Exact Solution of Discrete Two-Dimensional R^2 Gravity

TL;DR

This paper presents an exact solution to a matrix model of 2D quantum gravity with R^2 interactions, revealing that the system's infrared behavior remains that of pure gravity regardless of the R^2 coupling, and introduces novel analytical techniques.

Contribution

It provides the first exact solution of a matrix model for 2D quantum gravity with R^2 terms, exploring intermediate regimes and developing new large N expansion methods.

Findings

No flattening phase transition with respect to R^2 coupling

Infrared behavior remains that of pure gravity at finite R^2

A scaling function interpolates between pure gravity and a dilute curvature defect gas at infinite coupling

Abstract

We exactly solve a special matrix model of dually weighted planar graphs describing pure two-dimensional quantum gravity with an R^2 interaction. It permits us to study the intermediate regimes between the gravitating and flat metric. Flat space is modeled by a regular square lattice, while localised curvature is introduced through lattice defects. No ``flattening'' phase transition is found with respect to the R^2 coupling: the infrared behaviour of the system is that of pure gravity for any finite R^2 coupling. In the limit of infinite coupling, we are able to extract a scaling function interpolating between pure gravity and a dilute gas of curvature defects on a flat background. We introduce and explain some novel techniques concerning our method of large N character expansions and the calculation of Schur characters on big Young tableaux.