Saddles, singularities, and extrema in random phase fields

Four simple topological rules are derived that constrain the arrangement of critical points (saddles, singularities, and extrema) in a random phase field. These rules relate the signs of the singularities to the nature of the extrema (maxima or minima) and the topology of the saddles. Once the latter is fixed, only a single degree of freedom remains and if, for example, some extremum is chosen to be a maximum, this choice automatically determines the nature of all other extrema and the signs of all singularities. Thus, even in a random wave field there are extensive, topologically mandated correlations between all critical points. Higher-order gradient fields derived from the phase are considered and the rules and their induced correlations are shown to apply also to these fields. Other aspects of the phase field are discussed and it is shown, for example, that the number of saddles very nearly (but not necessarily exactly) equals the number of singularities plus the number of extrema and that during the evolution of the wave field, large numbers of different, specific features must appear simultaneously.