Related papers: A universality theorem for ratios of random charac…

We identify the Atkin polynomials in terms of associated Jacobi polynomials. Our identificationthen takes advantage of the theory of orthogonal polynomials and their asymptotics to establish many new properties of the Atkin polynomials.…

Consider fixed and bounded trace Gaussian orthogonal, unitary and symplectic ensembles, closely related to Gaussian ensembles without any constraint. For three restricted trace Gaussian ensembles, we prove universal limits of correlation…

We consider ensembles of random polynomials of the form $p(z)=\sum_{j = 1}^N a_j P_j$ where $\{a_j\}$ are independent complex normal random variables and where $\{P_j\}$ are the orthonormal polynomials on the boundary of a bounded simply…

We study the distribution of the extended binomial coefficients by deriving a complete asymptotic expansion with uniform error terms. We obtain the expansion from a local central limit theorem and we state all coefficients explicitly as…

We discuss various universality aspects of numerical computations using standard algorithms. These aspects include empirical observations and rigorous results. We also make various speculations about computation in a broader sense.

The weak convergence of orthogonal polynomials is given under conditions on the asymptotic behaviour of the coefficients in the three-term recurrence relation. The results generalize known results and are applied to several systems of…

We discuss universality in random matrix theory and in the study of Hamiltonian partial differential equations. We focus on universality of critical behavior and we compare results in unitary random matrix ensembles with their counterparts…

We study global distribution of zeros for a wide range of ensembles of random polynomials. Two main directions are related to almost sure limits of the zero counting measures, and to quantitative results on the expected number of zeros in…

Let P_{n,m} denote the graph taken uniformly at random from the set of all planar graphs on {1,2,..., n} with exactly m(n) edges. We use counting arguments to investigate the probability that P_{n,m} will contain given components and…

We show the convergence of the characteristic polynomial for random permutation matrices sampled from the generalized Ewens distribution. Under this distribution, the measure of a given permutation depends only on its cycle structure,…

We study the characteristic polynomial of random permutation matrices following some measures which are invariant by conjugation, including Ewens' measures which are one-parameter deformations of the uniform distribution on the permutation…

Some asymptotic notions for random variables are discussed. In particular, different versions of O and o for sequences of random variables are studied. The results are elementary and more or less well-known, but collected here for future…

We investigate some topological properties of random geometric complexes and random geometric graphs on Riemannian manifolds in the thermodynamic limit. In particular, for random geometric complexes we prove that the normalized counting…

A finite dimensional quantum system for which the quantum chaos conjecture applies has eigenstates, which show the same statistical properties than the column vectors of random orthogonal or unitary matrices. Here, we consider the different…

We study two types of probability measures on the set of integer partitions of $n$ with at most $m$ parts. The first one chooses the random partition with a chance related to its largest part only. We then obtain the limiting distributions…

Ratio asymptotics for matrix orthogonal polynomials with recurrence coefficients $A_n$ and $B_n$ having limits $A$ and $B$ respectively (the matrix Nevai class) were obtained by Dur\'an. In the present paper we obtain an alternative…

For a broad class of unitary ensembles of random matrices we demonstrate the universal nature of the Janossy densities of eigenvalues near the spectral edge, providing a different formulation of the probability distributions of the limiting…

For a random intersection graph with a power law degree sequence having a finite mean and an infinite variance we show that the global clustering coefficient admits a tunable asymptotic distribution.

In this document, we make a round up of the theory of asymptotic normality of sums of associated random variables, in a coherent approach in view of further contributions for new researchers in the field. (Version 01)

We derive an asymptotic expansion for the subgroup of arbitrary Fuchsian groups and some other classes of large groups. Moreover, the main conjecture for Random Walks on symmetric groups is established in full generality. Both problems…