Abstract
We introduce a family of models, which we name matrix models associated with children’s drawings—the so-called dessin d’enfant. Dessins d’enfant are graphs of a special kind drawn on a closed connected orientable surface (in the sky). The vertices of such a graph are small disks that we call stars. We attach random matrices to the edges of the graph and get multimatrix models. Additionally, to the stars we attach source matrices. They play the role of free parameters or model coupling constants. The answers for our integrals are expressed through quantities that we call the “spectrum of stars”. The answers may also include some combinatorial numbers, such as Hurwitz numbers or characters from group representation theory.
Highlights
Interest in matrix integrals arose in different contexts and at different times
We introduce a family of models, which we name matrix models associated with children’s drawings—the so-called dessin d’enfant
We offer a family of models that in a sense can be called exactly solvable. They are built according to the so-called children’s drawings, more precisely, clean children’s drawings, which in combinatorics are called maps. This is a graph drawn on a closed orientable surface, which has the following property: if we cut it along the edges, the surface decomposes into regions homeomorphic to disks
Summary
Interest in matrix integrals arose in different contexts and at different times. These are problems of statistics in biology (Wishart), and problems of quantum chaos (Wigner, Dyson, Gorkov, Eliashberg, Efetov), and purely mathematical problems of representation theory (see the textbook [1]). They are built according to the so-called children’s drawings, more precisely, clean children’s drawings, which in combinatorics are called maps This is a graph drawn on a closed orientable surface, which has the following property: if we cut it along the edges, the surface decomposes into regions homeomorphic to disks (that is, they can be turned into disks by continuous transformation). It turns out that studying the products of random matrices with sources is an easier and more natural task. Writing answers for such integrals turns out to be a faster task if we have these matrices.
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