centertableaux,boxsize=.5em 11institutetext: Stanford Institute for
Theoretical Physics, Stanford University, Stanford, CA 94305, USA
22institutetext: Simons Center for Geometry and Physics, SUNY, Stony Brook, NY
11794, USA 33institutetext: CERN, Theoretical Physics Department, 1211 Geneva
23, Switzerland

# The volume of the black hole interior at late times

Luca V. Iliesiu1, Márk Mezei2 and Gábor Sárosi3

###### Abstract

Understanding the fate of semi-classical black hole solutions at very late
times is one of the most important open questions in quantum gravity. In this
paper, we provide a path integral definition of the volume of the black hole
interior and study it at arbitrarily late times for black holes in various
models of two-dimensional gravity. Because of a novel universal cancellation
between the contributions of the semi-classical black hole spectrum and some
of its non-perturbative corrections, we find that, after a linear growth at
early times, the length of the interior saturates at a time, and towards a
value, that is exponentially large in the entropy of the black hole. This
provides a non-perturbative confirmation of the complexity equals volume
proposal since complexity is also expected to plateau at the same value and at
the same time.

CERN-TH-2021-102

##  1 Introduction

Among the greatest challenges in physics is obtaining a microscopic
understanding of the expanding universe. While this is presently out of reach
for the observed universe, an intriguing example of such an expanding
spacetime is the interior of black holes, which in turn can be studied in the
framework of the AdS/CFT correspondence Maldacena:1997re ; Witten:1998qj ;
Gubser:1998bc . It has been proposed by Susskind and collaborators that the
growth of the black hole interior is dual to the growth of complexity, the
minimal number of gates required for a quantum circuit to prepare the
corresponding time evolved state in the dual CFT Susskind:2014rva ;
Susskind:2014moa . One version of this proposal is that complexity equals the
volume of a maximal slice in the black hole interior Stanford:2014jda .111Both
the definition of complexity and the size of the black hole interior suffer
from many ambiguities, which have lead to many versions of this proposal
Brown:2015bva ; Brown:2015lvg ; Couch:2016exn ; Belin:2021 . However, these
ambiguities are believed not to affect the late time universal scaling
properties.  The most important rationale behind this conjecture is that the
black hole interior grows for a very long time, past when conventional probes,
like correlation functions or entanglement entropy, have already reached their
thermal equilibrium value. Complexity, on the other hand, is expected to grow
in chaotic systems at the same rate as the volume of the interior, until times
exponentially large in the entropy, after which it saturates in a complexity
plateau Susskind:2015toa ; Brown:2016wib ; Balasubramanian:2019wgd ;
Susskind:2020wwe ; Balasubramanian:2021mxo ; Haferkamp:2021uxo . Up to this
point, it has been an open problem to establish whether the volume of the
black hole interior exhibits a similar saturation at late times.

A priori, there is no reason to expect that a geometrical description of the
interior could capture the late time saturation of complexity. However, recent
developments show that the gravitational path integral has a surprisingly
large regime of validity, with the exponentially small corrections coming from
summing over topologies being capable of capturing fine details of spectral
statistics Saad:2019lba ; Cotler:2020ugk ; Belin:2020hea , or the unitary Page
curve of the entropy of an evaporating black hole Almheiri:2019qdq ;
Penington:2019kki . Emboldened by this success, in this paper, we calculate
the corrections coming from higher topologies to the volume of the black hole
interior in a wide class of models of two-dimensional gravity, including the
widely studied theory of JT gravity Teitelboim:1983ux ; Jackiw:1984je ;
Maldacena:2016upp . Importantly, this requires us to give a nonperturbative
definition of the length of the black hole interior and to complete the sum
over surfaces of higher topology to the matrix model of Saad-Shenker-Stanford
Saad:2019lba  (or its generalizations Witten:2020wvy ; Maxfield:2020ale ). We
indeed find that the length of the black hole interior saturates at a time and
length exponentially large in the entropy. Our result thus provides a non-
perturbative version of the complexity equals volume conjecture.

In section 2 we propose a natural candidate for the length
$\langle\ell\rangle$ of the Einstein-Rosen (ER) bridge. In a two-dimensional
theory, the volume of the black hole interior is given by length of the
geodesic going from one side to the other. While the definition of this length
is unambiguous on surfaces with trivial topology, on manifolds of higher genus
there are an infinite number of geodesic spanning between the two sides of the
ER bridge. We explore various definitions for this length and settle on one
that is well defined on surfaces of arbitrary topology, causes minimal
backreaction on the metric, and can be analytically continued between
Euclidean and Lorentzian geometries:

$\begin{split}\langle\ell\rangle=\lim_{{{\Delta}}\to
0}\expectationvalue{\sum_{{{\gamma}}}\ell_{{\gamma}}\,e^{-{{\Delta}}\ell_{{\gamma}}}}\,,\end{split}$
(1.1)

where ${{\gamma}}$ is labeling non self-intersecting geodesics,
$\langle\dots\rangle$ evaluates the gravitational path integral by summing
over surfaces with any topology and where ${{\Delta}}$ acts as a regulator.
Since the free field boundary-to-boundary propagator on hyperbolic surfaces is
given by the sum over geodesics (including self-intersecting ones),
$\langle\ell\rangle$ is closely related to the two-point function
$\langle\chi\chi\rangle$, with an operator of dimension ${{\Delta}}$ inserted
on each side of the two-sided black hole, as a probe for the black hole
interior.

In section 3, we turn to the analysis of this quantity. In the limit in which
$\chi$ is a probe field and when we approximate the two-point function by
summing over non self-intersecting geodesics, we prove that at arbitrary order
in $e^{S_{0}}$ (with $S_{0}$ being the leading order entropy of the black
hole) the two-point function is simply given by

$\begin{split}\langle\chi(x_{1})\chi(x_{2})\rangle_{\text{non-
int.}}&\sim\int_{0}^{\infty}dE_{1}dE_{2}\
\left\langle\rho(E_{1})\rho(E_{2})\right\rangle
e^{-E_{1}(x_{1}-x_{2})-E_{2}(\beta-
x_{1}+x_{2})}\,{{\mathcal{M}}}_{{\Delta}}(E_{1},E_{2})\,,\end{split}$ (1.2)

where ${{\mathcal{M}}}_{{\Delta}}(E_{1},E_{2})=\absolutevalue{\langle
E_{1}|\chi|E_{2}\rangle}^{2}$ can be computed analytically. This result can be
viewed as a generalization to arbitrary genus and dilaton potential of the
genus 1 result for JT gravity of Saad:2019lba . The expression (1.2) is also
consistent with a conjectured formula for the matter two-point function in
gravity that was obtained by studying operator insertions in the matrix
integral that is dual to the JT theory Saad:2019lba ; Blommaert:2020seb .

Figure 1: We show the numerical evaluation of the Einstein-Rosen bridge length
(see (4.6)) for $\beta=15$ in orange, while the early and late time
asymptotics (1.3) are shown by the blue dashed and gray dotted curves,
respectively.

In section 4 we obtain the main result of our paper, that is, up to an overall
additive constant222The linear growth of the black hole interior at time
scales $t\ll e^{S_{0}}$ but exactly in the temperature was first obtained in
Yang:2018gdb .

