Spatial Structure of Chaotic Wave Functions

For the first time the statistics of the spatial correlations of eigenfunctions in a random system is investigated. It is rigorously shown that the wave functions corresponding to different energy levels are uncorrelated in space. Within a given eigenstate the joint distribution of the electron density at two different space points is described by a one-parameter universal law, where the parameter is the Friedel function. It is found that this distribution is decoupled in the region of small fluctuations of the electron density and reveals an increasing spatial correlation for the large fluctuations.