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

Abstract

Rectangular N × 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 R 2 to R 2 is indicated. At different critical points, the scaling behaviour 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 Painlevé II equation arises (in conjuction 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.