Narrow Results

The fourth Standard Model (SM) family quarks and weak iso-singlet quarks predicted by E₆ GUT are considered. The spin-average of the pseudoscalar η₄(n¹S₀) and vector ψ₄(n³S₁) quarkonium binding masses of the new mesons formed by the fourth Standard Model (SM) family and iso-singlet E₆ with their mixings to ordinary quarks are investigated. Further, the fine and hyperfine mass splittings of the these states are also calculated. We solved the Schrödinger equation with logarithmic and Martin potentials using the shifted large-N expansion technique. Our results are compared with other models to gauge the reliability of the predictions and point out differences.

Using a formulation of quantum mechanics based on orthogonal polynomials in the energy and physical parameters, we present a method that gives the class of potential functions for exactly solvable problems corresponding to a given energy spectrum. In this work, we study the class of problems associated with the continuous dual Hahn polynomial. These include the one-dimensional logarithmic potential and the three-dimensional Coulomb plus linear potential.

We consider the so-called Cahn–Hilliard–Oono equation with singular (e.g. logarithmic) potential in a bounded domain of ${ℝ}^d$, $d\le 3$. The equation is subject to an initial condition and Neumann homogeneous boundary conditions for the order parameter as well as for the chemical potential. However, contrary to the Cahn–Hilliard equation, the total mass might not be conserved. The existence of a global finite energy solution to such a problem was proven by Miranville and Temam. We first establish some regularization properties in finite time of the (unique) solution. Then, in dimension two, we prove the so-called strict separation property, namely, we show that any finite energy solution stays away from pure phases, uniformly with respect to the initial energy and the total mass. Taking advantage of these results, we study the long-time behavior of solutions. More precisely, we establish the existence of the global attractor in both two and three dimensions. Due to the strict separation property in dimension two, we also prove the existence of exponential attractors and we show that a finite energy solution always converges to a single equilibrium even though the mass is not conserved.

We study the longtime behavior of the Caginalp phase-field model with a logarithmic potential and dynamic boundary conditions for both the order parameter and the temperature. Due to the possible lack of distributional solutions, we deal with a suitable definition of solutions based on variational inequalities, for which we prove well-posedness and the existence of global and exponential attractors with finite fractal dimension.

We investigate the universality of singular value and eigenvalue distributions of matrix-valued functions of independent random matrices and apply these general results in several examples. In particular we determine the limit distribution and prove universality under general conditions for singular value and eigenvalue distributions of products of independent matrices from spherical ensembles.

We consider the random Markov matrix obtained by assigning i.i.d. non-negative weights to each edge of the complete oriented graph. In this study, the weights have unbounded first moment and belong to the domain of attraction of an alpha-stable law. We prove that as the dimension tends to infinity, the empirical measure of the singular values tends to a probability measure which depends only on alpha, characterized as the expected value of the spectral measure at the root of a weighted random tree. The latter is a generalized two-stage version of the Poisson weighted infinite tree (PWIT) introduced by David Aldous. Under an additional smoothness assumption, we show that the empirical measure of the eigenvalues tends to a non-degenerate isotropic probability measure depending only on alpha and supported on the unit disk of the complex plane. We conjecture that the limiting support is actually formed by a strictly smaller disk.

We consider random $n\times n$ matrices of the form $Y_n=\frac{1}{\sqrt{d}}{A}_n\circ X_n$, where $A_n$ is the adjacency matrix of a uniform random $d$-regular directed graph on $n$ vertices, with $d=\lfloor pn\rfloor$ for some fixed $p\in (0,1)$, and $X_n$ is an $n\times n$ matrix of i.i.d. centered random variables with unit variance and finite $(4+\eta )$th moment (here ∘ denotes the matrix Hadamard product). We show that as $n\to \infty$, the empirical spectral distribution of $Y_n$ converges weakly in probability to the normalized Lebesgue measure on the unit disk.

