# On the fate of the Hoop Conjecture in quantum gravity

## TL;DR

This paper demonstrates that in 3D quantum space modeled by spin-networks, boundary states become thermal when the boundary-to-bulk edge ratio exceeds a threshold, linking quantum entanglement to horizon formation.

## Contribution

It introduces a quantum gravity analogue of Thorne's Hoop conjecture, showing thermal boundary states emerge from entanglement in spin-network models.

## Key findings

- Boundary states are thermal when boundary-to-bulk edge ratio exceeds threshold.
- Thermal entropy is proportional to boundary area.
- Entanglement between boundary and bulk drives the thermalization.

## Abstract

We consider a closed region $R$ of 3d quantum space modeled by $SU(2)$ spin-networks. Using the concentration of measure phenomenon we prove that, whenever the ratio between the boundary $\partial R$ and the bulk edges of the graph overcomes a finite threshold, the state of the boundary is always thermal, with an entropy proportional to its area. The emergence of a thermal state of the boundary can be traced back to a large amount of entanglement between boundary and bulk degrees of freedom. Using the dual geometric interpretation provided by loop quantum gravity, we interprete such phenomenon as a pre-geometric analogue of Thorne's "Hoop conjecture", at the core of the formation of a horizon in General Relativity.