Let $K(n)$ be the complete 2-dimensional simplicial complex on $n$ vertices. Give each triangle of $K(n)$ a uniformly random area in [0,1], independently from other triangles. What is the (random) total area of a subcomplex of $K(n)$ that contains all vertices and is homeomorphic to the 2-sphere? We determine its order of magnitude, but many questions remain.

This problem can be thought of as the 2-dimensional analogue of the Random Travelling Salesman Problem. But unlike its 1-dimensional counterpart, the area-minimiser comes with interesting geometry: it defines a random triangulation of the 2-sphere. Such random geometries have attracted a lot of interest from mathematicians and physicists following deep results of Angel & Schramm, Le Gall, Miermont, Miller & Sheffield, and others. I will survey this area and pose some open problems connecting the latter with our model.

Joint work with John Haslegrave and Joel Larsson Danielsson.