  We report the first large-scale statistical study of very high-lying
eigenmodes (quantum states) of the mushroom billiard proposed by L. Bunimovich
in this journal, vol. 11, 802 (2001). The phase space of this mixed system is
unusual in that it has a single regular region and a single chaotic region, and
no KAM hierarchy. We verify Percival's conjecture to high accuracy (1.7%). We
propose a model for dynamical tunneling and show that it predicts well the
chaotic components of predominantly-regular modes. Our model explains our
observed density of such superpositions dying as E^{-1/3} (E is the
eigenvalue). We compare eigenvalue spacing distributions against Random Matrix
Theory expectations, using 16000 odd modes (an order of magnitude more than any
existing study). We outline new variants of mesh-free boundary collocation
methods which enable us to achieve high accuracy and such high mode numbers
orders of magnitude faster than with competing methods.
