# Discrete Gravity on Random Tensor Network and Holographic R\'enyi Entropy

## TL;DR

This paper develops a framework connecting discrete gravity, tensor networks, and holographic entanglement entropy, demonstrating that random tensor networks can accurately reproduce CFT ground state Re9nyi entropies via holographic principles.

## Contribution

It introduces a novel approach to realize AdS$_3$/CFT$_2$ correspondence using random tensor networks and discrete gravity, providing a new method to approximate CFT ground states.

## Key findings

- Re9nyi entropy of the boundary state matches CFT results
- High-precision approximation of CFT ground state Re9nyi entropy
- Reproduction of the correct n-dependence in Re9nyi entropy

## Abstract

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 $|\Psi\rangle$ 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 R\'enyi entropy of $|\Psi\rangle$ is shown to holographically relate to the on-shell action of Einstein gravity on a branch cover bulk manifold. The resulting R\'enyi entropy $S_n$ of $|\Psi\rangle$ approximates with high precision the R\'enyi 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$_3$/CFT$_2$ correspondence on random tensor networks, and provide a new proposal to approximate CFT ground state.