$\begin{split}\langle\ell(t)\rangle\approx\begin{cases}C_{1}t+\dots\qquad&t\ll
e^{S_{0}}\,,\\\ C_{0}-\dots&t\gg e^{S_{0}}\,,\end{cases}\end{split}$ (1.3)

in a large class of theories of dilaton gravity, including JT gravity. Here,
the coefficients $C_{1}$ is $O(1)$ while $C_{0}$ is $O\left(e^{S_{0}}\right)$
and depend on the spectral density $\rho(E)$ in each dilaton gravity model and
on the inverse-temperature $\beta$ of the thermofield double state in which
the system is prepared at $t=0$. For a plot of $\langle\ell(t)\rangle$ see
figure 1.

In section 5 we remark on the late-time differences between the ER bridge
length, the spectral form factor, and the two-point function. While all three
quantities exhibit a period of growth and then a plateau at later times, the
physics of the growth and plateau periods for the bridge is widely different
from that for the latter two quantities. While the period of growth for the ER
bridge length is due to the contribution from the classical solution, the
“ramp” in the spectral form factor and two-point function is due to the
leading Euclidean wormhole geometry. More importantly, while the “plateau” in
the spectral form factor and two-point function occurs due to a cancellation
between the wormhole geometry and other non-perturbative corrections to the
spectral two-point function $\left\langle\rho(E_{1})\rho(E_{2})\right\rangle$,
the saturation of the bridge length is due to novel cancellations between the
classical disconnected contribution and non-perturbative contributions in
$\left\langle\rho(E_{1})\rho(E_{2})\right\rangle$. To our knowledge, these
types of cancellations have not been noticed in the random matrix literature
before and are universal in all one matrix models.

To further probe the relation between complexity and the late time length of
the ER bridge, we compute the variance of the latter quantity in section 6. We
start by deriving a geometric formula for the variance in terms of the
spectral four-point function. We then show that at late times

$\begin{split}\sigma_{\ell}\sim e^{S_{0}/2}\,,\qquad\text{for}\quad t\sim
e^{S_{0}}\,,\qquad\,\,\,\text{ and
}\,\,\,\qquad\sigma_{\ell}\sim\sqrt{t}\,,\qquad\text{for}\quad t\gg
e^{S_{0}}\,,\end{split}$ (1.4)

In contrast to the plateaus for the spectral form factor and the two-point
function, on the plateau for the ER bridge, the “signal” is much larger than
the “noise” in the regime $t\ll e^{2S_{0}}$. However, we find that as opposed
to $\langle\ell\rangle$, the variance $\sigma_{\ell}$ does not have a plateau,
instead it grows forever, in particular, it becomes the same order as the
signal when $t\propto e^{2S_{0}}$. We will discuss some possible explanations
for this unexpected behavior.

Finally, in section 7 we discuss the interpretation of our results beyond the
theories of dilaton gravity and their matrix integral dual. Our geometric
computation suggests the importance of a new spectral quantity, called
spectral complexity, which we expect to closely track the volume of the black
interior in any chaotic system with a holographic dual. We confirm this
expectation in a numerical study of the SYK model.

##  2 Nonperturbative length

We want to define the length $\expectationvalue{\ell}$ of the ER bridge in
two-dimensional models of gravity. Our definition needs to take into account
that in higher genus geometries, there are multiple geodesics connecting two
boundary points. Furthermore, when we probe this length we do not wish to
experience strong back-reaction from the measurement since that might spoil
the growth of the ER bridge.

One is first tempted to consider the minimal length geodesic spanning between
the two sides of the ER bridge as the definition of $\ell$. However, in a
gravitational path integral, we oftentimes have to work with complexified
geometries and complex geodesic lengths. Consequently, the minimum geodesic
length cannot be analytically continued. Alternatively, one might imagine
computing the minimal length of the ER bridge by considering the two-sided two
point function $\langle\chi_{L}(t)\chi_{R}(0)\rangle$ and taking the limit
$-\lim_{\Delta\to\infty}\frac{1}{\Delta}\log\langle\chi_{L}(t)\chi_{R}(0)\rangle=\lim_{\Delta\to\infty}\frac{1}{\Delta}\langle\sum_{{{\gamma}}}\,e^{-{{\Delta}}\ell_{{\gamma}}}\rangle$
which seemingly picks the geodesic with minimal real part from the
sum.333Here, $L/R$ denotes operators inserted on the left/right sides of the
bridge.  However, due to the backreaction of the operator insertion on the
metric the measurement of the ER bridge length is spoiled: as shown in
Yang:2018gdb  for any $\Delta>0$, the two-point function decays exponentially
up to a time $t\sim O(1)$, past which it transitions to a power law decay due
to quantum corrections coming from JT gravity ($\sim 1/t^{3}$, regardless of
the value of $\Delta$). Because of this,
$-\lim_{\Delta\to\infty}\frac{1}{\Delta}\log\langle\chi_{L}(t)\chi_{R}(0)\rangle$
does not see the growth of the ER bridge past the thermalization time (which
is $\sim\beta$). Therefore, we dismiss these candidates when defining the
length of the bridge.

A candidate that resolves all the issues above is the regularized sum444We
will specify the exact meaning of $\langle\dots\rangle$ shortly.

$\begin{split}\langle\ell\rangle=\lim_{{{\Delta}}\to
0}\expectationvalue{\sum_{{{\gamma}}}\ell_{{\gamma}}\,e^{-{{\Delta}}\ell_{{\gamma}}}}\,,\end{split}$
(2.1)

where ${{\gamma}}$ is labeling non self-intersecting geodesics and
$\langle\dots\rangle$ evaluates the sum over geodesics within the JT gravity
or dilaton gravity path integral, including the contribution of surfaces of
arbitrary genus. The prescription for choosing geodesics without self-
intersections is natural since, if we cut the manifold open along such curves,
we obtain a state in which the boundary points are separated by geodesic
length $\ell_{{\gamma}}$.555The cut that separates the manifold into two
disjoint parts will in general also create some accompanying baby universes.
From the quantum information perspective, this choice makes contact with the
original motivation of interpreting $\expectationvalue{\ell}$ as a length of a
minimal tensor network that can create the state. Furthermore, this quantity
can be analytically continued and has minimal backreaction on the metric.666We
can think about this quantity as inserting a tensionful defect and extracting
the leading correction in the tensionless limit. This is similar to how the
minimal surface is obtained from the gravitational Rényi entropies using the
replica trick Lewkowycz:2013nqa . However, one might worry that (2.1) could be
divergent since it involves an infinite number of geodesics on all surfaces
with $g>1$. As we show shortly, we find that the time-dependent piece in (2.1)
is finite, independent of the regularization scheme, and is precisely the
probe of the late time physics that we are interested in: i.e., it is
sufficient to study $\langle\ell(t)\rangle-\langle\ell(0)\rangle$.

