Effective Potential for Complex Langevin Equations

TL;DR

This paper develops an effective potential framework for the complex Langevin equation on a lattice, linking its minima to the average field configurations and predicting convergence behavior through loop and derivative expansions.

Contribution

It introduces an effective potential approach for analyzing the complex Langevin equation, connecting it with Schwinger-Dyson equations to predict stationary distributions.

Findings

Effective potential minima correspond to average field configurations.

Loop expansion matches derivative expansion of Schwinger-Dyson equations.

Predicts the stationary distribution of the complex Langevin process.

Abstract

We construct an effective potential for the complex Langevin equation on a lattice. We show that the minimum of this effective potential gives the space-time and Langevin time average of the complex Langevin field. The loop expansion of the effective potential is matched with the derivative expansion of the associated Schwinger-Dyson equation to predict the stationary distribution to which the complex Langevin equation converges.