相关论文: Fluctuations in the logarithmic energy for zeros o…

In this paper, we compute the expected logarithmic energy of solutions to the polynomial eigenvalue problem for random matrices. We generalize some known results for the Shub-Smale polynomials, and the spherical ensemble. These two…

In 2011, Armentano, Beltr\'an and Shub obtained in \cite{ABS11} a closed expression for the expected logarithmic energy of the random point process on the sphere given by the roots of random elliptic polynomials. We consider a different…

We describe several randomized collections of $3\times 3$ rotation matrices and analyze their associated logarithmic energy. The best one (i.e. the one attaining the lowest expected logarithmic energy) is constructed by choosing $r$…

The 17th of the problems proposed by Steve Smale for the 21st century asks for the existence of a deterministic algorithm computing an approximate solution of a system of $n$ complex polynomials in $n$ unknowns in time polynomial, on the…

This article is divided in two parts. In the first part we review some recent results concerning the expected number of real roots of random system of polynomial equations. In the second part we deal with a different problem, namely, the…

Distributing points on a (possibly high-dimensional) sphere with minimal energy is a long-standing problem in and outside the field of mathematics. This paper considers a novel energy function that arises naturally from statistics and…

In this paper, we investigate discrete logarithmic energy problems in the unit circle. We study the equilibrium configuration of $n$ electrons and $n-1$ pairs of external protons of charge $+1/2$. It is shown that all the critical points of…

It is well-known that the semi-circle law, which is the limiting distribution in the Wigner theorem, is the minimizer of the logarithmic energy penalized by the second moment. A very similar fact holds for the Girko and Marchenko--Pastur…

We analyse several constructions of random point sets on the sphere $\mathbb{S}^{3}\subset\mathbb{R}^4$ evaluating and comparing them through their discrete logarithmic energy: \begin{equation*} E_0(\omega_N) = \sum_{\substack{i, j=1\\ i…

We discuss a series of 8 energy scales, some of which just speculated by ourselves, and fit the logarithms of these energies as a straight line versus a quantity related to the dimensionalities of action terms in a way to be defined in the…

We study the stochastic homogenization and obtain a random fluctuation theory for semilinear elliptic equations with a rapidly varying random potential. To first order, the effective potential is the average potential and the nonlinearity…

We prove a Central Limit Theorem for the Critical Points of Random Spherical Harmonics, in the High-Energy Limit. The result is a consequence of a deeper characterizations of the total number of critical points, which are shown to be…

Let us consider i.i.d. random variables $\{a_k,b_k\}_{k \geq 1}$ defined on a common probability space $(\Omega, \mathcal F, \mathbb P)$, following a symmetric Rademacher distribution and the associated random trigonometric polynomials…

In this article we consider the distribution of $N$ points on the unit sphere $\mathbb{S}^{d-1}$ in $\mathbb{R}^d$ interacting via logarithmic potential. A characterization theorem of the stationary configurations is derived when $N=d+2$…

In this paper we explore the connections between minimizers of the discrete logarithmic energy on the 2-dimensional sphere, univariate polynomials with optimal condition number in the Shub-Smale sense and a quotient involving norms of…

In part I: We find a series physical scales such as 1) Planck scale, 2) Minimal approximate grand unification SU(5), 3) the mass scale of the see saw model right handed or Majorana neutrinoes, some invented scale with many scalar bosons,…

In this paper we obtain the central limit theorems, moderate deviations and the laws of the iterated logarithm for the energy \[H_n=\sum_{1\le j<k\le n}\omega_j\omega_k1_{\{S_j=S_k\}}\] of the polymer $\{S_1,...,S_n\}$ equipped with random…

We revisit a classical question: how large is the minimal logarithmic energy of $n$ points on $\mathbb{S}^2$ $$ \mathcal{E}_{\log}(n) = \min_{x_1, \dots, x_n \in \mathbb{S}^2} \quad \sum_{i,j =1 \atop i \neq j}^{n}{…

We obtain a Central Limit Theorem for the normalized number of real roots of a square Kostlan-Shub-Smale random polynomial system of any size as the degree goes to infinity. A study of the asymptotic variance of the number of roots is…

Using the multiplicities of the Laplace eigenspace on the sphere (the space of spherical harmonics) we endow the space with Gaussian probability measure. This induces a notion of random Gaussian spherical harmonics of degree $n$ having…