An Occupancy Approach to the Hard-Core Model

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" summary

Shows that the natural repulsion inequality—expected excluded volume increases when centers are conditioned non-overlapping—actually fails in dimension 24 via a Leech-lattice packing lower bound, contradicting the naive upper bound on the empty random geometric graph probability. Proves for triangle-free d‑regular graphs that the occupancy fraction is maximized by K_{d,d} with a short probabilistic convexity argument. Extends to matchings (monomer–dimer), where the repulsion inequality holds and yields partition‑function maxima. Conjectures exponential decay of correlations implies the repulsion inequality, which would locate phase transitions.

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" chapters (8)

1. 0:00Introduction to hard spheres
2. 2:13Excluded volume analysis
3. 5:17Repulsion inequality
4. 11:05The hard core model on graphs
5. 16:07Proof for graph occupancy
6. 21:44Translation invariant measures
7. 24:35Free volume interpretation
8. 27:42Monomer-dimer and conclusions