  We prove a new lower bound on the critical density $\rho_c$ of the hard disk
model, i.e., the density below which it is possible to efficiently sample
random configurations of $n$ non-overlapping disks in a unit torus. We use a
classic Markov chain which moves one disk at a time, but with an improved path
coupling analysis. Our main tool is an optimized metric on neighboring pairs of
configurations, i.e., configurations that differ in the position of a single
disk: we define a metric that depends on the difference in these positions, and
which approaches zero continuously as they coincide. This improves the previous
lower bound $\rho_c \ge 1/8$ to $\rho_c \ge 0.154$.
