Counting states in a model of replica wormholes

TL;DR

This paper analyzes the Hilbert space of multiple black holes influenced by replica wormholes, revealing a phase transition and aligning the Hilbert space dimension with Bekenstein-Hawking entropy.

Contribution

It introduces a gauge symmetry interpretation of the inner product induced by replica wormholes and counts the remaining states in a large n limit using a Coulomb gas analogy.

Findings

Identifies a third-order phase transition in the state count.

Shows the Hilbert space dimension matches Bekenstein-Hawking entropy.

Develops a collective Coulomb gas model for Young diagram shapes.

Abstract

We study the Hilbert space of a system of $n$ black holes with an inner product induced by replica wormholes. This takes the form of a sum over permutations, which we interpret in terms of a gauge symmetry. The resulting inner product is degenerate, with null states lying in representations corresponding to Young diagrams with too many rows. We count the remaining states in a large $n$ limit, which is governed by an emergent collective Coulomb gas description describing the shape of typical Young diagrams. This exhibits a third-order phase transition when the null states become numerous. We find that the dimension of the black hole Hilbert space accords with a microscopic interpretation of Bekenstein-Hawking entropy.