From polymers to quantum gravity: triple-scaling in rectangular matrix models

TL;DR

This paper analyzes rectangular matrix models in a triple-scaling limit, revealing complex phase transitions and scaling behaviors that interpolate between branched polymers and quantum gravity, with connections to differential equations and topology fluctuations.

Contribution

It introduces a triple-scaling regime for rectangular matrix models, uncovering new phase transition behaviors and linking them to differential equations and topology fluctuations.

Findings

Identification of three types of scaling behaviors at critical points

Emergence of Painlevé II equation in branched polymer regime

Dual weak/strong coupling expansions for certain critical points

Abstract

Rectangular $N\times M$ matrix models can be solved in several qualitatively distinct large $N$ limits, since two independent parameters govern the size of the matrix. Regarded as models of random surfaces, these matrix models interpolate between branched polymer behaviour and two-dimensional quantum gravity. We solve such models in a `triple-scaling' regime in this paper, with $N$ and $M$ becoming large independently. A correspondence between phase transitions and singularities of mappings from ${\bf R}^2$ to ${\bf R}^2$ is indicated. At different critical points, the scaling behavior is determined by: i) two decoupled ordinary differential equations; ii) an ordinary differential equation and a finite difference equation; or iii) two coupled partial differential equations. The Painlev\'e II equation arises (in conjunction with a difference equation) at a point associated with branched polymers. For critical points described by partial differential equations, there are dual weak-coupling/strong-coupling expansions. It is conjectured that the new physics is related to microscopic topology fluctuations.