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Fluctuations, correlations and the sign problem in QCD

We study the distribution of the phase angle and the magnitude of the fermion determinant as well as its correlation with the chiral condensate and the baryon number for QCD at non-zero quark chemical potential. Results are derived to one-loop order in Chiral Perturbation Theory (ChPT), as well as by analytical and numerical calculations in QCD in one Euclidean dimension. We find a qualitative change of the distribution of the phase of the fermion determinant when the quark mass enters the spectrum of the Dirac operator: it changes from a periodicized Gaussian distribution to a periodicized Lorentzian distribution. We also explore the possibility that some observables remain weakly correlated with the phase of the fermion determinant even though the sign problem is severe. We discuss the practical implications of our findings on lattice simulations of QCD at non-zero baryon chemical potential.