Necessary and sufficient conditions for correctness of complex Langevin

TL;DR

This paper establishes necessary and sufficient mathematical conditions to verify the correctness of complex Langevin simulations, aiding in detecting incorrect convergence without requiring exact solutions.

Contribution

It derives a comprehensive set of criteria based on Schwinger-Dyson equations and bounds, applicable to complex Langevin and general complex densities, for assessing simulation correctness.

Findings

Criteria can detect incorrect convergence in toy models

Conditions are necessary and sufficient mathematically

Useful for validation without exact solutions

Abstract

We derive a family of correctness conditions for complex Langevin simulations. In particular, we show that if in a given theory the expectation values of all observables within a particular space satisfy the theory's Schwinger-Dyson equations as well as certain bounds, then these expectation values are necessarily correct. In fact, these findings are not only valid in the context of complex Langevin simulations, but they also hold for general probability densities on complex manifolds, given an initial complex density on a real manifold. We stress that, while the proposed conditions are necessary and sufficient in a mathematical sense, their practical use is not to prove the correctness of obtained simulation results. Rather, they are mainly useful for detecting incorrect convergence. In particular, we test these criteria in a few simple one- and two-dimensional toy models and find that they are indeed capable of ruling out incorrect results without the need of exact solutions.