The Mechanism of Complex Langevin Simulations

TL;DR

This paper analyzes the conditions under which complex Langevin simulations yield correct expectation values, highlighting challenges in proving general convergence and illustrating results with a solvable example.

Contribution

It establishes specific conditions for convergence of complex Langevin processes for polynomial actions and discusses when these processes accurately reproduce integral averages.

Findings

Convergence depends on certain conditions of the process.

Failure to meet conditions may lead to incorrect results.

Illustrated with an exactly solvable harmonic oscillator.

Abstract

We discuss conditions under which expectation values computed from a complex Langevin process $Z$ will converge to integral averages over a given complex valued weight function. The difficulties in proving a general result are pointed out. For complex valued polynomial actions, it is shown that for a process converging to a strongly stationary process one gets the correct answer for averages of polynomials if $c_{\tau}(k) \equiv E(e^{ikZ(\tau)}) $ satisfies certain conditions. If these conditions are not satisfied, then the stochastic process is not necessarily described by a complex Fokker Planck equation. The result is illustrated with the exactly solvable complex frequency harmonic oscillator.