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We describe a simple lattice model of higher-curvature quantum gravity in twodimensions and study the phase structure of the theory as a function of thecurvature coupling. It is shown that the ensemble of flat graphs isentropically unstable to the formation of baby universes. In these simplifiedmodels the growth in graphs exhibits a branched polymer behaviour in the phasedirectly before the flattening transition.Comment: 18 pages LaTeX, 3 .eps figures, uses epsf.tex; clarifying comments  added and typos correcte

Bottleneck surfaces and worldsheet geometry of higher-curvature quantum gravity