Discrete gravity on random tensor network and holographic Renyi entropy

In this paper we apply the discrete gravity and Regge calculus to tensor networks and Anti-de Sitter/conformal field theory (AdS/CFT) correspondence. We construct the boundary many-body quantum state vertical bar Psi > using random tensor networks as the holographic mapping, applied to the Wheeler-deWitt wave function of bulk Euclidean discrete gravity in 3 dimensions. The entanglement Renyi entropy of vertical bar Psi > is shown to holographically relate to the on-shell action of Einstein gravity on a branch cover bulk manifold. The resulting Renyi entropy S-n of vertical bar Psi > approximates with high precision the Renyi entropy of ground state in 2-dimensional conformal field theory (CFT). In particular it reproduces the correct n dependence. Our results develop the framework of realizing the AdS(3)/CFT2 correspondence on random tensor networks, and provide a new proposal to approximate the CFT ground state.