Nodal domain statistics for quantum chaotic maps

We study the statistical distribution of nodal domains in the eigenvectors of quantum chaotic maps in the semiclassical limit. For generic quantum maps, which are believed to behave statistically like random matrices, the nodal domains are described by an uncorrelated critical percolation model. This leads to predictions for the distribution of the number, size and fractal dimension of the nodal domains, which are tested numerically. Furthermore, we find that the corresponding nodal lines can be modelled by stochastic Loewner evolution (SLE) with parameter K close to 6. Interestingly, the percolation model and SLE are also found to describe the statistical properties of the nodal domains for the quantum cat map, which is non-generic in that its spectral statistics do not fall into any of the random matrix universality classes.