  We study classical dimers on two-dimensional quasiperiodic Ammann-Beenker
(AB) tilings. Despite the lack of periodicity we prove that each infinite
tiling admits 'perfect matchings' in which every vertex is touched by one
dimer. We introduce an auxiliary 'AB$^*$' tiling obtained from the AB tiling by
deleting all 8-fold coordinated vertices. The AB$^*$ tiling is again
two-dimensional, infinite, and quasiperiodic. The AB$^*$ tiling has a single
connected component, which admits perfect matchings. We find that in all
perfect matchings, dimers on the AB$^*$ tiling lie along disjoint
one-dimensional loops and ladders, separated by 'membranes', sets of edges
where dimers are absent. As a result, the dimer partition function of the
AB$^*$ tiling factorizes into the product of dimer partition functions along
these structures. We compute the partition function and free energy per edge on
the AB$^*$ tiling using an analytic transfer matrix approach. Returning to the
AB tiling, we find that membranes in the AB$^*$ tiling become
'pseudomembranes', sets of edges which collectively host at most one dimer.
This leads to a remarkable discrete scale-invariance in the matching problem.
The structure suggests that the AB tiling should exhibit highly inhomogenous
and slowly decaying connected dimer correlations. Using Monte Carlo
simulations, we find evidence supporting this supposition in the form of
connected dimer correlations consistent with power law behaviour. Within the
set of perfect matchings we find quasiperiodic analogues to the staggered and
columnar phases observed in periodic systems.
