Does the complex Langevin method give unbiased results?

TL;DR

This paper examines whether the complex Langevin method produces unbiased expectation values by analyzing the Fokker-Planck equation and identifying an anomaly that can introduce bias, which appears common in one-dimensional problems.

Contribution

The paper reveals the presence of an anomaly in the Fokker-Planck equation of the complex Langevin method, challenging its reliability for unbiased results in one-dimensional cases.

Findings

Anomalous term in the Fokker-Planck equation can bias expectation values.

Anomaly is common when the Langevin walker reaches infinity in finite time.

The anomaly is likely prevalent in one-dimensional problems, but may diminish with more variables.

Abstract

We investigate whether the stationary solution of the Fokker-Planck equation of the complex Langevin algorithm reproduces the correct expectation values. When the complex Langevin algorithm for an action $S(x)$ is convergent, it produces an equivalent complex probability distribution $P(x)$ which ideally would coincide with $e^{-S(x)}$. We show that the projected Fokker-Planck equation fulfilled by $P(x)$ may contain an anomalous term whose form is made explicit. Such term spoils the relation $P(x)=e^{-S(x)}$, introducing a bias in the expectation values. Through the analysis of several periodic and non-periodic one-dimensional problems, using either exact or numerical solutions of the Fokker-Planck equation on the complex plane, it is shown that the anomaly is present quite generally. In fact, an anomaly is expected whenever the Langevin walker needs only a finite time to go to infinity and come back, and this is the case for typical actions. We conjecture that the anomaly is the rule rather than the exception in the one-dimensional case, however, this could change as the number of variables involved increases.