We will compute $\langle\ell(t)\rangle$ by taking the ${{\Delta}}$ derivative
of the two-sided correlation function, computed in an approximation, where we
drop self-intersecting geodesics. Explicitly,

$\begin{split}\langle\ell(t)\rangle=-\lim_{{{\Delta}}\to
0}{\frac{\partial\left\langle\chi_{L}(t)\chi_{R}(0)\right\rangle_{\text{non-
int.}}}{\partial{{\Delta}}}}\,.\end{split}$ (2.2)

Two-sided correlation functions can be obtained by analytic continuation from
Euclidean correlators, and it will be convenient to obtain
$\left\langle\chi_{L}\left(t\right)\chi_{R}\left(0\right)\right\rangle$ from a
Euclidean computation performed in section 3. The analytic continuation that
we use is valid in any quantum system

$\begin{split}\left\langle{{\mathcal{O}}}_{L}(t){{\mathcal{O}}}_{R}(0)\right\rangle&\equiv\bra{\text{TFD}_{\beta}}{{\mathcal{O}}}_{L}(t){{\mathcal{O}}}_{R}(0)\ket{\text{TFD}_{\beta}}=\bra{\text{TFD}_{\beta}}{{\mathcal{O}}}_{L}(t/2){{\mathcal{O}}}_{R}(-t/2)\ket{\text{TFD}_{\beta}}\\\
&=\Tr\left[{{\mathcal{O}}}\left({\beta/2}+it\right){{\mathcal{O}}}(0)e^{-\beta
H}\right]\equiv\left\langle{{\mathcal{O}}}\left({\beta/2}+it\right){{\mathcal{O}}}(0)\right\rangle_{\beta}\,.\end{split}$
(2.3)

where in the first line we used the two-sided (thermofield double) and in the
second the Euclidean representation. The last expression in the first line is
the most intuitively connected to the ER bridge length where we evolved both
the left and rightt boundary points forward in time equally by $t/2$.777Recall
that in the thermofield double in the right system the natural time direction
runs backwards. The equality between the first and second lines can be
established by inserting a complete set of states in the second line and using
that the thermofield double state is
$\ket{\text{TFD}_{\beta}}=\sum_{n}e^{-\beta
R_{n}/2}\ket{E_{n}}_{L}\ket{E_{n}^{*}}_{R}$. The dilaton gravity theories
considered in this paper are famously ensembles of quantum systems, random
matrix theories, so when we apply (2.3) in our context, the gravity
computation gives the ensemble average of the above quantities.

In section 3, we will shown that
$\left\langle\chi(x_{1})\chi(x_{2})\right\rangle_{\text{non-int.}}$ is a
particularly nice quantity in JT gravity and other theories of dilaton
gravity, as it can be computed to all orders in the genus expansion and can
even be computed non-perturbatively using the matrix model representation for
such gravitational theories. We will therefore use (2.2) to extract
$\langle\ell(t)\rangle-\langle\ell(0)\rangle$.

##  3 The two-point function in JT gravity

JT gravity is a theory of two-dimensional gravity with a metric $g_{\mu\nu}$,
coupled to a dilaton $\phi$, whose Euclidean action is given by
Teitelboim:1983ux ; Jackiw:1984je ; Maldacena:2016upp

$I_{JT}=-S_{0}\chi({\mathcal{M}})-\frac{1}{2}\int\sqrt{g}\phi(R+2)-\int_{\partial{\mathcal{M}}}\sqrt{h}\phi(K-1)\,,\qquad
Z=\int\frac{Dg_{\mu\nu}D\phi}{\text{Diffeos}}e^{-I_{JT}}.$ (3.1)

The first term is purely topological, as it only accounts for the Euler
characteristic of the manifold ${\mathcal{M}}$. The bulk term fixes the
geometry to be a patch of hyperbolic space whose boundary has fixed, large,
proper length $L=\beta/\epsilon$ and dilaton value $\phi=1/\epsilon$.888Since
we set the boundary value of the dilaton to one, times and inverse
temperatures in the rest of the paper are measured in these units. In
particular the Schwarzian theory is strongly coupled when $\beta\gg 1$. These
boundary conditions require the presence of the Gibbons-Hawking-York term,
which gives an action for the boundary fluctuations in each such hyperbolic
patch. While the model exhibits great simplicity since it only depends on
topological data and boundary fluctuations, it still has black hole solutions
and, in particular, we will be interested in studying two-sided black holes,
which have an ER bridge between the two sides.

In this section, our goal is to probe the length of the ER bridge when one
includes the contribution of surfaces with arbitrary topology to the
gravitational path integral in (3.1). The probe that we shall consider is the
boundary-to-boundary two-point function of a probe matter field $\chi$, with
scaling dimension $\Delta$, which only couples to the metric in the JT theory.
In the probe limit, the two-point function on any hyperbolic surface can be
schematically expressed as

$\langle\Tr_{\beta}\left(\chi(x_{1})\chi(x_{2})\right)\rangle=\sum_{\mathcal{M}}\sum_{\gamma}\langle\Tr_{\beta}(e^{-\Delta\ell_{\gamma}})\rangle_{\mathcal{M}}\,,$
(3.2)

where the sum over ${\mathcal{M}}$ is a sum over all surfaces, obeying the
boundary conditions discussed above, and the sum over $\gamma$ is a sum over
all geodesics connecting the points at boundary times $x_{1}$ and $x_{2}$,
each of length $\ell_{\gamma}$.

To clarify our notation for expectation values of observables in the
gravitational path integral, $\Tr_{\beta}(\dots)\equiv\Tr(e^{-\beta H}\dots)$
is a mnemonic for the insertion of a gravitational boundary with the boundary
conditions described above while the $\dots$ will indicate what operators are
inserted on each boundary. Finally, the $\langle\dots\rangle$ will represent
the evaluation of the gravitational path integral with a number of boundaries
given by the number of traces inside, each with its own specified operator
insertions.

On a hyperbolic surface of genus $g\geq 1$, there will be an infinite number
of geodesics $\gamma$, some of which will self-intersect. Since we are
interested in the two-point function in order to probe the length
$\ell_{\gamma}$ of the ER bridge in a two-sided black hole, we shall restrict
ourselves to geodesics that do not self-intersect and set $x_{1}=0$ and
$x_{2}=\frac{\beta}{2}+it$; for each surface ${\mathcal{M}}$ we will denote
such geodesics going from $x_{1}$ to $x_{2}$ by $\mathcal{G}_{x_{1},x_{2}}$
and will denote correlators that solely account for these geodesics by
$\langle\dots\rangle_{\text{non-int.}}$.999Since we discarded self-
intersecting geodesics,
$\langle\Tr_{\beta}\left(\chi(x_{1})\chi(x_{2})\right)\rangle_{\text{non-
int.}}$ is only an approximation to the full two-point function. It was argued
by Saad Saad:2019pqd  that this approximation gives the leading behavior of
the function for large Lorentzian times, and he computed this modified two-
point function to the first subleading order in $e^{S_{0}}$ given by the genus
1 geometry. The leading disk contribution (which does not suffer from this
ambiguity) was obtained in Mertens:2017mtv . Since in this paper, our focus is
on studying the length of the ER bridge and not on the full two-point
function, we will not further discuss whether self-intersecting geodesics are
important in its late-time behavior.

Figure 2: An example (with genus 2) of the type of hyperbolic surfaces which
the path integral (3.3) sums over. The green curves represent the wiggly
asymptotic boundaries, while the red curve represents the boundary to boundary
geodesic. While there are an infinite number of non-self-intersecting
boundary-to-boundary geodesics on such surfaces, above, we have drawn an
example of all possible surface topologies that can result by cutting a genus
two surface along the geodesics. The purple curves represent the closed
geodesics which we use to glue the trumpet wavefunctions in (3.7) to different
bordered Riemann surfaces.

