# Critical behavior of the black hole / black string transition

## TL;DR

This paper numerically studies the critical transition between localized black holes and black strings in higher dimensions, revealing a spiral phase diagram and complex critical exponents consistent with theoretical predictions.

## Contribution

It provides high-precision numerical results on the phase diagram and critical behavior of black hole/black string transitions in five and six dimensions, extending previous analyses.

## Key findings

- Phase diagram exhibits a spiral structure near the critical point.
- Physical quantities follow a discrete scaling symmetry with complex critical exponents.
- Numerical critical exponents match those from the double-cone metric theory.

## Abstract

We numerically construct static localized black holes in five and six spacetime dimensions which are solutions to Einstein's vacuum field equations with one compact periodic dimension. In particular, we investigate the critical regime in which the poles of the localized black hole are about to merge. A well adapted multi-domain pseudo-spectral scheme provides us with accurate results and enables us to investigate the phase diagram of those localized solutions within the critical regime, which goes far beyond previous results. We find that in this regime the phase diagram possesses a spiral structure adapting to the one recently found for non-uniform black strings. When approaching the common endpoint of both phases, the behavior of physical quantities is described by complex critical exponents giving rise to a discrete scaling symmetry. The numerically obtained values of the critical exponents agree remarkably well with those derived from the double-cone metric.