Nodal Domain Statistics for Quantum Maps, Percolation and SLE

Abstract

We develop a percolation model for nodal domains in the eigenvectors of quantum chaotic torus maps. Our model follows directly from the assumption that the quantum maps are described by random matrix theory. Its accuracy in predicting statistical properties of the nodal domains is demonstrated by numerical computations for perturbed cat maps and supports the use of percolation theory to describe the wave functions of general hamiltonian systems, where the validity of the underlying assumptions is much less clear. We also demonstrate that the nodal domains of the perturbed cat maps obey the Cardy crossing formula and find evidence that the boundaries of the nodal domains are described by SLE with $κ$ close to the expected value of 6, suggesting that quantum chaotic wave functions may exhibit conformal invariance in the semiclassical limit.