In JT gravity, the path integral over all metrics reduces to an integral over
boundary wiggles which we denote by $D{(\mathcal{W})}$ and an integral over
the moduli space of hyperbolic surfaces Saad:2019lba . Therefore, the two-
point function can be rewritten as

$\begin{split}\langle\Tr_{\beta}\left(\chi(x_{1})\chi(x_{2})\right)\rangle_{\text{non-
int.}}=\sum_{g}e^{S_{0}(1-2g)}\int_{\frac{\mathcal{T}_{g,1}}{\text{Mod}({\mathcal{M}}_{g,1})}}\omega\int
D{(\mathcal{W})}e^{-I_{JT,\text{
bdy}}(\mathcal{W})}\sum_{\gamma\in\mathcal{G}_{x1,x2}}e^{-\Delta\ell_{\gamma}}\,,\end{split}$
(3.3)

where $\omega=\sum_{j=1}^{3g-3+n}db\wedge d\tau$ is the Weil-Peterson
symplectic form on the moduli space of hyperbolic surfaces, and
$\mathcal{T}_{g,1}$ represents the moduli space of all hyperbolic surfaces
${\mathcal{M}}_{g,1}$ of genus $g$ with $1$ asymptotic boundary. Due to the
redundancy of large diffeomorphism in the gravitational path integral, we are
supposed to quotient this moduli space by the mapping class group
$\text{Mod}({\mathcal{M}}_{g,1})$ of ${\mathcal{M}}_{g,1}$. As in the example
shown in figure 2, cutting along each geodesic $\gamma$ in the sum above, we
can either obtain two disconnected manifolds, each with a single boundary,
${\mathcal{M}}_{h,1}$ and ${\mathcal{M}}_{g-h,1}$ with $0\leq h\leq g$, or a
single connected manifold with two boundaries ${\mathcal{M}}_{g-1,\,2}$. The
mapping class group acts by relating all boundary-to-boundary geodesics within
a given cutting class as well as all closed geodesics on ${\mathcal{M}}_{h,1}$
and ${\mathcal{M}}_{g-h,1}$ (or ${\mathcal{M}}_{g-1,\,2}$ in the latter case)
which do not intersect the boundary-to-boundary geodesic. Therefore, following
a similar strategy as in mirzakhani2007simple ; Saad:2019pqd ;
Blommaert:2020seb , we can rewrite the integral over the moduli space as

$\begin{split}\int_{\frac{\mathcal{T}_{g,1}}{\text{Mod}({\mathcal{M}}_{g,1})}}\omega\sum_{\gamma\in\mathcal{G}_{x1,x2}}e^{-\Delta\ell_{\gamma}}=e^{-\Delta\ell}\int_{\frac{\mathcal{T}_{g-1,2}}{\text{Mod}({\mathcal{M}}_{g-1,2})}}\omega+\sum_{h\geq
0}e^{-\Delta\ell}\int_{\frac{\mathcal{T}_{h,1}}{\text{Mod}({\mathcal{M}}_{h,1})}}\omega\int_{\frac{\mathcal{T}_{g-h,1}}{\text{Mod}({\mathcal{M}}_{g-h,1})}}\omega\,,\end{split}$
(3.4)

where $\ell$ is the length of a geodesic which is now part of the boundary of
${\mathcal{M}}_{h,1}$ and ${\mathcal{M}}_{g-h,1}$ (or part of both boundaries
of ${\mathcal{M}}_{g-1,2}$) and where for the remainder of the paper the sum
over $h$ is bounded such that disconnected surfaces of genus $h$ and $g-h$ are
not counted twice. When the genus $g=2$, we can use figure 2 to better
understand (3.4). The first term on the RHS sums over the moduli space of the
genus one surface with two boundaries obtained after cutting the rightmost
figure along the red geodesic. The $h=0$ term in (3.4) corresponds to the sum
over the moduli space of the leftmost figure after a cut along the red
geodesic which results into a one boundary genus zero and a one boundary genus
one surface, while the $h=1$ term correspond to summing over the moduli space
of the two genus one surfaces obtained after the cut in the center figure.

As drawn in figure 2, we can now cut each such surface along the closed
geodesic which is closest to the asymptotic boundary to separate the
manifolds, into two bordered Riemann surfaces with a geodesic boundary (or a
single such surface in the case of ${\mathcal{M}}_{g-1,2}$), and two trumpets
which, on one end, have a closed geodesic boundary, and, on the other end,
have a boundary with fixed proper length and fixed dilaton glued to a geodesic
of fixed length, $\ell$. Following the wavefunctional formalism discussed in
Yang:2018gdb , we denote the path integral for such an object as
$\psi_{\text{Trumpet},x}(\ell,b)$. The expression for the trumpet wavefunction
was derived in Saad:2019pqd  in terms of the Hartle-Hawking wavefunction for a
disk with a fixed ADM mass $E$, cut by a geodesic boundary of length $\ell$,
$\psi_{\text{Disk, E}}(\ell)$:

$\begin{split}\psi_{\text{Trumpet, }x}(\ell,b)=\leavevmode\hbox
to116.66pt{\vbox to70.34pt{\pgfpicture\makeatletter\hbox{\hskip
62.66956pt\lower-35.82727pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\ell$}}
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}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{44.94353pt}{26.86928pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$x$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\int_{0}^{\infty}dE\rho_{\text{Trumpet}}(E)\psi_{\text{Disk},\,E}(\ell)e^{-xE}\,,\end{split}$
(3.5)

where the density of states of the trumpet
$\rho_{\text{Trumpet}}(E)=\frac{\cos(b\sqrt{2E})}{\pi\sqrt{2E}}$ can be
computed from fermionic localization Stanford:2017thb , while
$\psi_{\text{Disk},E}(\ell)$ is given by
$\psi_{\text{Disk},E}(\ell)=4e^{-\ell/2}K_{i\sqrt{8E}}(e^{-\ell/2})$
Yang:2018gdb . Here, $E$ can be interpreted as the ADM mass observed on the
asymptotic boundary and the conservation of this mass follows from the
orthogonality relation

$\begin{split}\int_{\infty}^{\infty}d\ell
e^{\ell}\,\psi_{\text{Disk},E}(\ell)\psi_{\text{Disk},E^{\prime}}^{*}(\ell)=\frac{\delta(E-E^{\prime})}{\rho_{\text{Disk}}(E)}\,.\end{split}$
(3.6)

Above, $\rho_{\text{Disk}}(E)$ is the density of states in JT gravity on the
disk, given by $\rho_{\text{Disk}}(E)=\sinh(2\pi\sqrt{E})$Stanford:2019vob .
Using (3.4) we can thus rewrite the genus $g$ contribution,101010The integral
over $\omega$ in (3.4) can thus be rewritten by separating the integral over
the moduli of the two closed geodesics separating the trumpets from the
bordered Riemann surfaces,
$\int_{\frac{\mathcal{T}_{h,1}}{\text{Mod}({\mathcal{M}}_{h,1})}}\omega=\int_{0}^{\infty}db\int_{0}^{b}d\tau\int_{\frac{\tilde{\mathcal{T}}_{h,1}(b)}{\text{Mod}(\tilde{\mathcal{M}}_{h,1}(b))}}\omega$
$\bigg{(}$and
$\int_{\frac{\mathcal{T}_{g-1,2}}{\text{Mod}({\mathcal{M}}_{g-1,2})}}\omega=\int_{0}^{\infty}db_{1}\int_{0}^{b}d\tau_{1}\int_{0}^{\infty}db_{2}\int_{0}^{b}d\tau_{2}\int_{\frac{\tilde{\mathcal{T}}_{g-1,2}(b_{1},b_{2})}{\text{Mod}(\tilde{\mathcal{M}}_{g-1,2}(b_{1},b_{2}))}}\omega\bigg{)}$
where the $\tilde{\,}$ signifies that we are considering the moduli space of
surfaces with one (or two) geodesic boundary of fixed length $b$ (or $b_{1}$
and $b_{2}$).

