  We study single hole dynamics in the bilayer Heisenberg and Kondo Necklace
models. Those models exhibit a magnetic order-disorder quantum phase transition
as a function of the interlayer coupling J_perp. At strong coupling in the
disordered phase, both models have a single-hole dispersion relation with band
maximum at p = (\pi,\pi) and an effective mass at this p-point which scales as
the hopping matrix element t. In the Kondo Necklace model, we show that the
effective mass at p = (\pi,\pi) remains finite for all considered values of
J_perp such that the strong coupling features of the dispersion relation are
apparent down to weak coupling. In contrast, in the bilayer Heisenberg model,
the effective mass diverges at a finite value of J_perp. This divergence of the
effective mass is unrelated to the magnetic quantum phase transition and at
weak coupling the dispersion relation maps onto that of a single hole doped in
a planar antiferromagnet with band maximum at p = (\pi/2,\pi/2). We equally
study the behavior of the quasiparticle residue in the vicinity of the magnetic
quantum phase transition both for a mobile and static hole. In contrast to
analytical approaches, our numerical results do not unambiguously support the
fact that the quasiparticle residue of the static hole vanishes in the vicinity
of the critical point. The above results are obtained with a generalized
version of the loop algorithm to include single hole dynamics on lattice sizes
up to 20 X 20.
