  The present work describes the asymptotic local shape of a graph drawn
uniformly at random from all connected simple planar graphs with n labelled
vertices. We establish a novel uniform infinite planar graph (UIPG) as quenched
limit in the local topology as n tends to infinity. We also establish such
limits for random 2-connected planar graphs and maps as their number of edges
tends to infinity. Our approach encompasses a new probabilistic view on the
Tutte decomposition. This allows us to follow the path along the decomposition
of connectivity from planar maps to planar graphs in a uniformed way, basing
each step on condensation phenomena for random walks under subexponentiality
and Gibbs partitions. Using large deviation results, we recover the asymptotic
formula by Gim\'enez and Noy (2009) for the number of planar graphs.
