Matrix Models on Large Graphs

TL;DR

This paper investigates the behavior of multi-matrix models on large regular graphs, showing that in the limit of high degree, the free energy becomes universal and simplifies to contributions from tree-like surface structures.

Contribution

It demonstrates the universality of the free energy for matrix models on large graphs and provides an algebraic form for each order in the low temperature expansion.

Findings

Free energy becomes graph-independent as degree increases.

Universal free energy includes only tree-structured surface contributions.

Non-universal and non-tree contributions are suppressed by inverse powers of the degree.

Abstract

We consider the spherical limit of multi-matrix models on regular target graphs, for instance single or multiple Potts models, or lattices of arbitrary dimension. We show, to all orders in the low temperature expansion, that when the degree of the target graph $\Delta\to\infty$, the free energy becomes independent of the target graph, up to simple transformations of the matter coupling constant. Furthermore, this universal free energy contains contributions only from those surfaces which are made up of ``baby universes'' glued together into trees, all non-universal and non-tree contributions being suppressed by inverse powers of $\Delta$. Each order of the free energy is put into a simple, algebraic form.