# A Note on Subregion Holographic Complexity

## TL;DR

This paper analytically and numerically investigates the behavior of subregion holographic complexity, related to the volume inside Ryu-Takayanagi surfaces, across various black hole regimes, revealing insights into phase transitions and potential new features.

## Contribution

It provides a systematic analysis of subregion holographic complexity for different black hole backgrounds, including low/high temperature and extremal cases, highlighting new features and divergences.

## Key findings

- Complexity behaves similarly to entanglement entropy during phase transitions.
- Spurious divergences can occur in near horizon expansions at high charge.
- Numerical results show complexity captures phase transition information.

## Abstract

The volume inside a Ryu-Takayanagi surface has been conjectured to be related to the complexity of subregions of the boundary field theory. Here, we study the behaviour of this volume analytically, when the entangling surface has a strip geometry. We perform systematic expansions in the low and high temperature regimes for AdS-Schwarzschild and RN-AdS black holes. In the latter regime, we point out spurious divergences that might occur due to the limitations of a near horizon expansion. A similar analysis is performed for extremal black holes, and at large charge, we find that there might be some new features of the volume as compared to the area. Finally, we numerically study a four dimensional RN-AdS black hole in global AdS, the entangling surface being a sphere. We find that the holographic complexity captures essentially the same information as the entanglement entropy, as far as phase transitions are concerned.