Sharp concentration of the number of submaps in random planar triangulations (Q558223)

The maximum degree \(\Delta_n\) of a random rooted triangulation on \(n\) vertices has \[ \biggl| \Delta_n - {{\log n - {{1}\over{2}}\log \log n}\over{\log(4/3)}}\biggr| \geq \Omega_n \] with probability \(O(1/\log n + (3/4)^{\Omega_n})\) where \(\Omega_n\to\infty\) arbitrarily slowly. Similar sharp concentration results are given on the number of vertices of given degree in random planar maps.