  We study close-packed dimers on the quasiperiodic Ammann-Beenker (AB) graph,
that was recently shown to have the unusual feature that hard-core dimer
constraints are exactly reproduced at successive discrete length scales. This
observation led to a conjecture that it would be possible to construct an exact
real-space decimation scheme where each iteration preserves both the
quasiperiodic tiling structure and the constraint. Here, we confirm this
conjecture by explicitly constructing the corresponding renormalization group
transformation and show, using large-scale Monte Carlo simulations, that the
dimer distributions flow to a fixed point with non-zero dimer potentials. We
use the fixed-point Hamiltonian to demonstrate the existence of slowly decaying
dimer correlations. We thus identify a remarkable example of a classical
statistical mechanical model whose properties are controlled by the fixed point
of an exact renormalization group procedure exhibiting discrete scale
invariance but lacking translational and continuous rotational symmetries.

Statistical Mechanics;Disordered Systems and Neural Networks;Strongly Correlated Electrons