CLT for non-Hermitian random band matrices with variance profiles

We show that the fluctuations of the linear eigenvalue statistics of a non-Hermitian random band matrix of increasing bandwidth $b_{n}$ with a continuous variance profile $w_{\nu}(x)$ converges to a $N(0,\sigma_{f}^{2}(\nu))$, where $\nu=\lim_{n\to\infty}(2b_{n}/n)\in [0,1]$ and $f$ is the test function. When $\nu\in (0,1]$, we obtain an explicit formula for $\sigma_{f}^{2}(\nu)$, which depends on $f$, and variance profile $w_{\nu}$. When $\nu=1$, the formula is consistent with Rider, and Silverstein (2006). We also independently compute an explicit formula for $\sigma_{f}^{2}(0)$ i.e., when the bandwidth $b_{n}$ grows slower compared to $n$. In addition, we show that $\sigma_{f}^{2}(\nu)\to \sigma_{f}^{2}(0)$ as $\nu\downarrow 0$.Comment: Typos corrected; a few more explanations and a couple of pictures have been adde

A functional CLT for partial traces of random matrices

In this paper we show a functional central limit theorem for the sum of the first $\lfloor t n \rfloor$ diagonal elements of $f(Z)$ as a function in $t$, for $Z$ a random real symmetric or complex Hermitian $n\times n$ matrix. The result holds for orthogonal or unitarily invariant distributions of $Z$, in the cases when the linear eigenvalue statistic $\operatorname{tr} f(Z)$ satisfies a CLT. The limit process interpolates between the fluctuations of individual matrix elements as $f(Z)_{1,1}$ and of the linear eigenvalue statistic. It can also be seen as a functional CLT for processes of randomly weighted measures

Effects of NPK And Sulfur on the Yield and Absorption of Nutrients of Lepidium peruvianum Ch. in Field and Greenhouse

Increasing the quality and yields of maca are important goals to meet market demands. The objective of the research was to quantify the extraction of nutrients from the soil and evaluate the effect of three levels of nitrogen (N), phosphorus (P), potassium (K) and sulfur (S) on agronomic indicators of maca cultivation under field and greenhouse conditions. The experiment was conducted under the randomized complete block design with four repetitions per treatment. The levels were  240-180-210-60, 160-120-140-40 and 0-0-0-0 NPKS, respectively. Tukey's test was used for the comparison of means. Different indicators of plant growth and development were evaluated, such as plant height, hypocotyl diameter, hypocotyl weight, dry matter percentage and yield. The evaluation of nutrient extraction was evaluated according to the methodology used in the AGROLAB laboratory: Nitrogen: micro kjeldahl method. Phosphorus: Bray–Kurtz colorimetric method.  Potassium: Peech's turbidimetric method. Sulfur: Massoumi's turbidimetric method. Calcium and Magnesium: volumetric method of complexometry. Among the results, significant differences were found between S levels in all the variables evaluated. No statistical differences were found in nutrient extraction due to NPKS doses. It is concluded that higher doses of NPKS increases in the values of height, hypocotyl size, hypocotyl weight and yield, except for dry matter, in maca plants are obtained

Fluctuations for analytic test functions in the Single Ring Theorem

We consider a non-Hermitian random matrix $A$ whose distribution is invariant under the left and right actions of the unitary group. The so-called Single Ring Theorem, proved by Guionnet, Krishnapur and Zeitouni, states that the empirical eigenvalue distribution of $A$ converges to a limit measure supported by a ring $S$. In this text, we establish the convergence in distribution of random variables of the type $Tr (f(A)M)$ where $f$ is analytic on $S$ and the Frobenius norm of $M$ has order $\sqrt{N}$. As corollaries, we obtain central limit theorems for linear spectral statistics of $A$ (for analytic test functions) and for finite rank projections of $f(A)$ (like matrix entries). As an application, we locate outliers in multiplicative perturbations of $A$.Comment: 29 pages, 1 figure. In Version v2, we slightly modified the assumptions, in order to fix a problem un the control of the tails (see Assumption 2.3). In v3, some minors typos were corrected. In v4, some explanations were added in the introduction and some typos were corrected. To appear in Indiana Univ. Math.

Fluctuations of eigenvalues of random normal matrices

In this note, we prove Gaussian field convergence of fluctuations of eigenvalues of random normal matrices in the interior of a quantum droplet

Linear Statistics of Non-Hermitian Matrices Matching the Real or Complex Ginibre Ensemble to Four Moments

We prove that, for general test functions, the limiting behavior of the linear statistic of an independent entry random matrix is determined only by the first four moments of the entry distributions. This immediately generalizes the known central limit theorem for independent entry matrices with complex normal entries. We also establish two central limit theorems for matrices with real normal entries, considering separately functions supported exclusively on and exclusively away from the real line. In contrast to previously obtained results in this area, we do not impose analyticity on test functions.Comment: Preliminary versio