In the chaotic quantum map, the overlap distributions between the eigenstates and a probe state are well described by the random matrix theory (RMT) in most cases. However, if the probe state is localized on a fixed point, the overlap distribution shows a strong deviation from the RMT expectation, which is a result of the scarring phenomena in chaotic eigenstates . In our recent paper , a joint-probability distribution has been developed for the overlaps with the eigenstates of harmonic oscillator which act as the probe states. This is a kind of scarred RMT, and has explained successfully the scarring effect in tunneling rate distributions. This joint-probability distribution contains all information of wavefunction statistics near the fixed point, and has an interesting structure which is directly related to the corresponding classical dynamics. We find that one linear combination of 's is an optimized probe state. Following Vergini et al  we call this the scar state, and the statistics of its overlap with chaotic states shows maximum deviation from RMT. This scar state shows that the scarring appears along the stable and unstable manifolds. We show that this joint distribution works very well not only in the perturbed cat map, but also in the boundary function of billiard. This means that the chaotic wavefunction statistics are characterized by the Lyapunov exponent and the angle between the stable and unstable manifolds of the fixed point. In the billiard case, we investigate the relation between the statistics of the boundary functions and those of the real eigenfunctions. This study gives the way to see which eigenstate should be scarred or not in a statistical way.
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