Related papers: Poissonian correlation of higher order differences
Given a simple transient random walk $(S_n)_{n\geq 0}$ in $\mathbf{Z}$ and a stationary sequence of real random variables $(\xi(s))_{s\in \mathbf{Z}}$, we investigate the extremes of the sequence $(\xi(S_n))_{n\geq 0}$. Under suitable…
It is demonstrated that in a two-stage scenario with elementary Poissonian emitters of particles (colour strings) arbitrarily distributed in their number and average multiplicities, the forward- backward correlations are completely…
Let $E$ be an elliptic curve defined over $\mathbb{Q}$ and let $N$ be a positive integer. Now, $M_E(N)$ counts the number of primes $p$ such that the group $E_p(\mathbb{F}_p)$ is of order $N$. In an earlier joint work with Balasubramanian,…
Let $S_n$ denote the symmetric group of permutations acting on $n$ elements. We investigate the double sequence $\{N_{\ell}(n)\}$ counting the number of $\ell$ tuples of elements of the symmetric group $S_n$, where the components commute,…
We study the Kronecker sequence $\{n\alpha\}_{n\leq N}$ on the torus ${\mathbb T}^d$ when $\alpha$ is uniformly distributed on ${\mathbb T}^d.$ We show that the discrepancy of the number of visits of this sequence to a random box,…
We study the behavior of a polynomial sequence which is defined by iterating a polynomial pair under Thue-Morse dynamic. We show that in suitable sense, the sequence will behave like $\{2\cos 2^nx: n\ge 1\}$. Basing on this property we can…
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 $\lambda$,…
We study the distribution of large prime factors of a random element $u$ of arithmetic sequences satisfying simple regularity and equidistribution properties. We show that if such an arithmetic sequence has level of distribution $1$ the…
We prove that any increasing sequence of real numbers with average gap $1$ and Poisson pair correlations has some gap that is at least $3/2+10^{-9}$. This improves upon a result of Aistleitner, Blomer, and Radziwill.
In this paper we investigate the existence of a solution to the Poisson equation on complete manifolds with positive spectrum and Ricci curvature bounded from below. We show that if a function $f$ has decay $f=O(r^{-1-\varepsilon}) $ for…
We establish a lower bound on the entropy of weighted sums of (possibly dependent) random variables $(X_1, X_2, \dots, X_n)$ possessing a symmetric joint distribution. Our lower bound is in terms of the joint entropy of $(X_1, X_2, \dots,…
We present the necessary and sufficient conditions of the well-posedness of the initial value problem for certain fourth-order linear dispersive systems on the one-dimensional torus. This system is related with a dispersive flow for closed…
We prove results towards the equidistribution of certain families of periodic torus orbits on homogeneous spaces, with particular focus on the case of the diagonal torus acting on quotients of $\PGL_n(\R)$. After attaching to each periodic…
We consider the number of common edges in two independent random spanning trees of a graph $G$. For complete graphs $K_n$, we give a new proof of the fact, originally obtained by Moon, that the distribution converges to a Poisson…
We generalize the Poisson limit theorem to binary functions of random objects whose law is invariant under the action of an amenable group. Examples include stationary random fields, exchangeable sequences, and exchangeable graphs. A…
A Goellnitz-Gordon partition is one in which the parts differ by at least 2, and where the inequality is strict if a part is even. Let Q_i(n) denote the number of partitions of n into distinct parts not congruent to i mod 4. By attaching…
The maximal correlation coefficient is a well-established generalization of the Pearson correlation coefficient for measuring non-linear dependence between random variables. It is appealing from a theoretical standpoint, satisfying…
We consider the fully-coupled McKean-Vlasov equation with multi-time-scale potentials, and all the coefficients depend on the distributions of both the slow component and the fast motion. By studying the smoothness of the solution of the…
We give a sufficient condition for a random sequence in [0,1] generated by a $\Psi$-process to be equidistributed. The condition is met by the canonical example -- the $\max$-2 process -- where the $n$th term is whichever of two uniformly…
We consider the large-N limit of \mathcal{N}=6 U(N) \times U(N) superconformal Chern-Simons (ABJM) theory with fixed level k, which is conjectured to be dual to M-theory on AdS4\times (S^7/Z_k) background. We point out that the so-called…