Related papers: Gap statistics and higher correlations for geometr…

A permutation $\sigma$ describing the relative orders of the first $n$ iterates of a point $x$ under a self-map $f$ of the interval $I=[0,1]$ is called an \emph{order pattern}. For fixed $f$ and $n$, measuring the points $x\in I$ (according…

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,…

In this paper, we show how to use the framework of mod-Gaussian convergence in order to study the fluctuations of certain models of random graphs, of random permutations and of random integer partitions. We prove that, in these three…

A generic uniformly distributed sequence $(x_n)_{n \in \mathbb{N}}$ in $[0,1)$ possesses Poissonian pair correlations (PPC). Vice versa, it has been proven that a sequence with PPC is uniformly distributed. Grepstad and Larcher gave an…

Let G be a connected semisimple Lie group with finite center and without compact factors, P a minimal parabolic subgroup of G, and \Gamma a lattice in G. We prove that every \Gamma-orbits in the Furstenberg boundary G/P is equidistributed…

The Bayesian solution to a statistical inverse problem can be summarised by a mode of the posterior distribution, i.e. a MAP estimator. The MAP estimator essentially coincides with the (regularised) variational solution to the inverse…

The distribution of higher order level spacings, i.e. the distribution of $\{s_{i}^{(n)}=E_{i+n}-E_{i}\}$ with $n\geq 1$ is derived analytically using a Wigner-like surmise for Gaussian ensembles of random matrix as well as Poisson…

Answering an informal question of K. Park, we show that by fixing some irrational alpha to have a particular standard continued fraction expansion, we may force the associated discrepancy sequences for all x in [0,1), which track the…

We establish an equidistribution result for push-forwards of certain locally finite algebraic measures in the adelic extension of the space of lattices in the plane. As an application of our analysis we obtain new results regarding the…

In this article, we study the smallest gaps of the Gaussian orthogonal ensemble. The main result is that the smallest gaps, after normalized by $n$, will tend to a Poisson distribution, and the limiting density of the $k$-th normalized…

Considered are the large $N$, or large intensity, forms of the distribution of the length of the longest increasing subsequences for various models. Earlier work has established that after centring and scaling, the limit laws for these…

Let G=ASL(2,R) be the affine special linear group of the plane, and set Gamma=ASL(2,Z). Building on recent work of Str\"ombergsson we prove a rate of equidistribution for the orbits of a certain 1-dimensional unipotent flow of Gamma\G,…

Let $P^{\left(\frac 12\right)}(n)$ denote the middle prime factor of $n$ (taking into account multiplicity). More generally, one can consider, for any $\alpha \in (0,1)$, the $\alpha$-positioned prime factor of $n$, $P^{(\alpha)}(n)$. It…

Let L be a Lie group and Lambda a lattice in L. Suppose G is a non-compact simple Lie group realized as a Lie subgroup of L, and the image of G on L/Lambda is dense. Let c be a diagonalizable element of G not contained in a compact…

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…

A classical result of Kahn and Saks states that given any partially ordered set with two distinguished elements, the number of linear extensions in which the ranks of the distinguished elements differ by $k$ is log-concave as a function of…

Let $(x_n)_{n=1}^\infty$ be a sequence of integers. We study the number variance of dilations $(\alpha x_n)_{n=1}^\infty$ modulo 1 in intervals of length $S$, and establish pseudorandom (Poissonian) behavior for Lebesgue-almost all $\alpha$…

We elaborate the notions of Martin-L\"of and Schnorr randomness for real numbers in terms of uniform distribution of sequences. We give a necessary condition for a real number to be Schnorr random expressed in terms of classical uniform…

The Poisson distribution of order $k$ is a special case of a compound Poisson distribution. For $k=1$ it is the standard Poisson distribution. Our main result is a proof that for sufficiently small values of the rate parameter $\lambda$,…

The aim of the present article is to introduce a concept which allows to generalise the notion of Poissonian pair correlation, a second-order equidistribution property, to higher dimensions. Roughly speaking, in the one-dimensional setting,…