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

Thomas P. Hayes; Cristopher Moore

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arXiv · 1407.1930

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

Thomas P. Hayes·Cristopher Moore

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Abstract

We prove a new lower bound on the critical density $ρ_c$ 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 $ρ_c \ge 1/8$ to $ρ_c \ge 0.154$.

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BibTeX RIS

Thomas P. Hayes, Cristopher Moore. 2014-07-08. Lower Bounds on the Critical Density in the Hard Disk Model via Optimized Metrics. https://arxiv.org/abs/1407.1930

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