Related papers: A unitary extension of virtual permutations

Motivated by the developing mathematics of deep learning, we build universal functions approximators of continuous maps between arbitrary Polish metric spaces $\mathcal{X}$ and $\mathcal{Y}$ using elementary functions between Euclidean…

We define a way of approximating actions on measure spaces using finite graphs; we then show that in quite general settings these graphs form a family of expanders if and only if the action is expanding in measure. This provides a somewhat…

The second part of the paper mainly deals with convergence of infinite determinantal measures, understood as the convergence of the approximating finite determinantal measures. In addition to the usual weak topology on the space of…

Extremal spacings between eigenvalues of random unitary matrices of size N pertaining to circular ensembles are investigated. Explicit probability distributions for the minimal spacing for various ensembles are derived for N = 4. We study…

We deal with random processes obtained from a homogeneous random process with independent increments by replacement of the time scale and by multiplication by a norming constant. We prove the convergence in distribution of these processes…

We prove some extension theorems involving uniformly continuous maps of the universal Urysohn space. We also prove reconstruction theorems for certain groups of autohomeomorphisms of this space and of its open subsets.

One considers Hilbert space valued measures on the Borel sets of a compact metric space. A natural numerical valued integral of vector valued continuous functions with respect to vector valued functions is defined. Using this integral,…

Together with spaces of constant sectional curvature and products of a real line with a manifold of constant curvature, the socalled Egorov spaces and $\varepsilon$-spaces exhaust the class of $n$-dimensional Lorentzian manifolds admitting…

We study the Hausdorff dimension of self-similar sets and measures on the line. We show that if the dimension is smaller than the minimum of 1 and the similarity dimension, then at small scales there are super-exponentially close cylinders.…

The singular values of products of standard complex Gaussian random matrices, or sub-blocks of Haar distributed unitary matrices, have the property that their probability distribution has an explicit, structured form referred to as a…

On unitary compact groups the decomposition of a generic element into product of reflections induces a decomposition of the characteristic polynomial into a product of factors. When the group is equipped with the Haar probability measure,…

For any finite point set in $D$-dimensional space equipped with the 1-norm, we present random linear embeddings to $k$-dimensional space, with a new metric, having the following properties. For any pair of points from the point set that are…

The analysis of parametrised systems is a growing field in verification, but the analysis of parametrised probabilistic systems is still in its infancy. This is partly because it is much harder: while there are beautiful cut-off results for…

One of the major themes of random matrix theory is that many asymptotic properties of traditionally studied distributions of random matrices are universal. We probe the edges of universality by studying the spectral properties of random…

In this paper, we study the almost sure boundedness and the convergence of the stochastic approximation (SA) algorithm. At present, most available convergence proofs are based on the ODE method, and the almost sure boundedness of the…

It is known that there is a comeagre set of mutually conjugate measure preserving homeomorphisms of Cantor space equipped with the coinflipping probability measure, i.e., Haar measure. We show that the generic measure preserving…

We consider $N\times N$ symmetric or hermitian random matrices with independent, identically distributed entries where the probability distribution for each matrix element is given by a measure $\nu$ with a subexponential decay. We prove…

Consider the ensemble of Real Symmetric Toeplitz Matrices, each entry iidrv from a fixed probability distribution p of mean 0, variance 1, and finite higher moments. The limiting spectral measure (the density of normalized eigenvalues)…

We prove the conjecture of Baik, Deift, and Johansson which says that with respect to the Plancherel measure on the set of partitions of $n$, the 1st, 2nd, and so on, rows behave, suitably scaled, like the 1st, 2nd, and so on, eigenvalues…

The topic of this paper is the typical behavior of the spectral measures of large random matrices drawn from several ensembles of interest, including in particular matrices drawn from Haar measure on the classical Lie groups, random…