$R^2$ 2D Quantum Gravity and Dually Weighted Graphs

Abstract

A recently introduced model of dually weighted planar graphs is solved in terms of an elliptic parametrization for some particular collection of planar graphs describing the 2D $R^2$ quantum gravity. Along with the cosmological constant $\lambda$ one has a coupling $\beta$ in the model corresponding to the $R^2$-coupling constant. It is shown that for any value of $\beta$ the large scale behavior of the model corresponds to that of the standard pure 2D quantum gravity. On small distances it describes the dynamics of point-like curvature defects introduced into the flat 2D space. The scaling function in the vicinity of almost flat metric is obtained. The major steps of the exact solution are given.