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Investigating corrections to a Gaussian distribution of the complex phase

It has been suggested that the density of states approach to performing lattice simulations in QCD with nonzero chemical potential can be modified to improve the signal to noise ratio by performing a cumulant expansion of the complex phase of the fermion determinant, and then simplified by truncating the expansion after the first non-zero cumulant. This truncation corresponds to approximating the distribution of the complex phase of the fermion determinant by a Gaussian form. The crucial question is: how large are the other cumulants? We calculate the distribution of the complex phase from the hadron resonance gas model and from a combined lattice strong coupling and hopping expansion. In the case of the hadron resonance gas model the distribution takes a Gaussian form, but from the strong coupling and hopping expansion there are corrections. We discuss the implications to lattice simulations.