For a $k$ -uniform hypergraph $\mathcal{H}$ on vertex set $\{1, \ldots, n\}$ we associate a particular signed incidence matrix $M(\mathcal{H})$ over the integers. For $\mathcal{H} \sim \mathcal{H}_k(n, p)$ an Erdős–Rényi random $k$ -uniform hypergraph, ${\mathrm{coker}}(M(\mathcal{H}))$ is then a model for random abelian groups. Motivated by conjectures from the study of random simplicial complexes we show that for $p = \omega (1/n^{k - 1})$ , ${\mathrm{coker}}(M(\mathcal{H}))$ is torsion-free.

In this paper we analyze a simple spectral method (EIG1) for the problem of matrix alignment, consisting in aligning their leading eigenvectors: given two matrices A and B, we compute two corresponding leading eigenvectors $v_1$ and $v'_{\!\!1}$ . The algorithm returns the permutation $\hat{\pi}$ such that the rank of coordinate $\hat{\pi}(i)$ in $v_1$ and that of coordinate i in $v'_{\!\!1}$ (up to the sign of $v'_{\!\!1}$ ) are the same.

We consider a model of weighted graphs where the adjacency matrix A belongs to the Gaussian orthogonal ensemble of size $N \times N$ , and B is a noisy version of A where all nodes have been relabeled according to some planted permutation $\pi$ ; that is, $B= \Pi^T (A+\sigma H) \Pi $ , where $\Pi$ is the permutation matrix associated with $\pi$ and H is an independent copy of A. We show the following zero–one law: with high probability, under the condition $\sigma N^{7/6+\epsilon} \to 0$ for some $\epsilon>0$ , EIG1 recovers all but a vanishing part of the underlying permutation $\pi$ , whereas if $\sigma N^{7/6-\epsilon} \to \infty$ , this method cannot recover more than o(N) correct matches.

This result gives an understanding of the simplest and fastest spectral method for matrix alignment (or complete weighted graph alignment), and involves proof methods and techniques which could be of independent interest.

Let $\mathbf{X}$ be a $p\times n$ random matrix whose entries are independent and identically distributed real random variables with zero mean and unit variance. We study the limiting behaviors of the 2-normal condition number k(p,n) of $\mathbf{X}$ in terms of large deviations for large n, with p being fixed or $p=p(n)\rightarrow\infty$ with $p(n)=o(n)$ . We propose two main ingredients: (i) to relate the large-deviation probabilities of k(p,n) to those involving n independent and identically distributed random variables, which enables us to consider a quite general distribution of the entries (namely the sub-Gaussian distribution), and (ii) to control, for standard normal entries, the upper tail of k(p,n) using the upper tails of ratios of two independent $\chi^2$ random variables, which enables us to establish an application in statistical inference.

Let $F$ be a non-discrete non-Archimedean locally compact field and ${\mathcal{O}}_{F}$ the ring of integers in $F$ . The main results of this paper are the classification of ergodic probability measures on the space $\text{Mat}(\mathbb{N},F)$ of infinite matrices with entries in $F$ with respect to the natural action of the group $\text{GL}(\infty ,{\mathcal{O}}_{F})\times \text{GL}(\infty ,{\mathcal{O}}_{F})$ and the classification, for non-dyadic $F$ , of ergodic probability measures on the space $\text{Sym}(\mathbb{N},F)$ of infinite symmetric matrices with respect to the natural action of the group $\text{GL}(\infty ,{\mathcal{O}}_{F})$ .

This paper presents a theoremon universality on orthogonal polynomials/randommatrices under a weak local condition on the weight function $w$ . With a new inequality for polynomials and with the use of fast decreasing polynomials, it is shown that an approach of D. S. Lubinsky is applicable. The proof works at all points that are Lebesgue-points for both the weight function $w$ and $\log \,w$ .

Consider a quartic $q$-Weil polynomial $f$. Motivated by equidistribution considerations, we define, for each prime $\ell$, a local factor that measures the relative frequency with which $f$$ \bmod \,\ell $ occurs as the characteristic polynomial of a symplectic similitude over ${{\mathbb{F}}_{\ell }}$. For a certain class of polynomials, we show that the resulting infinite product calculates the number of principally polarized abelian surfaces over ${{\mathbb{F}}_{q}}$ with Weil polynomial $f$.

We establish asymptotics for Christoffel functions associated with multivariate orthogonal polynomials. The underlying measures are assumed to be regular on a suitable domain. In particular, this is true if they are positive a.e. on a compact set that admits analytic parametrization. As a consequence, we obtain asymptotics for Christoffel functions for measures on the ball and simplex under far more general conditions than previously known. As another consequence, we establish universality type limits in the bulk in a variety of settings.

We describe a probability distribution on isomorphism classes of principally quasi-polarized $p$-divisible groups over a finite field $k$ of characteristic $p$ which can reasonably be thought of as a ‘uniform distribution’, and we compute the distribution of various statistics ($p$-corank, $a$-number, etc.) of $p$-divisible groups drawn from this distribution. It is then natural to ask to what extent the $p$-divisible groups attached to a randomly chosen hyperelliptic curve (respectively, curve; respectively, abelian variety) over $k$ are uniformly distributed in this sense. This heuristic is analogous to conjectures of Cohen–Lenstra type for $\text{char~} k\not = p$, in which case the random $p$-divisible group is defined by a random matrix recording the action of Frobenius. Extensive numerical investigation reveals some cases of agreement with the heuristic and some interesting discrepancies. For example, plane curves over ${\mathbf{F} }_{3} $ appear substantially less likely to be ordinary than hyperelliptic curves over ${\mathbf{F} }_{3} $.

