Related papers: Distribution of gaps between eigenangles of Hecke …

The statistics of gap ratios between consecutive energy levels is a widely used tool, in particular in the context of many-body physics, to distinguish between chaotic and integrable systems, described respectively by Gaussian ensembles of…

This note reports the results of an undergraduate research project from the year 2013-14, concerning the convergence of iterated Steiner symmetrizations in the plane. The directions of symmetrization are chosen according to the Van der…

We survey some past conditional results on the distribution of large differences between consecutive primes and examine how the Hardy-Littlewood prime k-tuples conjecture can be applied to this question.

A generic uniformly distributed random sequence on the unit interval has Poissonian pair correlations. At the same time, there are only very few explicitly known examples of sequences with this property. Moreover, many types of…

For a random variable with a unimodal distribution and finite second moment Gau\ss \, (1823) proved a sharp bound on the probability of the random variable to be outside a symmetric interval around its mode. An alternative proof for it is…

Let $(\lambda_f(n))_{n\geq 1}$ be the Hecke eigenvalues of either a holomorphic Hecke eigencuspform or a Hecke-Maass cusp form $f$. We prove that, for any fixed $\eta>0$, under the Ramanujan-Petersson conjecture for $\rm GL_2$ Maass forms,…

This article studies a general divide-and-conquer algorithm for approximating continuous one-dimensional probability distributions with finite mean. The article presents a numerical study that compares pre-existing approximation schemes…

This article presents a limit theorem for the gaps $\widehat{G}_{i:n}:= X_{n-i+1:n} - X_{n-i:n}$ between order statistics $X_{1:n} \le \cdots \le X_{n:n}$ of a sample of size $n$ from a random discrete distribution on the positive integers…

A systematic study of avoidance of mesh patterns of length 2 was conducted by Hilmarsson et al., where 25 out of 65 non-equivalent cases were solved. In this paper, we give 27 distribution results for these patterns including 14…

We consider the Hankel determinant generated by the moments of the even weight function ${\rm e}^{-x^2}(A+B\theta(x^2-a^2)), x\in(-\infty,+\infty), a>0, A\ge0, A+B\ge0$. It is intimately related to the gap probability of the Gaussian…

Consider N equally-spaced points on a circle of circumference N. Choose at random n points out of $N$ on this circle and append clockwise an arc of integral length k to each such point. The resulting random set is made of a random number of…

The Nearest Neighbour Spacing (NNS) distribution can be computed for generalized symmetric 2x2 matrices having different variances in the diagonal and in the off-diagonal elements. Tuning the relative value of the variances we show that the…

Let $p$ be a prime number, $X$ be an absolutely irreducible affine plane curve over $\mathbb{F}_p$, and $g,f\in\mathbb{F}_p(x,y)$. We study the distribution of the values of the hybrid exponential sums S_n on $n\in\mathcal{I}$ for some…

Let $(\lambda_f(n))_{n\geqslant1}$ be the Hecke eigenvalues of a holomorphic cusp form $f$. We prove that the exponent of distribution of $\lambda_f*1$ in arithmetic progressions is as large as $\frac{1}{2}+\frac{1}{70}$ when the modulus…

Given a variety over $\mathbb{Q}$, we study the distribution of the number of primes dividing the coordinates as we vary an integral point. Under suitable assumptions, we show that this has a multivariate normal distribution. We generalise…

We propose the following model of a random graph on n vertices. Let F be a distribution in R_+^{n(n-1)/2} with a coordinate for every pair i$ with 1 \le i,j \le n. Then G_{F,p} is the distribution on graphs with n vertices obtained by…

Based on the primes less than $4 \times 10^{18}$, Oliveira e Silva et al. (2014) conjectured an asymptotic formula for the sum of the $k$th power of the gaps between consecutive primes less than a large number $x$. We show that the…

By a classical result of Weyl, for any increasing sequence $(n_k)_{k \geq 1}$ of integers the sequence of fractional parts $(\{n_k x\})_{k \geq 1}$ is uniformly distributed modulo 1 for almost all $x \in [0,1]$. Except for a few special…

In this note we compare two recent results about the distribution of eigenvalues for semi-classical pseudodifferential operators in two dimensions. For classes of analytic operators A. Melin and the author obtained a complex Bohr-Sommerfeld…

Let us denote the nth difference between consecutive primes by d_n. The Prime Number Theorem clearly implies that d_n is logn on average. Paul Erd\H{o}s conjectured about 60 years ago that the sequence d_n/logn is everywhere dense on the…