Almost Flat Planar Diagrams

TL;DR

This paper develops a new method for solving matrix models of dually weighted graphs, enabling exact solutions and insights into the transition from flat to quantum gravity surfaces.

Contribution

It introduces a general large N character expansion method that reduces complex matrix models to a solvable Cauchy-Riemann problem.

Findings

Exact solution for a new model with positive curvature defects

Method applicable to a broad class of matrix models

Insight into the crossover from flat space to quantum gravity

Abstract

We continue our study of matrix models of dually weighted graphs. Among the attractive features of these models is the possibility to interpolate between ensembles of regular and random two-dimensional lattices, relevant for the study of the crossover from two-dimensional flat space to two-dimensional quantum gravity. We further develop the formalism of large $N$ character expansions. In particular, a general method for determining the large $N$ limit of a character is derived. This method, aside from being potentially useful for a far greater class of problems, allows us to exactly solve the matrix models of dually weighted graphs, reducing them to a well-posed Cauchy-Riemann problem. The power of the method is illustrated by explicitly solving a new model in which only positive curvature defects are permitted on the surface, an arbitrary amount of negative curvature being introduced at a single insertion.