and exact probabilistic inference. It has been shown that PCs with a general directed acyclic graph (DAG)…
The notion of $r$-crossing and $r$-nesting of a complete matching was introduced and a symmetry property was proved by Chen et al. [Trans. Amer. Math. Soc. 359 (2007) 1555-1575]. We consider random matchings of large size and study their…
We derive Painlev\'e--type expressions for the distribution of the $m^{th}$ largest eigenvalue in the Gaussian Orthogonal and Symplectic Ensembles in the edge scaling limit. This work generalizes to general $m$ the $m=1$ results of Tracy…
Quantum counterparts of certain simple classical systems can exhibit chaotic behaviour through the statistics of their energy levels and the irregular spectra of chaotic systems are modelled by eigenvalues of infinite random matrices. We…
We study the fluctuations of eigenvalues from a class of Wigner random matrices that generalize the Gaussian orthogonal ensemble. We begin by considering an $n \times n$ matrix from the Gaussian orthogonal ensemble (GOE) or Gaussian…
The statistical distribution of levels of an integrable system is claimed to be a Poisson distribution. In this paper, we numerically generate an ensemble of N dimensional random diagonal matrices as a model for regular systems. We evaluate…
We show that the maximal value in a size $n$ sample from GEM$(\theta)$ distribution is distributed as a sum of independent geometric random variables. This implies that the maximal value grows as $\theta\log(n)$ as $n\to\infty$. For the…
Using a systematic approach to evaluate Fredholm determinants numerically, we provide convincing evidence that the Airy_1-process, arising as a limit law in stochastic surface growth, is not the limit law for the evolution of the largest…
Let $X$ be a $p\times n$ independent identically distributed real Gaussian matrix with positive mean $\mu $ and variance $\sigma^2$ entries. The goal of this paper is to investigate the largest eigenvalue of the noncentral sample covariance…
We derive expansions of the resolvent Rn(x;y;t)=(Qn(x;t)Pn(y;t)-Qn(y;t)Pn(x;t))/(x-y) of the Hermite kernel Kn at the edge of the spectrum of the finite n Gaussian Unitary Ensemble (GUEn) and the finite n expansion of Qn(x;t) and Pn(x;t).…
We survey a number of models from physics, statistical mechanics, probability theory and combinatorics, which are each described in terms of an orthogonal polynomial ensemble. The most prominent example is apparently the Hermite ensemble,…
A connection is made between the random turns model of vicious walkers and random permutations indexed by their increasing subsequences. Consequently the scaled distribution of the maximum displacements in a particular asymmeteric version…
In this paper we will analyze discrete probability distributions in which probabilities of particular outcomes of some experiment (microstates) can be represented by the ratio of natural numbers (in other words, probabilities are…
We consider a discrete-time model for random interface growth which admits exact formulas and converges to the Polynuclear growth model in a particular limit. The height of the interface is initially flat and the evolution involves the…
This paper details an observation that for more primitive organisms, such as some yeasts, the statistical distribution of the origins of replication sometimes looks remarkably like the distribution of eigenvalues from the Circular…
We compute analytically the probability density function (pdf) of the largest eigenvalue $\lambda_{\max}$ in rotationally invariant Cauchy ensembles of $N\times N$ matrices. We consider unitary ($\beta = 2$), orthogonal ($\beta =1$) and…
We define a class of random matrix ensembles that pertain to random looped polymers. Such random looped polymers are a possible model for bio-polymers such as chromatin in the cell nucleus. It is shown that the distribution of the largest…
We develop a new method for generating probability tables based on a solid theoretical foundation. The fluctuating cross sections are calculated using the GOE-$S$-matrix model, in which the Gaussian Orthogonal Ensemble (GOE) is incorporated…
We consider the eigenvalues of sample covariance matrices of the form $\mathcal{Q}=(\Sigma^{1/2}X)(\Sigma^{1/2}X)^*$. The sample $X$ is an $M\times N$ rectangular random matrix with real independent entries and the population covariance…
The averages of ratios of characteristic polynomials det(lambda - X) of N x N random matrices X, are investigated in the large N limit for the GUE, GOE and GSE ensemble. The density of states and the two-point correlation function are…