$R^2$ 2D Quantum Gravity and Dually Weighted Graphs
  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. 
V.A.Kazakov