Lower Bounds on the Critical Density in the Hard Disk Model via Optimized Metrics

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 ${\rho }_c\ge 1\mathrm{/}8$ to ${\rho }_c\ge 0.154$.