Microstructure in matrix elements

TL;DR

This work extends Pennington–Shenker–Stanford–Yang by adding dynamical EOW branes behind the horizon, enabling interior scattering that couples different brane flavors. It develops a dual description in terms of a deformed matrix integral, showing that strong interior interactions collapse the ensemble to a fixed nonrandom matrix C0, with microscopic data encoded in couplings g_{ij}. Off-diagonal density-matrix elements become relevant toward the end of evaporation, allowing the radiation to approach a pure state and yielding a unique analytic form for all Renyi entropies via planar resummation. The results demonstrate that realistic, non-ensemble gravity models can capture Hawking radiation microstructure through strong interior dynamics, with implications for higher-dimensional black holes and the interpretation of replica wormholes.

Abstract

We investigate the simple model of Pennington, Shenker, Stanford and Yang for modeling the density matrix of Hawking radiation, but further include dynamics for EOW branes behind the horizon. This allows interactions that scatter one interior state to another, and also allows EOW loops. At strong coupling, we find that EOW states are no longer random; the ensemble has collapsed, and coupling constants encode the microscopic matrix elements of Hawking radiation. This suggests strong interior dynamics are important for understanding evaporating black holes, without any ensemble average. In this concrete model the density matrix of the radiation deviates from the thermal state, small off-diagonal fluctuations encode equivalences between naively orthogonal states, and bound the entropy from above. For almost evaporated black holes the off-diagonal terms become as large as the diagonal ones, eventually giving a pure state. We also find the unique analytic formula for all Renyi entropies.

Figures (7)

• Figure 1: Normalized version of the density matrix \ref{['simpledensity']} for $k=20$ and $e^{\textsf{S}}=50\,,5\,,1$ (left, middle, right). Orange is positive and blue negative. Intensity of colors reflects the magnitude of individual matrix elements. Off-diagonal elements are less/not suppressed for black holes that have almost evaporated (middle/right).
• Figure 2: In the model of section \ref{['sec:interior_dynamics']} the off-diagonal terms come from EOW interactions where boundary particles can change flavor, the associated coupling constants depend on $\gamma$ and $\textsf{C}_0$ and the summations are over different flavors of the intermediate boundary particles (external flavor labels $i$ and $j$ were left implicit).
• Figure 3: Haar random unitaries. Wires contract indices, integrating over Haar random unitaries corresponds with inserting complete sets of wire states, Weingarten functions weight each bra-ket combination. Dominant terms for large $L$ have identical bra and ket (middle), the subleading terms have different bra and ket (right).
• Figure 4: Normalized density matrix with $k=20$ and $e^{\textsf{S}}=50$ (left). Orange is positive and blue negative. The diagonal is removed (right) and the intensity is rescaled to probe typical sizes $1/k\, e^{\textsf{S}/2}$. This reproduces similar plots obtained within SYK, with notably a different realization for the density matrix than \ref{['eqn:rho_singlemember']}Stanford:2020wkf.
• Figure 5: Purity $R_2$ as function of $k$ with $e^{\textsf{S}}=50$, showing \ref{['purpu']} (black dots) and the planar approximation $1/k+1/L$ (orange), log-log representation.