Scaled Correlations of Critical Points of Random Sections on Riemann Surfaces

In this paper we prove that as N goes to infinity, the scaling limit of the correlation between critical points z ₁ and z ₂ of random holomorphic sections of the N-th power of a positive line bundle over a compact Riemann surface tends to 2/(3π ²) for small . 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.