arXiv subjects
Publications and source records attributed to Afrim Bojnik.
We prove a central limit theorem for smooth linear statistics related to the zero divisors of Gaussian i.i.d. centered holomorphic sections of tensor powers of a Hermitian holomorphic line bundle over a non-compact Hermitian manifold.
This paper primarily establishes an asymptotic variance estimate for smooth linear statistics associated with zero sets of systems of random holomorphic sections in a sequence of positive Hermitian holomorphic line bundles on a compact Kähler manifold $(X, ω)$ in a general non-Gaussian setting. Using this variance estimate and the expected distribution, we derive an equidistribution result for zeros of these random systems, which proves that the smooth positive closed form $ω^{k}$ can be approximated by currents of integration along analytic subsets of $X$ of codimension $k$, $k \in \{1, \ldots, n\}$. The probability measures taken into consideration in this paper are sufficiently general to include a wide range of the measures commonly encountered in the literature, for which we give equidistribution results at the end, such as the standard Gaussian measure, Fubini-Study measure, the area measure of spheres, probability measures whose distributions have bounded densities with logarithmic decaying tails and locally moderate measures among others.
We study random multivariate $P$-polynomials in $\mathbb{C}^d$ with monomial supports constrained to $nP\cap\mathbb{Z}_+^d$ for a convex body $P\subset\mathbb{R}_+^d$, and deterministic coefficients admitting a uniform exponential profile $f$ on $P$. Assuming the tail condition $\mathbb{P}(\log(1+|ξ_0|)>t)=o(t^{-d})$ on the i.i.d. complex coefficients, we prove that the normalized potentials $\frac1n\log|\mathbf{P}_n|$ converge in probability in $L^1_{\mathrm{loc}}(\mathbb{C}^d)$ to a deterministic toric plurisubharmonic function $Φ_{P,f}$, and consequently the normalized zero currents $\frac1n[Z_{\mathbf{P}_n}]$ converge weakly to the closed positive $(1,1)$-current $dd^cΦ_{P,f}$. Under the stronger logarithmic moment assumption $\mathbb{E}[(\log(1+|ξ_0|))^d]<\infty$, we prove almost sure weak convergence of the zero currents along the full sequence for $d>2$, and along sparse subsequences for $d \le 2$. On $(\mathbb{C}^*)^d$, the limiting potential is given by $Φ_{P,f}(z)=I_{P,f}(\operatorname{Log} z)$, where $I_{P,f}$ is the Legendre-Fenchel transform of the profile over $P$ and $\operatorname{Log} (z)=(\log|z_1|,\dots,\log|z_d|)$. These results extend the exponential-profile mechanism of Kabluchko and Zaporozhets from one complex variable to the genuinely multivariate $P$-polynomial setting under relaxed probabilistic assumptions, directly connecting random zero hypersurfaces with convex-analytic data determined by $(P,f)$.
In this note, we prove a central limit theorem for the mass distribution of random holomorphic sections associated with a sequence of positive line bundles endowed with $\mathscr{C}^3$ Hermitian metrics over a compact Kähler manifold. In addition, we show that almost every sequence of such random holomorphic sections exhibits quantum ergodicity in the sense of Zelditch.
We prove a central limit theorem for smooth linear statistics associated with zero divisors of standard Gaussian holomorphic sections in a sequence of holomorphic line bundles with Hermitian metrics of class $\mathscr{C}^{3}$ over a compact Kähler manifold. In the course of our analysis, we derive first-order asymptotics and upper decay estimates for near and off-diagonal Bergman kernels, respectively. These results are essential for determining the statistical properties of the zeros of random holomorphic sections.