Quantum black holes, partition of integers and self-similarity

TL;DR

This paper explores the quantization of black hole horizons, linking their configurations to integer partitions, revealing self-similarity properties, and analyzing quantum corrections to entropy within a finite fermionic degrees of freedom framework.

Contribution

It introduces a novel connection between black hole configurations and integer partitions, demonstrating self-similarity and providing insights into quantum entropy corrections.

Findings

Black hole horizon area quantized as $A = l_P^2 (4 \, \ln 2) N$

A two-to-one mapping between configurations and integer partitions of N

Self-similarity of black hole configurations derived from partition properties

Abstract

We take the view that the area of a black hole's event horizon is quantized, $A = l_P^2 \, (4 \ln 2) \, N$, and the associated degrees of freedom are finite in number and of fermionic nature. We then investigate general aspects of the entropy, $S_{BH}$, our main focus being black-hole self-similarity. We first find a two-to-one map between the black hole's configurations and the ordered partitions of the integer $N$. Hence we construct from there a composition law between the sub-parts making the whole configuration space. This gives meaning to black hole self-similarity, entirely within a single description, as a phenomenon stemming from the well known self-similarity of the ordered partitions of $N$. Finally, we compare the above to the well-known results on the subleading (quantum) corrections, that necessarily require different (quantum) statistical weights for the various configurations.