The convergence of complex Langevin simulations

Abstract

It is proven that ensemble average computed from a complex Langevin (CL) simulation will necessarily converge to the correct values if the ensemble averages become time independent. This is illustrated with two model problems defined on the compact spaces U(1) and S 2, as well as with a lattice fermion model. For all three problems, the CL method is found to be, with few exceptions, applicable. For the U(1) problem, this is demonstrated via a semi-analytic solution for the expectation values. The difficulties of obtaining accurate numerical solutions of the stochastic differential equations are discussed.