Localised distributions and criteria for correctness in complex Langevin dynamics

TL;DR

This paper investigates the properties of distributions in complex Langevin dynamics, demonstrating conditions under which the method yields correct results by analyzing the Fokker-Planck equation and distribution support.

Contribution

It provides an analytical study linking the distribution support in complexified space to the correctness of complex Langevin results, enhancing understanding of its validity.

Findings

Distribution can be supported in a strip in complex space

Correct results are expected when support is limited to a strip

Analytical connection between distribution support and correctness

Abstract

Complex Langevin dynamics can solve the sign problem appearing in numerical simulations of theories with a complex action. In order to justify the procedure, it is important to understand the properties of the real and positive distribution, which is effectively sampled during the stochastic process. In the context of a simple model, we study this distribution by solving the Fokker-Planck equation as well as by brute force and relate the results to the recently derived criteria for correctness. We demonstrate analytically that it is possible that the distribution has support in a strip in the complexified configuration space only, in which case correct results are expected.