  We study the effects of dissipation on a randomly diluted transverse-field
Ising magnet close to the percolation threshold. For weak transverse fields, a
novel percolation quantum phase transition separates a superparamagnetic
cluster phase from an inhomogeneously ordered ferromagnetic phase. The
properties of this transition are dominated by large frozen and slowly
fluctuating percolation clusters. This leads to a discontinuous
magnetization-field curve and exotic hysteresis phenomena as well as highly
singular behavior of magnetic susceptibility and specific heat. We compare our
results to the smeared transition in generic dissipative random quantum Ising
magnets. We also discuss the relation to metallic quantum magnets and other
experimental realizations.
