  The nature of ordering in dilute dipolar interacting systems dates back to
the work of Debye and is one of the most basic, oldest and as-of-yet unsettled
problems in magnetism. While spin-glass order is readily observed in several
RKKY-interacting systems, dipolar spin-glasses are subject of controversy and
ongoing scrutiny, e.g., in ${{\rm LiHo_xY_{1-x}F_4}}$, a rare-earth randomly
diluted uniaxial (Ising) dipolar system. In particular, it is unclear if the
spin-glass phase in these paradigmatic materials persists in the limit of zero
concentration or not. We study an effective model of ${{\rm LiHo_xY_{1-x}F_4}}$
using large-scale Monte Carlo simulations that combine parallel tempering with
a special cluster algorithm tailored to overcome the numerical difficulties
that occur at extreme dilutions. We find a paramagnetic to spin-glass phase
transition for all Ho ion concentrations down to the smallest concentration
numerically accessible of 0.1%, and including Ho ion concentrations which
coincide with those studied experimentally up to 16.7%. Our results suggest
that randomly-diluted dipolar Ising systems have a spin-glass phase in the
limit of vanishing dipole concentration, with a critical temperature vanishing
linearly with concentration, in agreement with mean field theory.
