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Sign problems, noise, and chiral symmetry breaking in a QCD-like theory

The Nambu-Jona-Lasinio model reduced to 2+1 dimensions has two different path integral formulations: at finite chemical potential one formulation has a severe sign problem similar to that found in QCD, while the other does not. At large N, where N is the number of flavors, one can compute the probability distributions of fermion correlators analytically in both formulations. In the former case one finds a broad distribution with small mean; in the latter one finds a heavy tailed positive distribution amenable to the cumulant expansion techniques developed in earlier work. We speculate on the implications of this model for QCD.