Bending the Padic Tensor Network and Emergent Einstein Equation

We take the tensor network describing explicit p-adic CFT partition functions proposed in 1902.01411, and consider boundary conditions of the network describing a deformed Bruhat-Tits (BT) tree geometry. We demonstrate that this geometry satisfies an emergent graph Einstein equation in a unique way that is consistent with the bulk effective matter action encoding the same correlation function as the tensor network, at least in the perturbative limit order by order away from the pure BT tree. Moreover, the (perturbative) definition of the graph curvature in the Mathematics literature naturally emerges from the consistency requirements of the emergent Einstein equation. The emergent "metric" in the tenor network can be interpreted as a Fisher information between states.