Related papers: Crystallization of random trigonometric polynomial…

We develop a new method for studying the asymptotics of symmetric polynomials of representation-theoretic origin as the number of variables tends to infinity. Several applications of our method are presented: We prove a number of theorems…

We study the asymptotic distribution of the number $Z_{N}$ of zeros of random trigonometric polynomials of degree $N$ as $N\to\infty$. It is known that as $N$ grows to infinity, the expected number of the zeros is asymptotic to…

With many pretreatment covariates and treatment factors, the classical factorial experiment often fails to balance covariates across multiple factorial effects simultaneously. Therefore, it is intuitive to restrict the randomization of the…

The radial distribution function is a characteristic geometric quantity of a point set in Euclidean space that reflects itself in the corresponding diffraction spectrum and related objects of physical interest. The underlying combinatorial…

We study the characteristic polynomial of Haar distributed random unitary matrices. We show that after a suitable normalization, as one increases the size of the matrix, powers of the absolute value of the characteristic polynomial as well…

This paper investigates asymptotic distribution of complex zeros of random polynomials $P_n(z):=\sum_{k=0}^{n}b(k)\xi_k z^k$, as $n\to\infty$, where $b$ is a regularly varying function at infinity with index $\alpha\in \mathbb{R}$ and…

We experimentally investigate the crystallization of a uniformly heated quasi-2D granular fluid as a function of filling fraction. Our experimental results for the Lindemann melting criterion, the radial distribution function, the bond…

Crystallization, a prototypical self-organization process during which a disordered state spontaneously transforms into a crystal characterized by a regular arrangement of its building blocks, usually proceeds by nucleation and growth. In…

In this article we describe the crystallization conjecture. It states that, in appropriate physical conditions, interacting particles always place themselves into periodic configurations, breaking thereby the natural translation-invariance…

We study the asymptotic laws for the spatial distribution and the number of connected components of zero sets of smooth Gaussian random functions of several real variables. The primary examples are various Gaussian ensembles of real-valued…

Consider the ensemble of Real Symmetric Toeplitz Matrices, each entry iidrv from a fixed probability distribution p of mean 0, variance 1, and finite higher moments. The limiting spectral measure (the density of normalized eigenvalues)…

We establish a uniform approximation result for the Taylor polynomials of the xi function of Riemann which is valid in the entire complex plane as the degree grows. In particular, we identify a domain growing with the degree of the…

We study the distribution of normalized spacings between the fractional parts of an^2, n=1,2,.... We conjecture that if a is "badly approximable" by rationals, then the sequence of fractional parts has Poisson spacings, and give a number of…

In this paper, we study the asymptotic macroscopic behavior of the root sets of iterated, randomized derivatives of polynomials. The randomization depend on a parameter of inverse temperature $\beta \in (0, \infty]$, the case $\beta =…

In this paper we study a number of conjectures on the behavior of the value distribution of eigenfunctions. On the two dimensional torus we observe that the symmetry conjecture holds in the strongest possible sense. On the other hand we…

We derive conditions under which random sequences of polarizations (two-point symmetrizations) converge almost surely to the symmetric decreasing rearrangement. The parameters for the polarizations are independent random variables whose…

In this paper, we consider sequences of polynomials that satisfy differential--difference recurrences. Our interest is motivated by the fact that polynomials satisfying such recurrences frequently appear as generating polynomials of integer…

Granular crystallisation is an important phenomenon whereby ordered packing structures form in granular matter under vibration. However, compared with the well-developed principles of crystallisation at the atomic scale, crystallisation in…

Polynomials whose coefficients, roots, and critical points lie in the ring of rational integers are called nice polynomials. In this paper, we present a general method for investigating such polynomials. We extend our results from the ring…

We investigate the behavior of some thin sets of integers defined through random trigonometric polynomial when one replaces Gaussian or Rademacher variables by p-stable ones, with 1 < p < 2. We show that in one case this behavior is…