Fluctuations in the Entropy of Hawking Radiation

TL;DR

This paper calculates the fluctuations of Hawking radiation entropy around the Page curve using the gravitational path integral, revealing different behaviors before and after the Page time and suggesting an ensemble interpretation of the GPI.

Contribution

It provides a detailed computation of entropy fluctuations in a 2D model, highlighting the ensemble nature of the gravitational path integral and its implications for black hole information.

Findings

Entropy fluctuations before Page time are proportional to e^{-S}/√2.

After Page time, fluctuations scale as e^{-S} with a logarithmic correction.

The results support an ensemble interpretation of the gravitational path integral.

Abstract

We use the gravitational path integral (GPI) to compute the fluctuations of the Hawking radiation entropy around the Page curve, in a two-dimensional model introduced by Penington \emph{et al}. Before the Page time, we find that $\delta S = e^{-S}/\sqrt{2}$, where $S$ is the black hole entropy. This result agrees with the Haar-averaged entropy fluctuations of a bipartite system, which we also compute at leading order. After the Page time, we find that $\delta S \sim e^{-S}$, up to a prefactor that depends logarithmically on the width of the microcanonical energy window. This is not symmetric under exchange of subsystem sizes and so does not agree with the Haar average for a subsystem of fixed Hilbert space dimension. The discrepancy can be attributed to the fact that the black hole Hilbert space dimension is not fixed by the state preparation: even in a microcanonical ensemble with a top-hat smearing function, the GPI yields an additive fluctuation in the number of black hole states. This result, and the fact that the Page curve computed by the GPI is smooth, all point towards an ensemble interpretation of the GPI.