  Let $P(n,m)$ be a graph chosen uniformly at random from the class of all
planar graphs on vertex set $[n]:=\left\{1, \ldots, n\right\}$ with $m=m(n)$
edges. We show that in the sparse regime, when $m/n\leq 1$, with high
probability the maximum degree of $P(n,m)$ takes at most two different values.
In contrast, this is not true anymore in the dense regime, when $m/n>1$, where
the maximum degree of $P(n,m)$ is not concentrated on any subset of $[n]$ with
bounded size.
