Lower Bounds on the Critical Density in the Hard Disk Model via Optimized Metrics
summary
Improves the rigorous lower bound for rapid mixing of a global‑move hard disk Glauber chain from 0.125 to 0.154 density by designing an optimized continuous metric for path coupling. Uses a mirror‑image coupling, geometric “danger crescent” area bounds, and a linear program that discretizes the intercenter distance (256 intervals) to find a distance function d(l) that contracts in expectation; shows n log n mixing up to that ρ and gives a route toward spatial mixing, though far from the physicists’ ~0.7 transition.
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