In this extended abstract we determine a normal limiting distribution for the number of triangles in a uniformly at random 3-connected cubic planar graph, as well as the precise expectation and variance values. Further comments towards the more complicated problem of studying both the limiting distribution of triangles in random cubic planar graphs, and the (asymptotic) number of triangle-free cubic planar graphs are discussed as well.
Requilé, C., Rue, J. Triangles in random cubic planar graphs. A: European Conference on Combinatorics, Graph Theory and Applications. "Eurocomb 2015". Bergen: 2015, p. 383-391.