Almost all rooted maps have large representativity

Let M be a map on a surface S . The edge‐width of M is the length of a shortest noncontractible cycle of M . The face‐width (or, representativity) of M is the smallest number of intersections a noncontractible curve in S has with M . (The edge‐width and face‐width of a planar map may be defined to be infinity.) A map is a large‐edge‐width embedding (LEW‐embedding) if its maximum face valency is less than its edge‐width. For several families of rooted maps on a given surface, we prove that there are positive constants C 1 and C 2 , depending on the family and the surface, such that1 almost all maps with n edges have face‐width and edge‐width greater than c 1 log n , and 2 the fraction of such maps that are LEW‐embeddings and the fraction that are not LEW‐embeddings both exceed n − >C2 .