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Random lattices and random sphere packings: Typical properties. (English) Zbl 0999.11032
The random quantity $$\Delta(\sigma)$$ is considered, where $$\sigma$$ is a typical random sphere packing and $$\Delta(\sigma)$$ is the density of $$\sigma$$. The authors firstly review the results on the typical lattice packings and show that the density is of order $$2^{-n}$$. Then the corresponding problem for random (non-lattice) packings is investigated. In this case some “improvement” of the packing is needed and the resulting packings have densities of order $$C(\nu)2^{-n}$$. The term $$C(\nu)$$ is estimated as well.

##### MSC:
 11H31 Lattice packing and covering (number-theoretic aspects) 52C17 Packing and covering in $$n$$ dimensions (aspects of discrete geometry) 82B05 Classical equilibrium statistical mechanics (general)
##### Keywords:
sphere packing; density; random field
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