  We introduce and solve a family of discrete models of 2D Lorentzian gravity
with higher curvature, which possess mutually commuting transfer matrices, and
whose spectral parameter interpolates between flat and curved space-times. We
further establish a one-to-one correspondence between Lorentzian triangulations
and directed Random Walks. This gives a simple explanation why the Lorentzian
triangulations have fractal dimension 2 and why the curvature model lies in the
universality class of pure Lorentzian gravity. We also study integrable
generalizations of the curvature model with arbitrary polygonal tiles. All of
them are found to lie in the same universality class.
