Submaps of maps. I. General 0–1 laws

AbstractLet Mn be the set of n edge maps of some class on a surface of genus g. When g = 0 (planar maps) we show how to prove that limn → ∞ |Mn|1/n exists for many classes of maps. Let P be a particular map that can appear as a submap of maps in our class. There is often a strong 0–1 law for the property that P is a submap of a randomly chosen map in Mn: If P is planar, then almost all Mn contain at least cn disjoint copies of P for small enough c; while if P is not planar, almost no Mn contain a copy of P. We show how to establish this for various classes of maps. For planar P, the existence of limn → ∞ |Mn|1n suffices. For nonplanar P, we require more detailed asymptotic information

Advances in Discrete Differential Geometry

Differential Geometr

Advances in Discrete Differential Geometry

Differential Geometr