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Complex Langevin: Correctness criteria, boundary terms and spectrum
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abstract
The Complex Langevin (CL) method to simulate `complex probabilities', ideally produces expectation values for the observables that converge to a limit equal to the expectation values obtained with the original complex `probability' measure. The situation may be spoiled in two ways: failure to converge and convergence to the wrong limit. It was found long ago that `wrong convergence' is caused by boundary terms; non-convergence may arise from bad spectral properties of the various evolution operators related to the CL process. Here we propose a class of criteria which allow to rule out boundary terms and at the same time bad spectrum. Ruling out boundary terms in the equilibrium distribution arising from a CL simulation implies that the so-called convergence conditions are fulfilled. This in turn has been shown to guarantee that the expectation values of holomorphic observables are given by complex linear combinations of $\exp(-S)$ over various integration cycles. If the spectrum is pathological, however, the CL simulation in general does not reproduce the integral over the desired real cycle.
Forward citations
Cited by 5 Pith papers
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A single compact Lefschetz thimble restores correct complex Langevin convergence, and a Dyson-Schwinger bias correction recovers the original expectation values.
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Complex Langevin results in one- and two-dimensional toy models match a linear combination of integration cycles when boundary terms vanish, and the kernel choice controls which cycles contribute.
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Diffusion models learn distributions generated by complex Langevin dynamics
Diffusion models reproduce the distributions sampled by complex Langevin dynamics in a Gaussian and a quartic toy model with complex mass.
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Energy-based diffusion models trained on complex Langevin data produce an explicit energy function for the sampled distribution, enabling MCMC without re-simulation.
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Thermodynamic Diagnostics for Complex Langevin Simulations: The Role of Configurational Temperature
Configurational temperature from action gradients and Hessians offers a sensitive new correctness diagnostic for complex Langevin simulations, reproducing input temperature to 0.2-3% in 1D PT-symmetric models.
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