A random hyperbolic surface, chosen uniformly at random from the moduli space of genus-g surfaces, is almost surely maximally connected. Its Cheeger constant — a measure of how hard it is to cut the surface into disconnected pieces — approaches the theoretical maximum as the genus increases.
This inverts the standard expectation about randomness. Random structures are usually assumed to be generic, mediocre, average. A random graph has average connectivity. A random number has average divisibility properties. Randomness, in most settings, produces typicality — and typicality is far from optimal.
Nalini Anantharaman and Laura Monk showed that for hyperbolic surfaces, typicality is optimality. The typical surface is not just well-connected; it is as well-connected as any surface could be. Randomness doesn't produce mediocrity here. It produces excellence.
The mechanism is that the constraints of negative curvature are so strong that they leave almost no room for poorly connected surfaces. In flat or positively curved geometry, there are many ways to build a surface with bottlenecks — narrow passages that, if cut, would disconnect it. In hyperbolic geometry, the exponential growth of area with radius makes bottlenecks geometrically expensive. A narrow passage in a hyperbolic surface must connect regions that are growing exponentially, and maintaining the narrowness against that growth requires a precise, fragile construction. Almost all surfaces fail to maintain it. Almost all surfaces are therefore maximally connected.
The result is about the shape of the space of possibilities. When constraints are weak, the space is large, most points are mediocre, and finding optimal structures requires search. When constraints are strong enough, the space collapses: almost every point is near-optimal because the constraints eliminate the mediocre configurations. The optimal isn't found. The suboptimal is forbidden.
This pattern — constraints so severe that they force excellence — appears in other settings. Error-correcting codes approaching the Shannon limit. Random matrices having universal eigenvalue statistics. Physical systems near critical points displaying universal scaling. In each case, the structure is not designed for optimality. It is constrained into optimality by the geometry of the space it inhabits. The typical and the optimal converge because the space itself has no room for anything else.