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.

A novel functorial relationship in perturbative quantum field theory is pointed out that associates Feynman diagrams (FD) having no external line in one theory $$\mathbf{Th}_1$$Th1 with singlet operators in another one $$\mathbf{Th}_2$$Th2 having an additional $$U(\mathcal{N})$$U(N) symmetry and is illustrated by the case where $$\mathbf{Th}_1$$Th1 and $$\mathbf{Th}_2$$Th2 are respectively the rank $$r-1$$r-1 and the rank r complex tensor model. The values of FD in $$\mathbf{Th}_1$$Th1 agree with the large $$\mathcal{N}$$N limit of the Gaussian average of those operators in $$\mathbf{Th}_2$$Th2 . The recursive shift in rank by this FD functor converts numbers into vectors, then into matrices, then into rank 3 tensors and so on. This FD functor can straightforwardly act on the d dimensional tensorial quantum field theory (QFT) counterparts as well. In the case of rank 2-rank 3 correspondence, it can be combined with the geometrical pictures of the dual of the original FD, namely, equilateral triangulations (Grothendieck’s dessins d’enfant) to form a triality which may be regarded as a bulk-boundary correspondence.

The paper contains a new nonperturbative representation for the subleading contribution to the free energy of a multicut solution for a Hermitian matrix model. This representation is a generalization

Explicit expressions for multimatrix models with complex and unitary matrices allows to couple these models with well-known unitary, orthogonsl and sympletic ensembles. We consider examples of such mixed ensembles which are solvable in the sense that the partition functions of such ensembles can be considered as tau functions of the classical integrable equations.