Matriх models on graphs

I will explain how matrix models, which I call "solvable" models, are constructed from an embedded graph. By solvable, I mean models whose partition function can be written as a sum over partitions. (This terminology was proposed by Kazakov. This is the case when a matrix integral, i.e., an integral over $\sim nN^2$ variables, is transformed into an integral or a sum over $N$ variables, where $n$ is the number of matrices and $N$ is the size of the matrices.) In such models, random matrices correspond to the edges of the graph, and either constant or random matrices are assigned to the corners at the vertices of the graph. All the solvable models described are generating functions for Hurwitz numbers. I will highlight "exactly-solvable" models, i.e., models whose partition function transforms into a tau function of one of the hierarchies: KP, DKP, or BKP.