$\begin{split}\langle\Tr_{\beta}\left(\chi(x_{1})\chi(x_{2})\right)\rangle_{\text{non-
int.},\,g}&\sim e^{S_{0}(1-2g)}\int d\ell\,e^{\ell}\int
db_{1}b_{1}\,db_{2}b_{2}\,\psi_{\text{Trumpet},x}(\ell,b_{1})\psi_{\text{Trumpet},\beta-x}(\ell,b_{2})e^{-\Delta\ell}\\\
&\times\bigg{[}{{\mathrm{Vol}}}_{g-1,2}(b_{1},b_{2})+\sum_{h\geq
0}{{\mathrm{Vol}}}_{g-h,1}(b_{1}){{\mathrm{Vol}}}_{h,1}(b_{2})\bigg{]}\,,\end{split}$
(3.7)

where, in order to compactly write (3.7), and not have to consider cuts which
involve the disk wavefunction separately, we have formally chosen
${{\mathrm{Vol}}}_{0,1}(b)$ to be such that, after the integral over $b$, the
disk wavefunction is recovered111111This is possible since
${{\mathrm{Vol}}}_{0,1}(b)$ (or its Laplace transform) does not enter the
topological recursion used to determine ${{\mathrm{Vol}}}_{g,n}$.

$\psi_{\text{Disk,x}}(\ell)\equiv\int_{0}^{\infty}dE\rho_{\text{Disk}}(E)\psi_{\text{Disk,
E}}(\ell)e^{-xE}\equiv\int_{0}^{\infty}dbb\,\psi_{\text{Trumpet},x}(\ell,b){{\mathrm{Vol}}}_{0,1}(b)\,.$
(3.8)

This formula can be seen as a generalization of the result from Saad:2019lba
which computed the boundary-to-boundary two-point function up to genus one, up
to arbitrary genus. Inserting (3.5) into (3.7), using the orthogonality
relation (3.6) and performing the integral over $\ell$ explicitly, we can more
conveniently rewrite (3.7) as

$\begin{split}\langle\Tr_{\beta}&\left(\chi(x_{1})\chi(x_{2})\right)\rangle_{\text{non-
int.}}=\\\ &={4e^{-S_{0}}\over Z(\beta)}\int_{0}^{\infty}dE_{1}dE_{2}\
\left\langle\rho(E_{1})\rho(E_{2})\right\rangle^{\text{JT}}e^{-E_{1}(x_{1}-x_{2})-E_{2}(\beta-
x_{1}+x_{2})}\,{{\mathcal{M}}}_{{\Delta}}(E_{1},E_{2})\,,\\\
{{\mathcal{M}}}_{{\Delta}}(E_{1},E_{2})&={\absolutevalue{{{\Gamma}}\left({{\Delta}}+i(s_{1}+s_{2})\right)}^{2}\over
2^{2\Delta+1}}\times{\absolutevalue{{{\Gamma}}\left({{\Delta}}+i(s_{1}-s_{2})\right)}^{2}\over{{\Gamma}}(2{{\Delta}})}\,,\hskip
14.22636pt\text{with}\hskip
14.22636pts_{1,2}\equiv\sqrt{2E_{1,2}}\,.\end{split}$ (3.9)

where $\left\langle\rho(E_{1})\rho(E_{2})\right\rangle^{\text{JT}}$ is the
spectral two-point function in JT gravity. The leading disk contribution of
(3.9) has been derived using a variety of techniques Yang:2018gdb ;
Mertens:2017mtv ; Blommaert:2018oro ; Kitaev:2018wpr ; Iliesiu:2019xuh , while
the arbitrary genus generalization found in (3.9) was hinted at by studying
two-point function insertions directly in the matrix integral dual to JT
gravity Saad:2019pqd ; Blommaert:2020seb . In these works
${{\mathcal{M}}}_{{\Delta}}(E_{1},E_{2})$ was identified with the squared
matrix element $\absolutevalue{\chi}^{2}_{E_{1},E_{2}}=\absolutevalue{\langle
E_{1}|\chi|E_{2}\rangle}^{2}$, evaluated in the energy basis. The formula
(3.9) can also be derived using a matrix model description of JT gravity
coupled to a scalar field jafferismatrix .121212We thank Daniel Jafferis for
explaining this to us.

The spectral two-point function has been the subject of intense study in JT
gravity and its matrix integral dual Saad:2019lba ; Blommaert:2019wfy ;
Blommaert:2020seb  (and in more general matrix integrals: see mehta2004random
; Guhr:1997ve ; millerRMT  for useful reviews) and, as we shall review
shortly, when $E_{1}\to E_{2}$, it can be estimated even at the non-
perturbative level.

A small consistency check of this formula is that for ${{\Delta}}=0$, at
leading order in the genus expansion we should get
$\langle\chi(x_{1})\chi(x_{2})\rangle_{\text{non-int.}}^{{{\Delta}}\to 0}=1$,
since for the disk there is a unique geodesic.131313At higher genus in JT
gravity there are an infinite number of geodesics and therefore the insertion
of the sum over geodesics is not equivalent to the insertion of the identity
operator. Indeed,

$\begin{split}{{\mathcal{M}}}_{0}(s_{1},s_{2})&={\absolutevalue{{{\Gamma}}\left(i(s_{1}+s_{2})\right)}^{2}\over
2}\times 2\pi{{\delta}}(s_{1}-s_{2})={{{\delta}}(s_{1}-s_{2})\over
4\,r\left({s_{1}+s_{2}\over 2}\right)}\,,\\\ r(s)&\equiv{s\over
2\pi^{2}}\sinh(2\pi s)\,,\end{split}$ (3.10)

where the latter coincides with the leading order density of states in JT
gravity, $\rho^{\text{(JT)}}_{0}(s)=e^{S_{0}}r(s)$, with
$\rho^{\text{(JT)}}_{0}(s)ds=e^{S_{0}}r(s)ds=e^{S_{0}}\rho_{\text{Disk}}(E)dE$.141414Even
though $r(s)$ happens to be related to the JT density of states, it will turn
out to have a more general role in our story and we will therefore denote it
differently.  Since to leading order in $e^{S_{0}}$ (LO) the spectral two-
point function factorizes, we get

$\begin{split}\langle\chi(x_{1})\chi(x_{2})\rangle^{\,\Delta\to 0}_{\text{non-
int.}}|_{\text{LO}}&={4e^{-S_{0}}\over
Z_{0}(\beta)}\int_{0}^{\infty}ds_{1}ds_{2}\
\rho_{0}^{\text{(JT)}}(s_{1})\rho_{0}^{\text{(JT)}}(s_{2})e^{-\frac{1}{2}\left(s_{1}^{2}(x_{1}-x_{2})+s_{2}^{2}(\beta-
x_{1}+x_{2})\right)}\,{{{\delta}}(s_{1}-s_{2})\over
4\,r\left({s_{1}+s_{2}\over 2}\right)}\\\ &={1\over
Z_{0}(\beta)}\int_{0}^{\infty}ds_{1}\
\rho_{0}^{\text{(JT)}}(s_{1})\exp\left[-\beta{s_{1}^{2}\over
2}\right]=1\,.\end{split}$ (3.11)

