Phase correlations in non-Gaussian fields

We present the general relationship between phase correlations and the hierarchy of polyspectra in the Fourier space, and a new theoretical understanding of the phase information is provided. Phase correlations are related to the polyspectra only through the nonuniform distributions of the phase sum theta(k1)+...+theta(kN) with closed wave vectors k(1)+...+k(N) = 0. The exact relationship is given by the infinite series, which one can truncate in a consistent manner. The method to calculate the series to arbitrary order is explained, and the explicit expression of the first-order approximation is given. A numerical demonstration proves that the distribution of the phase sum is a robust estimator and provides an alternative statistic to search for the non-Gaussianity.