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Abstract

It is well known that the Complex Langevin (CL) method sometimes fails to converge or converges to the wrong limit. We identified one reason for this long ago: insufficient decay of the probability density either near infinity or near poles of the drift, leading to boundary terms that spoil the formal argument for correctness. To gain a deeper understanding of this phenomenon, we analyze the emergence of such boundary terms thoroughly in a simple model, allowing some analytic results for comparing with numerics. We also show how some simple modification stabilizes the CL process in such a way that it produces results agreeing with direct integration.