An optimal Poincar\'e inequality for the complex Ginibre log-gas

Abstract

We establish an optimal Poincar\'e inequality for real-valued symmetric observables of the complex Ginibre log-gas. Equality is attained by the real and imaginary parts of the center-of-mass observable. Equivalently, we determine the exact spectral gap of the associated overdamped Langevin dynamics, for real symmetric observables. The Hessian of the energy of this log-gas is unbounded below, so standard convexity arguments do not directly apply. Our proof instead combines a Vandermonde transform, a holomorphic projection, and a complex Gaussian d-bar spectral-gap estimate, corresponding to the constant-curvature case of the H\"ormander-Berndtsson estimate.