# Ensemble averaging Semiclassical quantum gravity often exhibits a feature of [[0249 Factorisation problem|non-factorisation]],${Z\left(\beta_1,\beta_2\right)} \neq {Z\left(\beta_1\right)} {Z\left(\beta_2\right)},$where the LHS is computed from a bulk path integral with two boundaries, having periodicities $\beta_1$ and $\beta_2$ respectively, and the RHS is a product of bulk path integrals computed for spacetimes with a single boundary. The inequality arises due to the presence of [[0278 Euclidean wormholes|Euclidean wormholes]] as saddles of the gravitational path integral, which connects the two boundaries. This equation does not make sense if taken at face value, since, by holography, the boundary calculations of the partition trivially factorises as a basic property of the partition function. This puzzle is resolved if both sides of the equation are interpreted as an **ensemble average**, denoted now with an overline:$\overline{Z\left(\beta_1,\beta_2\right)} \neq \overline{Z\left(\beta_1\right)}\overline{Z\left(\beta_2\right)}.$Then this equation is no longer paradoxical. Rather, such correlators tell us about statistical properties of the ensemble. This is the case in low dimensions such as [[0050 JT gravity|JT gravity]], whose low-energy limit is known to be dual to [[0201 Sachdev-Ye-Kitaev model|SYK]] with ensemble averaging. Similar statements are true for [[0002 3D gravity|3D gravity]], pure or with some defects allowed. ## Refs - [[2019#Saad, Shenker, Stanford]] - [[2020#Engelhardt, Fischetti, Maloney]] - [[2020#Stanford]] - [[2020#Marolf, Maxfield (a)]] - [[2020#Bousso, Wildenhain]] ## Understanding it - finding the duality before ensemble averaging <!--(according to Zhengdi Sun) --> - [[AshwinkumarDodelsonKidambiLeedomYamazaki2021]][](https://arxiv.org/pdf/2104.14710.pdf) - there might be explicit averaging over boundary metrics (i.e. bdy is a 2d QG itself) - [[Nguyen2021]][](https://arxiv.org/pdf/2108.01095.pdf) - [[BenjaminCollierFitzpatrickMaloneyPerlmutter2021]][](https://arxiv.org/pdf/2107.10744.pdf) - supports [[0308 Half-wormhole]] idea by finding them in CFT directly - [[0268 Holevo information]] - [[2021#Qi, Shangnan, Yang]] - from single system - [[2024#Haneder, Urbina, Moreno, Weber, Richter]] ## Models - [[0050 JT gravity|JT gravity]] - [[2019#Saad]] - Narain ensemble of free bosons - [[2020#Afkhami-Jeddi, Cohn, Hartman, Tajdini]] - [[2020#Maloney, Witten]] - [[2021#Collier, Maloney]] - rational CFT - [[2023#Raeymaekers, Rossi]] - SYK with more dof - [[GotoSuzukiUgajin2021]][](https://arxiv.org/abs/2111.11705) - BCFT ensemble - [[2025#Wang, Wang, Wei]] - [[2025#Hung, Jiang, Lao]] ## New papers - [[Cheng2020]]@[pdf](https://arxiv.org/pdf/2010.11192.pdf) - [[BlommaertMertensVerschelde2020]]@[](https://arxiv.org/pdf/2005.13058.pdf) ## Issues - [[JanssenMirbabayiZograf2021]] - perturbative JT fails when the number of boundaries is infinite (which is needed for the uniqueness problem) ## Average over states rather than theories - [[FreivogelNikolakopoulouRotundo2021]][](https://arxiv.org/abs/2105.12771) ## Resolutions - [[0059 Gravity from averaging quantum theory]] - [[2022#Schlenker, Witten]]