Temporal breakdown and Borel resummation in the complex Langevin method

TL;DR

This paper investigates the complex Langevin method for complex measures, demonstrating that Borel resummation of short-time asymptotics accurately predicts moments at all times, but numerical simulations diverge after a finite breakdown time.

Contribution

It shows that Borel summation of asymptotics captures the full time evolution of moments in the complex Langevin approach, revealing limitations of numerical simulations.

Findings

Borel transform reproduces moments for all times including equilibrium.

Numerical simulations diverge from the correct moments after a finite time t_c.

Breakdown time t_c depends on the noise's imaginary part strength.

Abstract

We reexamine the Parisi-Klauder conjecture for complex e^{i\theta/2} \phi^4 measures with a Wick rotation angle 0 <= \theta/2 < \pi/2 interpolating between Euclidean and Lorentzian signature. Our main result is that the asymptotics for short stochastic times t encapsulates information also about the equilibrium aspects. The moments evaluated with the complex measure and with the real measure defined by the stochastic Langevin equation have the same t -> 0 asymptotic expansion which is shown to be Borel summable. The Borel transform correctly reproduces the time dependent moments of the complex measure for all t, including their t -> infinity equilibrium values. On the other hand the results of a direct numerical simulation of the Langevin moments are found to disagree from the `correct' result for t larger than a finite t_c. The breakdown time t_c increases powerlike for decreasing strength of the noise's imaginary part but cannot be excluded to be finite for purely real noise. To ascertain the discrepancy we also compute the real equilibrium distribution for complex noise explicitly and verify that its moments differ from those obtained with the complex measure.