The Statistical Mechanics of Map Enumeration and its Universal Characteristics
Maps are cellular networks on surfaces, As combinatorial objects they are natural extensions of graphs but with non-trivial geometric and topological structure. The well established study of random graphs, or networks, motivates the study of random maps as a natural analogue. The analysis of graphs is based on the study of its adjacency matrix and it's spectrum. In the case of a random graph the adjacency matrix is a random matrix and its spectrum is a special kind of point process. For random maps we have developed an analogue of the adjacency matrix which is also a random matrix. Exploiting this construction we have been able to resolve some long standing conjectures concerning combinatorial ensembles of maps and their enumeration as a function of size. These results reveal remarkable universal characteristics across a broad range of map ensembles. This talk will present an overview of the motivations for our study and of our results as well as some particular potential applications to spin systems on random lattices and conformal field theory on surfaces. This is joint work with Patrick Waters.