  The Kondo and Periodic Anderson models describe many of the qualitative
features of local moments coupled to a conduction band, and thereby the physics
of materials such as the heavy fermions. In particular, when the exchange
coupling $J$ or hybridization $V$ between the moments and the electrons of the
metallic band is large, singlets form, quenching the magnetism. In the
opposite, small $J$ or $V$, limit, the moments survive, and the conduction
electrons mediate an effective interaction which can trigger long range, often
antiferromagnetic, order. In the case of the Kondo model, where the moments are
described by local spins, Nozi\`eres considered the possibility that the
available conduction electrons within the Kondo temperature of the Fermi
surface would be insufficient in number to accomplish the screening. Much
effort in the literature has been devoted to the study of the temperature
scales in the resulting `exhaustion' problem, and how the `coherence
temperature' where a heavy Fermi liquid forms is related to the Kondo
temperature. In this paper, we study a version of the Periodic Anderson model
in which some of the conduction electrons are removed in a way which avoids the
fermion sign problem and hence allows low temperature Quantum Monte Carlo
simulations which can access both singlet formation and magnetic ordering
temperature scales. We are then able to focus on a somewhat different aspect of
exhaustion physics than previously considered: the effect of dilution on the
critical $V$ for the singlet-antiferromagnetic transition.
