On Products of Random Matrices

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....

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Bibliographic Details
Main Authors: Natalia Amburg, Aleksander Orlov, Dmitry Vasiliev
Format: Article
Language:English
Published: MDPI AG 2020-08-01
Series:Entropy
Subjects:
Online Access:https://www.mdpi.com/1099-4300/22/9/972
Description
Summary: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.
ISSN:1099-4300