From quantum disorder to quantum chaos

We study the statistics of wave functions in a ballistic chaotic system. The statistical ensemble is generated by adding weak smooth random potential, which allows us to apply the ballistic $\sigma$-model approach. We analyze conditions of applicability of the $\sigma$-model, emphasizing the role played by the single-particle mean free path and the Lyapunov exponent due to the random potential. In particular, we present a resolution of the puzzle of repetitions of periodic orbits counted differently by the $\sigma$-model and by the trace formula.