Scaled Correlations of Critical Points of Random Sections on Riemann Surfaces

TL;DR

This paper investigates the asymptotic behavior of correlations between critical points of random holomorphic sections on Riemann surfaces, revealing a universal scaling limit as the degree grows large.

Contribution

It provides the first rigorous derivation of the scaling limit of critical point correlations for random sections on Riemann surfaces, using advanced probabilistic and geometric techniques.

Findings

The correlation tends to 2/(3pi^2) for small scaled distances.

The scaling limit is explicitly computed using a generalized Kac-Rice formula.

Results apply to a broad class of random holomorphic sections.

Abstract

In this paper we prove that as N goes to infinity, the scaling limit of the correlation between critical points z1 and z2 of random holomorphic sections of the N-th power of a positive line bundle over a compact Riemann surface tends to 2/(3pi^2) for small sqrt(N)|z1-z2|. The scaling limit is directly calculated using a general form of the Kac-Rice formula and formulas and theorems of Pavel Bleher, Bernard Shiffman, and Steve Zelditch.