arXiv subjects
Publications and source records attributed to Aleksandr Yu. Orlov.
I present a generalization of our joint works with John Harnad (2021) that relates Schur functions, KP tau functions and KP correlation functions to Schur's $Q$-functions, BKP tau functions and BKP correlation functions, respectively.
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.
We obtain Hurwitz numbers as the number of Feynman diagrams of a certain type divided by the order of the automorphism group of the diagram.
We explain how Gaussian integrals over ensemble of complex matrices with source matrices generate Hurwitz numbers of the most general type, namely, Hurwitz numbers with arbitrary orientable or non-orientable base surface and arbitrary profiles at branch points. We use the Feynman diagram approach. The connections with topological theories and also with certain classical and quantum integrable theories in particular with Witten's description of two-dimensional gauge theory are shown.