Georgia Institute of Technology College of Sciences

The Walsh-quantized baker's maps are models for quantum chaos on the torus. We show that for all baker's map scaling factors D\ge2 except for D=4, typically (in the sense of Haar measure on the eigenspaces, which are degenerate) the distribution of the matrix element fluctuations for a randomly chosen eigenbasis looks Gaussian in the semiclassical limit N\to\infty, with variance given in terms of classical baker's map correlations. This determines the precise rate of convergence in the quantum ergodic theorem for these eigenbases. The presence of the classical correlations highlights that these eigenstates, while random, have microscopic correlations that differentiate them from Haar random vectors. For the single value D=4, the Gaussianity of the matrix element fluctuations depends on the values of the classical observable on a fractal subset of the torus.