In this review we present the current state-of-the-art on complex Langevin simulations and their implications for the QCD phase diagram. After a short summary of the complex Langevin method, we present and discuss recent developments. Here we focus on the explicit computation of boundary terms, which provide an observable that can be used to check one of the criteria of correctness explicitly. We also present the method of Dynamic Stabilization and elaborate on recent results for fully dynamical QCD.
One of the main challenges in simulations on Lefschetz thimbles is
the computation of the relative weights of contributing thimbles. In
this paper we propose a solution to that problem by means of computing
those weights using a reweighting procedure. Besides we present recipes
for finding parametrizations of thimbles and anti-thimbles for a given
theory. Moreover, we study some approaches to combine the Lefschetz
thimble method with the Complex Langevin evolution. Our numerical
investigations are carried out by using toy models among which we
consider a one-site z^4z4
model as well as a U(1)U(1)
one-link model.
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