Tensor Models: extending the matrix models structures and methods

TL;DR

This paper reviews structural properties of matrix models and explores their generalization to tensor models, highlighting the role of topological recursion and discussing challenges in extending existing solutions.

Contribution

It introduces the application of blobbed topological recursion to tensor models and discusses the difficulties in extending matrix model techniques to tensors.

Findings

Tensor models can satisfy a version of topological recursion.

Blobbed topological recursion applies to certain tensor models.

Challenges remain in extending matrix model solutions to tensor models.

Abstract

In this text we review a few structural properties of matrix models that should at least partly generalize to random tensor models. We review some aspects of the loop equations for matrix models and their algebraic counterpart for tensor models. Despite the generic title of this review, we, in particular, invoke the Topological Recursion. We explain its appearance in matrix models. Then we state that a family of tensor models provides a natural example which satisfies a version of the most general form of the topological recursion, named the blobbed topological recursion. We discuss the difficulties of extending the technical solutions existing for matrix models to tensor models. Some proofs are not published yet but will be given in a coming paper, the rest of the results are well known in the literature.