Finally, we should mention that the above analysis can be extended for more
general theories of dilaton gravity, with a dilaton potential $V(\phi)$, such
as those considered in Witten:2020wvy ; Maxfield:2020ale . As we shall explain
shortly, this will be particularly useful in exhibiting the generality of our
results for the time dependence of the ER bridge. For a particular class of
dilaton potentials, these theories were shown to have a matrix integral dual,
with a matrix potential that is modified from the one encountered in JT
gravity. This results in a modified spectral two-point function
$\left\langle\rho(E_{1})\rho(E_{2})\right\rangle^{V(\phi)}$. In appendix A, we
show that, by restricting to the same class of dilaton potentials as those
considered in Witten:2020wvy ; Maxfield:2020ale , the above result for the
two-point function that only accounts for non-intersecting geodesics (3.9),
still holds in these more general class of dilaton gravity theories, provided
we replace $\left\langle\rho(E_{1})\rho(E_{2})\right\rangle^{JT}$ with
$\left\langle\rho(E_{1})\rho(E_{2})\right\rangle^{V(\phi)}$. Since (3.9) is
now correct for more generic matrix integrals, we will use
$\left\langle\rho(E_{1})\rho(E_{2})\right\rangle$ to denote the spectral two-
point function in a generic one-cut matrix integral.151515The attentive reader
may have noticed that (3.11) does not give $1$ for dilaton gravities beyond
JT, since we get ${1\over Z_{0}(\beta)}\int_{0}^{\infty}ds\
{\rho_{0}(s)^{2}\over r(s)}\exp\left[-\beta{s^{2}\over 2}\right]\neq 1$. The
reason for this is that inserting defects to the disk geometry leads to
multiple geodesics, hence this quantity gives an average of the number of
geodesics weighted by the number of defect insertions. Nevertheless we
continue to normalize $\expectationvalue{\ell(t)}$ with $Z(\beta)$, since it
is the norm of the Hartle-Hawking state, and not by
$\langle\chi(x_{1})\chi(x_{2})\rangle^{\,\Delta\to 0}_{\text{non-int.}}$.

##  4 The length of the Einstein-Rosen bridge in dilaton gravity

We are primarily interested in the time dependence of

$\langle\ell(t)\rangle\equiv\langle
Tr_{\beta}(\hat{\ell})\rangle\equiv-\lim_{{{\Delta}}\to
0}{\frac{\partial\left\langle\Tr_{\beta}\left(\chi\left(\frac{\beta}{2}+it\right)\chi\left(0\right)\right)\right\rangle_{\text{non-
int.}}}{\partial{{\Delta}}}}$ (4.1)

Since $\left\langle\chi(x_{1})\chi(x_{2})\right\rangle_{\text{non-int.}}$ only
depends on ${{\Delta}}$ through ${{\mathcal{M}}}_{{\Delta}}$, we take the
${{\Delta}}$ derivative inside the integral in (3.9) and evaluate it on
${{\mathcal{M}}}_{{\Delta}}$:

$\begin{split}\omega&\equiv s_{1}-s_{2}\,,\qquad\bar{s}\equiv{s_{1}+s_{2}\over
2}\,,\\\ -\lim_{{{\Delta}}\to
0}{\partial{{\mathcal{M}}}_{{\Delta}}\over\partial{{\Delta}}}&=\\#{{\delta}}(\omega)-{1\over
16\pi^{2}r(\bar{s})r\left({\omega\over 2}\right)}\,.\end{split}$ (4.2)

From the exercise in (3.11) we learned that the ${{\delta}}(\omega)$ only
gives time independent contributions, which we do not compute
explicitly.161616This constant piece actually diverges as $1/{{\Delta}}$. In
the computation where we take ${{\Delta}}\to 0$ first, this divergence shows
up as $[{{\delta}}(\omega)]^{2}$, where one factor of ${{\delta}}(\omega)$
comes from (4.2), while the other comes from a contact term in
$\left\langle\rho(s_{1})\rho(s_{2})\right\rangle$, see (4.4). A divergence was
expected from the start: we had to work with a regularized sum (2.1) because
higher genus surfaces have infinitely many geodesics of increasing length, and
when the regulator ${{\Delta}}$ is removed a divergence arises.  Finally, we
obtain a compact expression for $\langle\ell(t)\rangle$:

$\begin{split}\langle\ell(t)\rangle=\text{const}-{e^{-S_{0}}\over
4\pi^{2}Z(\beta)}\int_{0}^{\infty}ds_{1}ds_{2}\
{\left\langle\rho(s_{1})\rho(s_{2})\right\rangle\over
r(\bar{s})r\left({\omega\over
2}\right)}\exp\left[-\beta\left({\bar{s}^{2}\over 2}+{\omega^{2}\over
8}\right)-{i\bar{s}\,\omega t}\right]\,.\end{split}$ (4.3)

For all one-cut matrix integrals, including those that describe the theories
of dilaton gravity discussed in section 3, the form of
$\left\langle\rho(s_{1})\rho(s_{2})\right\rangle$ is universal as $s_{1}\to
s_{2}$ ($E_{1}\to E_{2}$). Ref. Saad:2019lba  give the following approximate
formula for the spectral correlations:

$\begin{split}\left\langle\rho(s_{1})\rho(s_{2})\right\rangle\,=\,\,\underbrace{e^{2S_{0}}\hat{\rho}(s_{1})\hat{\rho}(s_{2})}_{\begin{subarray}{c}\text{Semi-
classical}\\\ \text{disconnected
piece}\end{subarray}}\,\,\,+\,\,\,\underbrace{e^{S_{0}}\hat{\rho}(\bar{s}){{\delta}}(\omega)}_{\text{Contact
term}}\,\,-\,\,\underbrace{{1\over\pi^{2}\omega^{2}}\,\sin^{2}\left(e^{S_{0}}\pi\hat{\rho}(\bar{s})\,\omega\right)}_{\text{Non-
perturbative contribution}}\,,\end{split}$ (4.4)

where we made all factors of $e^{S_{0}}$ manifest by defining
$\rho_{0}(s)=e^{S_{0}}\hat{\rho}(s)$, we used that $E=s^{2}/2$ and $\rho$
transforms as a density, and in the last term that
$\absolutevalue{s_{1}-s_{2}}\ll 1$. While there could be different non-
perturbative completions for the matrix model associated to JT or other models
of dilaton gravity, we emphasize that the formula (4.4) is universal and
common among all non-perturbative completions. Furthermore, given that (4.3)
only depends on the spectral two-point function, one could in principle
compute it even beyond random matrix theory in any quantum mechanical system
and we will speculate further about the role that it has in chaotic systems in
our discussion.

We can now perform the integrals in (4.3). The second term in (4.4) gives a
$t$-independent contribution, and can be absorbed into the constant term. In
fact this constant is divergent, since it is proportional to $\int d\omega\
{{\delta}}(\omega)/\omega^{2}$. As previously mentioned, a convenient
renormalization prescription is to consider the quantity $\ell(t)-\ell(0)$,
which is free of divergences. While this prescription will play a very
important role in the computation of the variance in section 6, for ease of
presentation here we will proceed without keeping track of the time-
independent constant.

