In any dimension $D$, the Euclidean Einstein-Hilbert action, which describes gravity in the absence of matter, can be discretized over random discrete spaces obtained by gluing families of polytopes together in all possible ways. In the physical limit of small Newton constant, only the spaces which maximize the mean curvature survive. In two dimensions, this results in a theory of random discrete spheres, which converge in the continuum limit towards the Brownian sphere, a random fractal space interpreted as a quantum random space-time. In this limit, the continuous Liouville theory of $D=2$ quantum gravity is recovered. Previous results in higher dimension regarded triangulations - gluings of tetrahedra or $D$-dimensional generalizations, leading to the continuum random tree, or gluings of simple colored building blocks of small sizes, for which multi-trace matrix model results are recovered. This work aims at providing combinatorial tools which would allow a systematic study of riche 查看全文>>