Random Lattices and Random Sphere Packings: Typical Properties

Senya Shlosman (1, 2) , Michael A. Tsfasman (3)

1 CPT - Centre de Physique Théorique - UMR 6207
2 CPT - Centre de Physique Théorique - UMR 7332
3 IML - Institut de mathématiques de Luminy

We review results about the density of typical lattices in $R^n.$ They state that such density is of the order of $2^{-n}.$ We then obtain similar results for random packings in $R^n$: after taking suitably a fraction $\nu$ of a typical random packing $\sigma$, the resulting packing $\tau$ has density $C(\nu) 2^{-n},$ with a reasonable $C(\nu).$ We obtain estimates on $C(\nu).$

Domaines: Physique mathématique [math-ph]; Combinatoire [math.CO]

Senya Shlosman, Michael A. Tsfasman. Random Lattices and Random Sphere Packings: Typical Properties. Moscow Mathematical Journal, 2001, 1, pp.73-89. ⟨hal-00117569⟩