Monte Carlo methods on compact symplectic manifolds

We build an unbiased Monte Carlo estimator of the integral of any $C^1$ function on a prequantized compact symplectic manifold against a smooth Riemannian volume form, taking for quadrature nodes the determinantal point process associated with an appropriate spectral projection of the Bochner-Schrödinger operator. We show that the estimator satisfies a central limit theorem, and the decay rate of the mean squared error reaches the optimal worst-case rate investigated by Bakhvalov in Euclidean spaces. These results extend previous results of Lemoine and Bardenet on Monte Carlo methods on compact complex manifolds.