You need to know: Basic topology, homeomorphism, topological surface, orientable surface, genus of a surface, notation.
Background: A (two-dimensional, orientable) triangulation is a way to glue together a collection of oriented triangles, called the faces, along their edges in a connected way that matches the orientations. If the number of triangles is finite, then is homeomorphic to an orientable topological surface . Let us define the genus of as the genus of . The triangulation is called rooted if it has a distinguished oriented edge called the root edge. Let be the number of rooted triangulations with faces and genus .
Let . For any , let be such that , and let
. For any , let be the unique solution to the equation , and let . Also, let and .
The Theorem: On 1st February 2019, Thomas Budzinski and Baptiste Louf submitted to arxiv a paper in which that proved that for any sequence such that for every and , one has .
Short context: Triangulations are important in many applications, for example, as finite representations of a surface to a computer, or in the theory of random surfaces. In the last application, it is important to understand how a uniformly random triangulation looks like, and, for this, we need to count triangulations with various properties. In 1986, Bender and Canfield established the asymptotic growth of when and is fixed. However, the important case when and both go to infinity remained open. The Theorem estimates up to sub-exponential factors in the case when the ratio converges to a constant.