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Typicality versus thermality: An analytic distinction

Vijay Balasubramanian, Bartlomiej Czech, Veronika Hubeny, Klaus Larjo, Mukund Rangamani, Joan Simon

TL;DR

The paper shows that for systems with large entropy $S$, the variance of finitely local correlation functions across microstates is exponentially suppressed by $e^{-S}$, making microstate distinctions difficult for typical measurements. By exploiting the analytic structure of correlators in imaginary time, the authors argue that microstate differences can be amplified at imaginary times $\tau$, providing a principled route to differentiate pure microstates from the thermal ensemble. They illustrate the general result with a free chiral boson toy model and with the D1-D5 CFT dual to BTZ black holes, finding that imaginary-time observables reveal distinctions at scales $\tau \sim S^2$ while real-time observables remain universal to leading order. The findings imply that, for semiclassical observers, black-hole microstates are effectively described by a coarse-grained geometry (eternal black hole or BTZ-like spacetime) despite an underlying ensemble of microstate spacetimes, and that nonlocal probes may offer more sensitivity to microstructure.

Abstract

In systems with a large degeneracy of states such as black holes, one expects that the average value of probe correlation functions will be well approximated by the thermal ensemble. To understand how correlation functions in individual microstates differ from the canonical ensemble average and from each other, we study the variances in correlators. Using general statistical considerations, we show that the variance between microstates will be exponentially suppressed in the entropy. However, by exploiting the analytic properties of correlation functions we argue that these variances are amplified in imaginary time, thereby distinguishing pure states from the thermal density matrix. We demonstrate our general results in specific examples and argue that our results apply to the microstates of black holes.

Typicality versus thermality: An analytic distinction

TL;DR

The paper shows that for systems with large entropy , the variance of finitely local correlation functions across microstates is exponentially suppressed by , making microstate distinctions difficult for typical measurements. By exploiting the analytic structure of correlators in imaginary time, the authors argue that microstate differences can be amplified at imaginary times , providing a principled route to differentiate pure microstates from the thermal ensemble. They illustrate the general result with a free chiral boson toy model and with the D1-D5 CFT dual to BTZ black holes, finding that imaginary-time observables reveal distinctions at scales while real-time observables remain universal to leading order. The findings imply that, for semiclassical observers, black-hole microstates are effectively described by a coarse-grained geometry (eternal black hole or BTZ-like spacetime) despite an underlying ensemble of microstate spacetimes, and that nonlocal probes may offer more sensitivity to microstructure.

Abstract

In systems with a large degeneracy of states such as black holes, one expects that the average value of probe correlation functions will be well approximated by the thermal ensemble. To understand how correlation functions in individual microstates differ from the canonical ensemble average and from each other, we study the variances in correlators. Using general statistical considerations, we show that the variance between microstates will be exponentially suppressed in the entropy. However, by exploiting the analytic properties of correlation functions we argue that these variances are amplified in imaginary time, thereby distinguishing pure states from the thermal density matrix. We demonstrate our general results in specific examples and argue that our results apply to the microstates of black holes.

Paper Structure

This paper contains 26 sections, 78 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Ensembles and variances of observables
  3. The variance in quantum observables
  4. Variances in quantum mechanics
  5. Variances in quantum field theory
  6. Entropic suppression of variance
  7. Summary:
  8. Free scalar toy model
  9. Correlation functions in a typical state
  10. Thermal state correlators:
  11. Variance in the microcanonical ensemble
  12. Variance in $\mathcal{M}_{bas}$:
  13. Variance in $\mathcal{M}_{sup}$:
  14. Summary for the free chiral boson:
  15. D1-D5 system and the BTZ $M=0$ black hole
  16. ...and 11 more sections

Figures (3)

  • Figure 1:
  • Figure 2: (a): Variance as a function of Euclidean time. The 'temperatures' are $\frac{1}{\beta} = 100$ (left) and $1000$ (right). (b): Standard deviation to mean--ratio as a function of Euclidean time for 'temperatures' $\frac{1}{\beta} = \frac{100}{3}$ (upper) and $100$ (lower).
  • Figure 3: The standard deviation to mean ratio $\mathcal{F}(t)$ in real time. (a): Short time scale behaviour, plotted for $\frac{1}{\beta} = 100$ (upper) and $\frac{1}{\beta} = 1000$ (lower)). (b): Long time behaviour plotted for $\frac{1}{\beta} = 1000$)