Convergence of the Langevin simulation for complex Gaussian integrals

Haymaker, Richard W.; Peng, Yingcai

doi:10.1103/physrevd.41.1269

Cited by 9 publications

(11 citation statements)

References 9 publications

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“…We begin by searching for solutions of Pϕ 0 = E 0 ϕ 0 in the form where we set A R = cosh 2 α/4, A I = sinh 2 α/4. For α = 0 this reduces to the result in [7,6]. The solution becomes non-normalizable for θ → π − as ϕ 0 (x, y) → 1.…”

Section: A: the Langevin Methods For Quadratic Complex Actionsmentioning

confidence: 71%

Temporal breakdown and Borel resummation in the complex Langevin method

Duncan

Niedermaier

2013

Annals of Physics

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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.Comment: title changed, results on parameter dependence of t_c added, exposition improved. 39 pages, 7 figure

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Section: A: the Langevin Methods For Quadratic Complex Actionsmentioning

confidence: 71%

Temporal breakdown and Borel resummation in the complex Langevin method

Duncan

Niedermaier

2013

Annals of Physics

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“…[25][26][27][28][29][30]. This method was successful in several cases including statistical mechanics problems with complex chemical potentials,…”

Section: The Langevin Methodsmentioning

confidence: 99%

Numerical simulations of PT-symmetric quantum field theories

2001

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Many non-Hermitian but PT -symmetric theories are known to have a real positive spectrum. Since the action is complex for there theories, Monte Carlo methods do not apply. In this paper the first field-theoretic method for numerical simulations of PT -symmetric Hamiltonians is presented. The method is the complex Langevin equation, which has been used previously to study complex Hamiltonians in statistical physics and in Minkowski space. We compute the equal-time one-point and two-point Green's functions in zero and one dimension, where comparisons to known results can be made. The method should also be applicable in four-dimensional space-time. Our approach may also give insight into how to formulate a probabilistic interpretation of PTsymmetric theories.

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“…Refs. [15,16,17,18]) and hence the analysis of P(x, y) requires its explicit construction using the Langevin equation.…”

Section: Complex Langevin Dynamicsmentioning

confidence: 99%

QCD at nonzero chemical potential: Recent progress on the lattice

Aarts

Attanasio

Jäger

et al. 2016

AIP Conference Proceedings

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Abstract. We summarise recent progress in simulating QCD at nonzero baryon density using complex Langevin dynamics. After a brief outline of the main idea, we discuss gauge cooling as a means to control the evolution. Subsequently we present a status report for heavy dense QCD and its phase structure, full QCD with staggered quarks, and full QCD with Wilson quarks, both directly and using the hopping parameter expansion to all orders.

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Convergence of the Langevin simulation for complex Gaussian integrals

Abstract: We solve the two-variable Fokker-Planck equation for the real probability distribution generated by the complex Langevin equation for a complex Gaussian integral. We find the eigenvalues, eigenvectors, and time-dependent behavior.

Cited by 9 publications

References 9 publications

Temporal breakdown and Borel resummation in the complex Langevin method

Temporal breakdown and Borel resummation in the complex Langevin method

Numerical simulations of PT-symmetric quantum field theories

QCD at nonzero chemical potential: Recent progress on the lattice

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