  We show that the interplay of geometric criticality and quantum fluctuations
leads to a novel universality class for the percolation quantum phase
transition in diluted magnets. All critical exponents involving dynamical
correlations are different from the classical percolation values, but in two
dimensions they can nonetheless be determined exactly. We develop a complete
scaling theory of this transition, and we relate it to recent experiments in
La$_{2}$Cu$_{1-p}$(Zn,Mg)$_{p}$O$_{4}$. Our results are also relevant for
disordered interacting boson systems.
