  Ocean rogue waves are large and suddenly appearing surface gravity waves,
which may cause severe damage to ships and other maritime structures. Despite
years of research, the exact origin of rogue waves is still disputed. Linear
interference of waves with random phase has often been cited as one possible
explanation, but apparently does not satisfactorily explain the probability of
extreme events in the ocean. Other explanations therefore suggested a decisive
role of a nonlinearity in the system. Here we show that linear interference of
a finite and variable number of waves may very well explain the heavy tail in
the wave height distribution. Our model can explain all prototypical ocean
rogue waves reported so far, including the "three sisters" as well as rogue
holes. We further suggest nonlinear time series analysis for estimation of the
characteristic number of interfering waves for a given sea state. If ocean
dynamics is ruled by interference of less than ten waves, rogue waves cannot
appear as a matter of principle. In contrast, for larger numbers, their
appearance is much more likely than predicted by parameterless models or
longterm observation. The pronounced threshold behavior of our model enables
effective forecasting of extreme ocean waves.
