Randomly diluted quantum boson and spin models in two dimensions combine the physics of classical percolation with the well-known dimensionality dependence of ordering in quantum lattice models. This combination is rather subtle for models that order in two dimensions but have no true order in one dimension, as the percolation cluster near threshold is a fractal of dimension between 1 and 2: two experimentally relevant examples are the
O(2) quantum rotor and the Heisenberg antiferromagnet. We study two analytic descriptions of the
O(2) quantum rotor near the percolation threshold. First a spin-wave expansion is shown to predict long-ranged order, but there are statistically rare points on the cluster that violate the standard assumptions of spin-wave theory. A real-space renormalization group (RSRG) approach is then used to understand how these rare points modify ordering of the
O(2) rotor. A new class of fixed points of the RSRG equations for disordered one-dimensional bosons is identified and shown to support the existence of long-range order on the percolation backbone in two dimensions. These results are relevant to experiments on bosons in optical lattices and superconducting arrays, and also (qualitatively) for the diluted Heisenberg antiferromagnet ${\mathrm{La}}_{2}{(\mathrm{Zn},\mathrm{Mg})}_{x}{\mathrm{Cu}}_{1\ensuremath{-}x}{\mathrm{O}}_{4}$.
The percolation transition in arbitrary dimension is studied in real space by generalizing the block-cluster theory. For both site and bond percolation on a decorated hypercubic lattice, this theory yields analytic renormalization-group equations in any number of dimensions. For bond percolation, these analytic equations are most remarkably simple. Results are presented for the thermal exponent and compare very favorably with results obtained by other methods. This is the first successful renormalization-group theory in real space in which the dimension of space enters as a variable parameter.