Formal matrix integrals and combinatorics of maps

TL;DR

This paper reviews the connections between matrix integrals and combinatorics of maps, highlighting their differences and summarizing 30 years of developments and classical models in physics.

Contribution

It clarifies the relationship between formal and convergent matrix integrals and their role as generating functions for maps, providing a historical overview.

Findings

Formal matrix integrals are generating functions for maps.

Formal and convergent matrix integrals are generally very different.

Classical matrix models have played a significant role in physics.

Abstract

This article is a short review on the relationship between convergent matrix integrals, formal matrix integrals, and combinatorics of maps. We briefly summarize results developed over the last 30 years, as well as more recent discoveries. We recall that formal matrix integrals are identical to combinatorial generating functions for maps, and that formal matrix integrals are in general very different from convergent matrix integrals. Finally, we give a list of the classical matrix models which have played an important role in physics in the past decades. Some of them are now well understood, some are still difficult challenges.