Asymptotic normality of linear statistics of zeros of random polynomials

In this note, we prove a central limit theorem for smooth linear statistics of zeros of random polynomials which are linear combinations of orthogonal polynomials with iid standard complex Gaussian coefficients. Along the way, we obtain Bergman kernel asymptotics for weighted $L^2$-space of polynomials endowed with varying measures of the form $e^{-2n\varphi_n(z)}dz$ under suitable assumptions on the weight functions $\varphi_n$.Comment: Minor revisions, references added. To appear in Proc. of Amer. Math. So

Equidistribution of zeros of random holomorphic sections

We study asymptotic distribution of zeros of random holomorphic sections of high powers of positive line bundles defined over projective homogenous manifolds. We work with a wide class of distributions that includes real and complex Gaussians. As a special case, we obtain asymptotic zero distribution of multivariate complex polynomials given by linear combinations of orthogonal polynomials with i.i.d. random coefficients. Namely, we prove that normalized zero measures of m i.i.d random polynomials, orthonormalized on a regular compact set $K\subset \Bbb{C}^m,$ are almost surely asymptotic to the equilibrium measure of $K$.Comment: Final version incorporates referee comments. To appear in Indiana Univ. Math.

An Off-Axis Galaxy Cluster Merger: Abell 0141

We present structural analysis results of Abell 0141 ($z$ = 0.23) based on X-ray data. The X-ray luminosity map demonstrates that Abell 0141 (A0141) is a bimodal galaxy cluster, which is separated on the sky by $\sim$ 0.65 Mpc with an elongation along the north-south direction. The optical galaxy density map also demonstrates this bimodality. We estimate sub-cluster ICM temperatures of 5.17$^{+0.20}_{-0.19}$ keV for A0141N and 5.23$^{+0.24}_{-0.23}$ keV for A0141S. We obtain X-ray morphological parameters w = 0.034$\pm{0.004}$, c = 0.113$\pm{0.004}$ and w = 0.039$\pm{0.004}$, c = 0.104$\pm{0.005}$ for A0141N and A0141S, respectively. The resulting X-ray morphological parameters indicate that both sub-clusters are moderately disturbed non-cool core structures. We find a slight brightness jump in the bridge region, and yet, there is still an absence of strong X-ray emitting gas between sub-clusters. We discover a significantly hotspot ($\sim$ 10 keV) between sub-clusters, and a Mach number $\textit{M}$ = 1.69$^{+0.40}_{-0.37}$ is obtained by using the temperature jump condition. However, we did not find direct evidence for shock-heating between sub-clusters. We estimate the sub-clusters' central entropies as $K_{0}$ > 100 keV cm$^{2}$, which indicates that the sub-clusters are not cool cores. We find some evidence that the system undergoes an off-axis collision; however, the cores of each sub-clusters have not yet been destroyed. Due to the orientation of X-ray tails of sub-clusters, we suggest that the northern sub-cluster moves through the south-west direction, and the southern cluster moves through the north-east direction. In conclusion, we are witnessing an earlier phase of close core passage between sub-clusters.Comment: 9 pages, 4 figure