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.

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

Thermodynamics of regular black holes with cosmic strings

In this article, the thermodynamics of regular black holes with a cosmic string passing through it is studied. We will observe that the string has no effect on the temperature as well as the relation between entropy S and horizon area A.Comment: Accepted for publication in EPJ Plus, 6 page

Regularity of the Optimal Stopping Problem for Jump Diffusions

The value function of an optimal stopping problem for jump diffusions is known to be a generalized solution of a variational inequality. Assuming that the diffusion component of the process is nondegenerate and a mild assumption on the singularity of the L\'{e}vy measure, this paper shows that the value function of this optimal stopping problem on an unbounded domain with finite/infinite variation jumps is in $W^{2,1}_{p, loc}$ with $p\in(1, \infty)$. As a consequence, the smooth-fit property holds.Comment: To Appear in the SIAM Journal on Control and Optimizatio