  We propose locally stable sparse hard-disk packings, as introduced by
B\"or\"oczky, as a model for the analysis and benchmarking of Markov-chain
Monte Carlo (MCMC) algorithms. We first generate such packings in a square box
with periodic boundary conditions and analyze their properties. We then study
how local MCMC algorithms, namely the Metropolis algorithm and several versions
of event-chain Monte Carlo (ECMC), escape from configurations that are obtained
by slightly reducing all disk radii by a relaxation parameter. A scaling
analysis is confirmed by simulation results. We obtain two classes of ECMC, one
in which the escape time varies algebraically with the relaxation parameter (as
for the local Metropolis algorithm) and another in which the escape time scales
as the logarithm of the relaxation parameter. We discuss the connectivity of
the hard-disk sample space, the ergodicity of local MCMC algorithms, as well as
the meaning of packings in the context of the NPT ensemble. Our work is
accompanied by open-source, arbitrary-precision software for B\"or\"oczky
packings (in Python) and for straight, reflective, forward, and Newtonian ECMC
(in Go).
