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Schwinger-Dyson equations and line integrals
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The Complex Langevin (CL) method sometimes shows convergence to the wrong limit, even though the Schwinger-Dyson Equations (SDE) are fulfilled. We analyze this problem in a more general context for the case of one complex variable. We prove a theorem that shows that under rather general conditions not only the equilibrium measure of CL but any linear functional satisfying the SDE on a space of test functions is given by a linear combination of integrals along paths connecting the zeroes of the underlying measure and noncontractible closed paths. This proves rigorously a conjecture stated long ago by one us (L.~L.~S.) and explains a fact observed in nonergodic cases of CL. one us (L.~L.~S.) and explains a fact observed in nonergodic cases of CL.
Forward citations
Cited by 2 Pith papers
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The Role of Integration Cycles in Complex Langevin Simulations
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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Combining complex Langevin dynamics with score-based and energy-based diffusion models
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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