Nodal domain distributions for quantum maps

The statistics of the nodal lines and nodal domains of the eigenfunctions of quantum billiards have recently been observed to be fingerprints of the chaoticity of the underlying classical motion by Blum et al These statistics were shown to be computable from the random wave model of the eigenfunctions. We here study the analogous problem for chaotic maps whose phase space is the two-torus. We show that the distributions of the numbers of nodal points and nodal domains of the eigenvectors of the corresponding quantum maps can be computed straightforwardly and exactly using random matrix theory. We compare the predictions with the results of numerical computations involving quantum perturbed cat maps. In a recent article Blum et al (2002) observed that the number-distributions of the nodal domains of quantum wavefunctions of billiards whose classical dynamics is integrable are different from those for chaotic billiards and argued that the latter are universal. Thus, the number-distribution of nodal domains appears to be a new criterion for quantum chaos that complements the usual ones based on spectral fluctuations. Blum et al computed these distributions for some integrable (and separable) systems, but no analytic formula exists for the number of nodal domains of a chaotic billiard. Berry (1977) has conjectured that the wavefunctions of quantum systems with a chaotic classical limit behave like Gaussian random functions. Supported by numerical evidence, Blum et al found that the limiting distribution of the number of nodal domains can be reproduced assuming Berry's conjecture. Bogomolny and Schmit (2002) developed a percolation model for nodal domains of Gaussian random functions and showed that their number is Gaussian distributed. They computed the mean and variance of this distribution, which are both proportional to the mean spectral counting function. Their results agree with the numerical computations reported by Blum et al for chaotic