The Kadomtsev–Petviashvili equation, an extension of the Korteweg–
de Vries equation in two horizontal dimensions, is here used to study the statistical
properties of random shallow water waves in constant depth for crossing sea
states. Numerical simulations indicate that the interaction of two crossing wave
trains generates steep and high amplitude peaks, thus enhancing the deviation of
the surface elevation from the Gaussian statistics. The analysis of the skewness and
the kurtosis shows that the statistical properties depend on the angle between the
two wave trains.