Triangulated surfaces are compact hyperbolic Riemann surfaces that admit a conformal triangulation by equilateral triangles. Brooks and Makover started the
study of random triangulated surfaces in the large genus setting, and proved results about the systole, diameter and Cheeger constant of random
triangulated surfaces. Subsequently Mirzakhani proved analogous results for random hyperbolic surfaces. These results, along with many others,
suggest that the geometry of random triangulated surfaces mirrors the geometry of random hyperbolic surfaces in the case of large genus asymptotics.
In this talk, I will describe an approach to show that triangulated surfaces are asymptotically well-distributed in moduli space.