Universal Structures and Emergent Geometry from Large-$c$ BCFT Ensemble

In this paper, we study the ensemble average of boundary CFT (BCFT) data consistent with the bootstrap equations. We apply the results to computing ensemble average of copies of multi-point correlation functions of boundary changing operators (BCO), and find the results in agreement with one copy of the Virasoro TQFT. Further, we consider ensemble average of CFT path-integrals expressed as tensor networks of BCO correlation functions using the formalism developed in arXiv:2210.12127, arXiv:2311.18005 and arXiv:2403.03179. We find a natural emergence of locality and a loop-sum structure reminiscent of lattice integrable models. We illustrate this universal structure through explicit examples at genus zero and genus one. Moreover, we provide strong evidence that, at leading order in large-$c$, the results match those of three-dimensional Einstein gravity. In the presence of closed CFT operator insertions, generalized free fields emerge, with their correlation functions governed by the shortest paths connecting the insertions.