Typicality versus thermality: An analytic distinction

TL;DR

This paper investigates how correlation functions in individual microstates differ from thermal averages, showing that variances are exponentially suppressed but amplified in imaginary time, helping distinguish pure states from thermal states, especially in black holes.

Contribution

It provides an analytic framework to differentiate microstates from thermal states using correlation function variances and their behavior in imaginary time.

Findings

Variance between microstates is exponentially suppressed in entropy.

Variances are amplified in imaginary time, revealing differences from thermal states.

Results apply to black hole microstates, aiding their identification.

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.