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We apply the Lefschetz thimble formulation of field theories to a couple ofdifferent problems. We first address the solution of a complex 0-dimensionalphi^4 theory. Although very simple, this toy-model makes us appreciate a fewkey issues of the method. In particular, we will solve the model by a correctaccounting of all the thimbles giving a contribution to the partition functionand we will discuss a number of algorithmic solutions to simulate this (simple)model. We will then move to a chiral random matrix (CRM) theory. This is asomehow more realistic setting, giving us once again the chance to tackle thesame couple of fundamental questions: how many thimbles contribute to thesolution? how can we make sure that we correctly sample configurations on thethimble? Since the exact result is known for the observable we study (acondensate), we can verify that, in the region of parameters we studied, onlyone thimble contributes and that the algorithmic solution that we set up workswell, despite its very crude nature. The deviation of results from phasequenched results highlights that in a certain region of parameter space thereis a quite important sign problem. In view of this, the success of our thimbleapproach is quite a significant one.

Thimble regularization at work: from toy models to chiral random matrix theories