  We introduce high staircase infinite measure preserving transformations and
prove that they are mixing under a restricted growth condition. This is used to
(i) realize each subset $E\subset\Bbb N\cup\{\infty\}$ as the set of essential
values of the multiplicity function for the Koopman operator of a mixing
ergodic infinite measure preserving transformation, (ii) construct mixing power
weakly mixing infinite measure preserving transformations, (iii) construct
mixing Poissonian automorphisms with a simple spectrum, etc.