We consider products of independent $n\times n$ non-Hermitian random matrices $X^{(1)},\dots ,X^{(m)}$. Assume that their entries, $X_{jk}^{(q)},1\le j,k\le n,q=1,\dots ,m$, are independent identically distributed random variables with zero mean, unit variance. Götze and Tikhomirov [On the asymptotic spectrum of products of independent random matrices, preprint (2010), arXiv:1012.2710] and O’Rourke and Sochnikov [Products of independent non-Hermitian random matrices, Electron. J. Probab.16 (2011) 2219–2245] proved that under these assumptions the empirical spectral distribution (ESD) of $X^{(1)}\cdots X^{(m)}$ converges to the limiting distribution which coincides with the distribution of the $m$th power of random variable uniformly distributed in the unit circle. In this paper, we provide a local version of this result. More precisely, assuming additionally that $𝔼|X_{11}^{(q)}{|}^{4+\delta }<\infty$ for some $\delta >0$, we prove that ESD of $X^{(1)}\cdots X^{(m)}$ converges to the limiting distribution on the optimal scale up to $n^{-1+2a},0<a<1/2$ (up to some logarithmic factor). Our results generalize the recent results of Bourgade et al. [Local circular law for random matrices, Probab. Theory Related Fields159 (2014) 545–595], Tao and Vu [Random matrices: Universality of local spectral statistics of non-Hermitian matrices, Ann. Probab. 43 (2015) 782–874] and Nemish [Local law for the product of independent non-hermitian random matrices with independent entries, Electron. J. Probab.22 (2017) 1–35]. We also give further development of Stein’s type approach to estimate the Stieltjes transform of ESD.

The aim of this paper is to investigate the Kolmogorov distance of the Circular Law to the empirical spectral distribution of non-Hermitian random matrices with independent entries. The optimal rate of convergence is determined by the Ginibre ensemble and is given by $n^{-1/2}$. A smoothing inequality for complex measures that quantitatively relates the uniform Kolmogorov-like distance to the concentration of logarithmic potentials is shown. Combining it with results from Local Circular Laws, we apply it to prove nearly optimal rate of convergence to the Circular Law in Kolmogorov distance. Furthermore, we show that the same rate of convergence holds for the empirical measure of the roots of Weyl random polynomials.

We study the empirical spectral distribution (ESD) for complex $n\times n$ matrix polynomials of degree k under relatively mild assumptions on the underlying distributions, thus highlighting universality phenomena. In particular, we assume that the entries of each matrix coefficient of the matrix polynomial have mean zero and finite variance, potentially allowing for distinct distributions for entries of distinct coefficients. We derive the almost sure limit of the ESD in two distinct scenarios: (1) $n\to \infty$ with k constant and (2) $k\to \infty$ with n bounded by $O(k^P)$ for some $P>0$; the second result additionally requires that the underlying distributions are continuous and uniformly bounded. Our results are universal in the sense that they depend on the choice of the variances and possibly on k (if it is kept constant), but not on the underlying distributions. The results can be specialized to specific models by fixing the variances, thus obtaining matrix polynomial analogues of results known for special classes of scalar polynomials, such as Kac, Weyl, elliptic and hyperbolic polynomials.

It has been known for some time that the existing asymptotic methods for integrals and differential equations are not applicable in the case of Stieltjes–Wigert polynomials with degree going to infinity. Using the recently introduced nonlinear steepest descent method for Riemann–Hilbert problems, here we not only derive an asymptotic expansion for these polynomials, but we also show that the result holds uniformly in the complex plane except for a sector containing the real axis from −∞ to 1/4. Furthermore, we give an asymptotic formula for the zeros of these polynomials, which approximates the true values of the zeros closely.

On sait depuis longtemps que les méthodes asymptotiques existantes pour les intégrales et les équations différentielles ne sont pas applicables aux polynômes de Stieltjes–Wigert lorsque le degré tend vers l'infini. En utilisant des méthodes de descente non lineaires, introduites récemment pour les problèmes de Riemann–Hilbert, nous obtenons non seulement un développement asymptotique pour ces polynômes, mais nous montrons également que le résultat est uniforme dans le plan complexe, sauf pour un secteur contenant l'axe réel de −∞ à 1/4. De plus, nous démontrons une formule asymptotique qui donne une bonne approximation des racines de ces polynômes.