Poincaré–Lelong Approach to Universality and Scaling of Correlations Between Zeros
Pavel Bleher, Bernard Shiffman, and Steve Zelditch
This note is concerned with the scaling limit as $N \to \infty$ of $n$-point correlations between zeros of random holomorphic polynomials of degree $N$ in $m$ variables. More generally we study correlations between zeros of holomorphic sections of powers $L^N$ of any positive holomorphic line bundle $L$ over a compact Kähler manifold. Distances are rescaled so that the average density of zeros is independent of $N$. Our main result is that the scaling limits of the correlation functions and, more generally, of the "correlation forms" are universal, i.e., independent of the bundle $L$, manifold $M$ or point on $M$.