Considering quantum random walks, we construct discrete-time approximations of the eigenvalues processes of minors of Hermitian Brownian motion. It has been recently proved by Adler, Nordenstam, and van Moerbeke that the process of eigenvalues of two consecutive minors of a Hermitian Brownian motion is a Markov process; whereas, if one considers more than two consecutive minors, the Markov property fails. We show that there are analog results in the noncommutative counterpart and establish the Markov property of eigenvalues of some particular submatrices of Hermitian Brownian motion.

Integrable operators arise in random matrix theory, where they describe the asymptotic eigenvalue distribution of large self-adjoint random matrices from the generalized unitary ensembles. We consider discrete Tracy–Widom operators and give sufficient conditions for a discrete integrable operator to be the square of a Hankel matrix. Examples include the discrete Bessel kernel and kernels arising from the almost Mathieu equation and the Fourier transform of Mathieu's equation.

For p ≤ n, let b1(n),...,bp(n) be independent random vectors in $\mathbb{R}^n$ with the same distribution invariant by rotation and without mass at the origin. Almost surely these vectors form a basis for the Euclidean lattice they generate. The topic of this paper is the property of reduction of this random basis in the sense of Lenstra-Lenstra-Lovász (LLL). If $\widehat b_{1}^{(n)},\ldots, \widehat b_p^{(n)}$ is the basis obtained from b1(n),...,bp(n) by Gram-Schmidt orthogonalization, the quality of the reduction depends upon the sequence of ratios of squared lengths of consecutive vectors $r_j^{(n)} = \Vert \widehat b^{(n)}_{n-j+1}\Vert^2 / \Vert \widehat b^{(n)}_{n-j} \Vert^2$ , j = 1,...,p - 1. We show that as n → +∡ the process $(r_j^{(n)}-1,j\geq 1)$ tends in distribution in some sense to an explicit process $({\mathcal R}_j -1,j\geq 1)$ ; some properties of the latter are provided. The probability that a random random basis is s-LLL-reduced is then showed to converge for p = n - g, and g fixed, or g = g(n) → +∞.

Following Füredi and Komlós, we develop a graph theory method to study the high moments of large random matrices with independent entries. We apply this method to sparse N × N random matrices AN,p that have, on average, p non-zero elements per row. One of our results is related to the asymptotic behaviour of the spectral norm ∥AN,p∥ in the limit 1 ≪ p ≪ N. We show that the value pc = log N is the critical one for lim ∥AN,p/√p∥ to be bounded or not. We discuss relations of this result with the Erdős–Rényi limit theorem and properties of large random graphs. In the proof, the principal issue is that the averaged vertex degree of plane rooted trees of k edges remains bounded even when k → ∞. This observation implies fairly precise estimates for the moments of AN,p. They lead to certain generalizations of the results by Sinai and Soshnikov on the universality of local spectral statistics at the border of the limiting spectra of large random matrices.

We consider the recursive equation x(n + 1)= A(n)⊗x(n), where x(n + 1) and x(n) are ℝk-valued vectors and A(n) is an irreducible random matrix of size k × k. The matrix-vector multiplication in the (max, +) algebra is defined by (A(n)⊗x(n))= maxj (Aij (n) + xj(n)). This type of equation can be used to represent the evolution of stochastic event graphs which include cyclic Jackson networks, some manufacturing models and models with general blocking (such as Kanban). Let us assume that the sequence {A(n), n ∈ ℕ} is i.i.d. or, more generally, stationary and ergodic. The main result of the paper states that the system couples in finite time with a unique stationary regime if and only if there exists a set of matrices such that and the matrices have a unique periodic regime.

We give a unified presentation of stability results for stochastic vector difference equations based on various choices of binary operations and , assuming that are stationary and ergodic. In the scalar case, under standard addition and multiplication, the key condition for stability is E[log |A0|] < 0. In the generalizations, the condition takes the form γ< 0, where γis the limit of a subadditive process associated with . Under this and mild additional conditions, the process has a unique finite stationary distribution to which it converges from all initial conditions.

The variants of standard matrix algebra we consider replace the operations + and × with (max, +), (max,×), (min, +), or (min,×). In each case, the appropriate stability condition parallels that for the standard recursions, involving certain subadditive limits. Since these limits are difficult to evaluate, we provide bounds, thus giving alternative, computable conditions for stability.

We consider the composition of random i.i.d. affine maps of a Hilbert space to itself. We show convergence of the nth composition of these maps in the Wasserstein metric via a contraction argument. The contraction condition involves the operator norm of the expectation of a bilinear form. This is contrasted with the usual contraction condition of a negative Lyapunov exponent. Our condition is stronger and easier to check. In addition, our condition allows us to conclude convergence of second moments as well as convergence in distribution.