  Let $P(n,m)$ be a graph chosen uniformly at random from the class of all
planar graphs on vertex set $\left\{1, \ldots, n\right\}$ with $m=m(n)$ edges.
We determine the (Benjamini-Schramm) local weak limit of $P(n,m)$ in the sparse
regime when $m\leq n+o\left(n\left(\log n\right)^{-2/3}\right)$. Assuming that
the average degree $2m/n$ tends to a constant $c\in[0,2]$ the local weak limit
of $P(n,m)$ is a Galton-Watson tree with offspring distribution $Po(c)$ if
$c\leq 1$, while it is the Skeleton tree if $c=2$. Furthermore, there is a
smooth transition between these two cases in the sense that the local weak
limit of $P(n,m)$ is a linear combination of a Galton-Watson tree and the
Skeleton tree if $c\in\left(1,2\right)$.
