Related papers: On the reduction of a random basis

Following S\"odergren, we consider a collection of random variables on the space $X_n$ of unimodular lattices in dimension $n$: Normalizations of the angles between the $N = N(n)$ shortest vectors in a random unimodular lattice, and the…

We construct a point set in the Euclidean plane that elucidates the relationship between the fine-scale statistics of the fractional parts of $\sqrt n$ and directional statistics for a shifted lattice. We show that the randomly rotated, and…

Complex bases, along with direct-sums defined by rings of imaginary quadratic integers, induce algebraic lattices. In this work, we study such lattices and their reduction algorithms. Firstly, when the lattice is spanned over a two…

We prove that the number of Siegel-reduced bases for a randomly chosen $n$-dimensional lattice becomes, for $n \rightarrow \infty$, tightly concentrated around its mean. We also show that most reduced bases behave as in the worst-case…

The $d$-process generates a graph at random by starting with an empty graph with $n$ vertices, then adding edges one at a time uniformly at random among all pairs of vertices which have degrees at most $d-1$ and are not mutually joined. We…

While an explicit basis is common in the study of Euclidean spaces, it is usually implied in the study of inertial relativistic systems. There are some conceptual advantages to including the basis in the study of special relativistic…

Several important families of computational and statistical results in machine learning and randomized algorithms rely on uniform bounds on quadratic forms of random vectors or matrices. Such results include the Johnson-Lindenstrauss (J-L)…

We determine the joint distribution of the lengths of, and angles between, the N shortest lattice vectors in a random n-dimensional lattice as n tends to infinity. Moreover we interpret the result in terms of eigenvalues and eigenfunctions…

Let $L_n(k)$ denote the least common multiple of $k$ independent random integers uniformly chosen in $\{1,2,\ldots ,n\}$. In this note, using a purely probabilistic approach, we derive a criterion for the convergence in distribution as…

In this paper we improve the probabilistic approach to compositions of Ehrenborg, Levin and Readdy by introducing a simpler but more general probabilistic model. As consequence we get some new estimates on the behavior of a uniform random…

Let $\a$ be a complex random variable with mean zero and bounded variance $\sigma^{2}$. Let $N_{n}$ be a random matrix of order $n$ with entries being i.i.d. copies of $\a$. Let $\lambda_{1}, ..., \lambda_{n}$ be the eigenvalues of…

We consider properties of determinants of some random symmetric matrices issued from multivariate statistics: Wishart/Laguerre ensemble (sample covariance matrices), Uniform Gram ensemble (sample correlation matrices) and Jacobi ensemble…

Let $P_r(n)$ be the set of partitions of n with non negative rth differences. Let $\lambda$ be a partition chosen uniformly at random among the set $P_r(n)$. Let $d(\lambda)$ be a positive rth difference chosen uniformly at random in…

We study the asymptotic distribution of norm ball averages along orbits of a lattice $\Gamma \subset \text{SO}(n,1)$ acting on the moduli space of pairs of orthogonal discrete subgroups of $\mathbb{R}^{n+1}$ up to homothety. Our main result…

In the context of mod-Gaussian convergence, as defined previously in our work with J. Jacod, we obtain lower bounds for local probabilities for a sequence of random vectors which are approximately Gaussian with increasing covariance. This…

Let $\{V_{i,j}; (i,j)\in\N^2\}$ be a two-dimensional array of i.i.d.\ random variables. The limit laws of the sum of independent random products $$ Z_n=\sum_{i=1}^{N_n} \prod_{j=1}^{n} e^{V_{i,j}} $$ as $n,N_n\to\infty$ have been…

Classical random matrix ensembles with orthogonal symmetry have the property that the joint distribution of every second eigenvalue is equal to that of a classical random matrix ensemble with symplectic symmetry. These results are shown to…

If $\alpha,\beta>0$ are distinct and if $A$ and $B$ are independent non-degenerate positive random variables such that $$S=\tfrac{1}{B}\,\tfrac{\beta A+B}{\alpha A+B}\quad \mbox{and}\quad T=\tfrac{1}{A}\,\tfrac{\beta A+B}{\alpha A+B} $$ are…

Consider $N\times N$ hermitian or symmetric random matrices $H$ with independent entries, where the distribution of the $(i,j)$ matrix element is given by the probability measure $\nu_{ij}$ with zero expectation and with variance…

A recent SICOMP paper on classical and quantum algorithms for the shortest vector problem introduced a lattice-dependent parameter \(\gamma(L)\), bounded universally in the exponential sense by \(2^{0.402n+o(n)}\), and conjectured that this…