From Random Polynomials to Symplectic Geometry

Abstract

We review some recent results on random polynomials and their generalizations in complex and symplectic geometry. The main theme is the universality of statistics of zeros and critical points of (generalized) polynomials of degree $N$ on length scales of order $\frac{D}{\sqrt{N}}$ (complex case), resp. $\frac{D}{N}$ (real case).

Expository article for Proceedings ICMP 2000

References in corpus (6)

• Distribution of zeros of random and quantum chaotic sections of positive line bundles
• Universality and scaling of correlations between zeros on complex manifolds
• Quantum Chaotic Dynamics and Random Polynomials
• Poincare-Lelong approach to universality and scaling of correlations between zeros
• Weak Quantum Ergodicity
• Random almost holomorphic sections of ample line bundles on symplectic manifolds

Cited by in corpus (5)

• Local number variances and hyperuniformity of the Heisenberg family of determinantal point processes
• Asymptotics of polynomials and eigenfunctions
• Partial Isometries, Duality, and Determinantal Point Processes
• Topics in the Value Distribution of Random Analytic Functions
• Determinantal Point Processes, Stochastic Log-Gases, and Beyond