The contribution of the first term to the length of the ER bridge was
considered in Yang:2018gdb , where it was evaluated using a saddle point
approximation and found to grow linearly in time. We proceed differently.
First, we notice that because of the $\exp\left[-{i\bar{s}\,\omega t}\right]$
factor for large $t$ the integral will be dominated by $\omega\sim 1/t$, and
implement this through the rescaling $\omega=\hat{\omega}/t$. Using that
$r\left({\omega\over 2}\right)\approx\hat{\omega}^{2}/(4\pi t^{2})$, also
defining $t=e^{S_{0}}\hat{t}$ and dropping the bar from $s$, we arrive at the
expression valid at leading order in $t$ for both nontrivial terms in (4.4):

$\begin{split}\langle\ell(t)\rangle=\text{const}-{e^{2S_{0}}\hat{t}\over\pi
Z_{0}(\beta)}\int_{0}^{\infty}d{s}\int_{-\infty}^{\infty}&d\hat{\omega}\
{\hat{\rho}^{2}({s})\over
r(s)\hat{\omega}^{2}}e^{-\beta{{s}^{2}/2}-{i{s}\,\hat{\omega}}}\\\
&\times\bigg{[}\underbrace{1}_{\begin{subarray}{c}\text{Semi-classical}\\\
\text{disconnected}\\\
\text{piece}\end{subarray}}-\,\,\underbrace{\left({\hat{t}\over\pi\hat{\rho}({s})\hat{\omega}}\right)^{2}\,\sin^{2}\left({\pi\hat{\rho}({s})\,\hat{\omega}\over\hat{t}}\right)}_{\text{Non-
perturbative contribution}}\bigg{]}\,.\end{split}$ (4.5)

The $\hat{\omega}$ integral can now be evaluated exactly. Remarkably, the
semi-classical and non-perturbative contributions conspire such that the
integrand is regular at $\hat{\omega}=0$. Because of this remarkable
cancellation, for $\pi\frac{\hat{\rho}({s})}{s}<\hat{t}$ (i.e.
$\sinh(2\pi{s})<\pi\hat{t}$ in the case of JT gravity) the integral can be
evaluated by closing the contour for the $\hat{\omega}$ integral using the arc
at complex infinity in the upper-half plane. Since the arc does not contribute
to the contour integral and the contour does not enclose any poles, we
conclude that for $\pi\frac{\hat{\rho}({s})}{s}<\hat{t}$ the $\hat{\omega}$
integral vanishes for any density of states $\hat{\rho}(s)$, hence any dilaton
gravity model.

For $\pi\frac{\hat{\rho}({s})}{s}>\hat{t}$ one can start by slightly deforming
the contour of $\hat{\omega}$ in the lower half-plane to avoid the point
$\hat{\omega}=0$.171717This does not have any effect on the value of the
integral since the integrand is regular at $\hat{\omega}=0$.  Expanding the
sine-kernel into exponentials, one can now enclose the $\hat{\omega}$ integral
in the upper or lower half plane in such a way that the arc at complex
infinity never contributes; i.e. for terms containing $e^{i\\#\omega}$ with
$\\#<0$ the contour is enclosed in the lower-half plane while for $\\#>0$ it
is enclosed in the upper half plane. Each term can contain a pole at
$\hat{\omega}=0$ however only the integrals enclosing the upper-half plane
receive contributions from the pole. Adding the contribution of the
appropriate residues we find that

$\begin{split}\langle\ell(t)\rangle&=\text{const}-{2\pi e^{2S_{0}}\over
3Z_{0}(\beta)}\int_{s_{*}(\hat{t})}^{\infty}d{s}\ {\hat{\rho}^{3}(s)\over
r(s)}\exp\left[-\beta{{s}^{2}/2}\right]\left(1-X(\hat{t},s)\right)^{3}\,,\\\
X(\hat{t},s)&\equiv{s\hat{t}\over 2\pi\hat{\rho}(s)}\,,\qquad
Z_{0}(\beta)={e^{S_{0}}\int_{0}^{\infty}ds\hat{\rho}(s)e^{-\beta s^{2}\over
2}}\end{split}$ (4.6)

where $s_{*}(\hat{t})$ is such that
$\pi\frac{\hat{\rho}({s_{*}(\hat{t})})}{s_{*}(\hat{t})}=\hat{t}$. In the
specific case of JT gravity,

$X(\hat{t},s)\equiv{\pi\hat{t}\over\sinh(2\pi{s})}\,,\qquad
s_{*}(\hat{t})={1\over 2\pi}{\rm arcsinh}(\pi\hat{t})\,.$ (4.7)

We can easily evaluate this expression for $\langle\ell(t)\rangle$
numerically, see Fig. 1 for the case of JT gravity.

We can also compute the asymptotics of $\langle\ell(t)\rangle$ for small and
large $\hat{t}$. To be concrete, we can consider the particular example of JT
gravity even though the asymptotics for small and large times will
qualitatively be the same.181818The large asymptotics will be the same if
$s_{*}(\hat{t})\to\infty$ for $\hat{t}\to\infty$. This occurs if
$\hat{\rho}(s)$ grows faster than linearly as $s\to\infty$.  For large
$\hat{t}$ the integral goes to zero, since the lower limit of the integral
$s_{*}(\hat{t})\to\infty$ and the integrand decays as
$\exp\left[-\beta{{s}^{2}/2}\right]$ for large $s$.191919 Note that $0\leq
X(s)\leq 1$ and hence $0\leq\left(1-X(s)\right)^{3}\leq 1$, and
$\hat{\rho}^{2}(s)\sim\exp(4\pi s)$. The leading late time asymptotics is
obtained by replacing the $\sinh$ functions with exponentials, and evaluating
the integral explicitly to obtain:

$\begin{split}\langle\ell(t)\rangle&=\text{const}-e^{S_{0}}A\,\frac{\hat{t}^{2}\exp\left[-\frac{\beta\log^{2}(2\pi\hat{t})}{8\pi^{2}}\right]}{\log^{2}(2\pi\hat{t})}+\dots\,,\\\
A&=\frac{32\sqrt{2}\,\pi^{9/2}e^{-{2\pi^{2}/\beta}}}{\beta^{5/2}}\end{split}$
(4.8)

At small times, we evaluate the integral by changing variables to
$\xi=\hat{t}X(s)$, expanding the resulting integrand for small $\hat{t}$ and
evaluating the resulting integrals. We get that
$\langle\ell(t)\rangle=\text{const}-C_{0}+\hat{C}_{1}\hat{t}+O(\hat{t}^{2})$.
We then absorb $C_{0}$ in const to obtain the final result:

$\begin{split}\langle\ell(t)-\ell(0)\rangle\approx\begin{cases}\hat{C}_{1}\hat{t}+\dots\qquad&\hat{t}\ll
1\,,\\\
C_{0}-e^{S_{0}}A\,\frac{\hat{t}^{2}\exp\left[-\frac{\beta\log^{2}(2\pi\hat{t})}{8\pi^{2}}\right]}{\log^{2}(2\pi\hat{t})}+\dots\qquad&\hat{t}\gg
1\,,\end{cases}\end{split}$ (4.9)

where the explicit formulas for the constants are:

$\begin{split}C_{0}&={e^{S_{0}}\over
12\pi^{2}}\left[e^{{6\pi^{2}/\beta}}\left(1+{16\pi^{2}\over\beta}\right)-e^{-{2\pi^{2}/\beta}}\right]\,,\\\
\hat{C}_{1}&=e^{S_{0}}\left[{e^{-{2\pi^{2}/\beta}}\over\sqrt{\pi\beta/2}}+{{\rm
erf}\left(\sqrt{2}\pi\over\sqrt{\beta}\right)\over
2\pi}\left(1+{4\pi^{2}\over\beta}\right)\right]\,.\end{split}$ (4.10)

In particular, the early-time linear growth is controlled by $\hat{C}_{1}$. At
temperatures high compared to the boundary value of the dilaton (which we have
set to one), we have $\hat{C}_{1}=2\pi e^{S_{0}}/\beta$, and hence
$\langle\ell(t)\rangle\approx\frac{2\pi}{\beta}t$, which is the expected
linear growth of the extremal volume in the classical regime Harlow:2018tqv
.202020By early-time here we mean $\hat{t}=e^{-S_{0}}t\ll 1$, but $t\gg 1$.
Transient effects at $t\sim\beta$ have been dropped in (4.5). At low
temperatures, the Schwarzian theory becomes strongly coupled, and the early
time growth coefficient receives loop corrections. In particular, at low
temperatures, $\langle\ell(t)\rangle\approx\frac{4}{\sqrt{2\pi\beta}}t$ at
early times.

##  5 A comparison to the spectral-form factor and the two-point function

It is insightful to compare $\langle\ell(t)\rangle$ to other related
quantities, the spectral form factor (SFF) and the two-point function. The
approximation to the two-point function is given in (3.9), while the spectral
form factor (with inverse temperature $\beta/2$) is proportional to the same
formula with the replacement ${{\mathcal{M}}}_{{\Delta}}(s_{1},s_{2})\to 1$.
Applying the same logic as for $\langle\ell(t)\rangle$, we can expand the
integrand for small $\omega=\hat{\omega}/t$, and obtain the formula as in
(4.5)

$\begin{split}F(t)\propto\text{const}+\int_{0}^{\infty}d{s}\int_{-\infty}^{\infty}d\hat{\omega}\
&\hat{\rho}^{2}({s})\left[1-\left({\hat{t}\over\pi\hat{\rho}({s})\hat{\omega}}\right)^{2}\,\sin^{2}\left({\pi\hat{\rho}({s})\,\hat{\omega}\over\hat{t}}\right)\right]e^{-\beta{{s}^{2}/2}-{i{s}\,\hat{\omega}}}\,\,\times\\\
&\times\,\begin{cases}{1\over r(s)\hat{\omega}^{2}}\hskip 82.51282pt\text{for
$\langle\ell(t)\rangle$,}\\\ 1\qquad\hskip 82.51282pt\text{for
SFF${}_{\beta/2}(t)$,}\\\
{\absolutevalue{{{\Gamma}}\left({{\Delta}}+2is\right)}^{2}\over
2^{2\Delta+1}}\times{{{\Gamma}}\left({{\Delta}}\right)^{2}\over{{\Gamma}}(2{{\Delta}})}\qquad\text{for
$\expectationvalue{\chi_{L}(t)\chi_{R}(0)}$.}\end{cases}\end{split}$ (5.1)

While these formulas are very similar, the presence of the
$1/\hat{\omega}^{2}$ singularity in the integrand of $\langle\ell(t)\rangle$
leads to dramatic differences from the case of the SFF and the two-point
function. For these latter two, the disconnected contribution represented by
the $1$ in the brackets vanishes after performing the $\hat{\omega}$
integral.212121The $\hat{\omega}$ integral gives a ${{\delta}}(s)$, which gets
multiplied by an $s$ dependent function that vanishes at $s=0$. Famously, it
is the sine kernel that gives rise to both the ramp and plateau in the SFF and
the two-point function. In contrast, for $\langle\ell(t)\rangle$, the initial
linear growth was given by the disconnected contribution, and the role of the
sine kernel was to cancel this contribution at very late times, giving rise to
the plateau.222222 One may wonder whether our approximation scheme can capture
the slope of the SFF through the disconnected contribution to the spectral
correlator. While, as explained above, at leading order in the
$\hat{\omega}/t$ expansion we got zero, at fourth-order in this expansion we
encounter the Fourier integral $\int d\hat{\omega}\
\hat{\omega}^{4}\exp\left[-{i{s}\,\hat{\omega}}\right]=2\pi{{\delta}}^{(4)}(s)$,
which integrated against $\hat{\rho}^{2}({s})=s^{4}/\pi+\dots$ gives the first
nonzero contribution. This is indeed the $1/t^{3}$ asymptotics of the slope of
the SFF and higher orders of the $1/t$ expansion can be similarly recovered.

Curiously, after adding the disconnected and sine kernel contributions for all
our quantities, we again get very similar $s$ integrals. Compare

$\begin{split}F(t)&\propto\text{const}-\int_{s_{*}(\hat{t})}^{\infty}d{s}\
\hat{\rho}^{2}(s)\exp\left[-\beta{{s}^{2}/2}\right]\left(1-X(\hat{t},s)\right)\times\begin{cases}1\qquad\hskip
82.51282pt\text{for SFF${}_{\beta/2}(t)$,}\\\
{\absolutevalue{{{\Gamma}}\left({{\Delta}}+2is\right)}^{2}\over
2^{2\Delta+1}}\times{{{\Gamma}}\left({{\Delta}}\right)^{2}\over{{\Gamma}}(2{{\Delta}})}\qquad\text{for
$\expectationvalue{\chi_{L}(t)\chi_{R}(0)}$,}\end{cases}\end{split}$ (5.2)

to (4.6). The close connection between the behavior of the SFF and the two
point function was emphasized in Saad:2019pqd ; Cotler:2019egt .

Figure 3: We show $\langle\ell(t)\rangle_{\text{micro}}$ (blue) and
$\text{SFF}_{\text{micro}}(t)$ (orange) for JT gravity with $s=1$ (or
$E=1/2$). We are plotting the expressions in (5.5), so we have dropped
inessential prefactors. $\langle\ell(t)\rangle$ is a cubic polynomial in time,
while the SFF is its second time derivative (up to the addition of a
constant). We have normalized $\langle\ell(t)\rangle_{\text{micro}}$ and
$\text{SFF}_{\text{micro}}(t)$ in such a way that their plateau values are the
same.

We also note that we were surprised to find how slowly the integrals (4.6) and
(5.2) reach their asymptotic values for small $\beta$. It is easy to repeat
the analysis leading to (4.8), and we find
$\exp\left[-\frac{\beta\log^{2}(2\pi\hat{t})}{8\pi^{2}}\right]$ asymptotics
for all three quantities, as has been found before for the spectral form
factor in the SYK model in Cotler:2016fpe .

Finally, we analyze the microcanonical versions of these quantities. These are
obtained by replacing the canonical density matrix $e^{-\beta H}$ with
averaging over a microcanonical window $(E-{{\delta}},E+{{\delta}})$ with
${{\delta}}\ll E$. Exploiting the identity

$\begin{split}{1\over 2\pi}\int_{c+i\mathbb{R}}d\beta\ e^{\beta
E+\beta^{2}{{\delta}}^{2}}\,e^{-\beta{\cal E}}=2\pi\,{e^{-(E-{\cal
E})^{2}/(4{{\delta}}^{2})}\over\sqrt{4\pi{{\delta}}^{2}}}\approx{{\delta}}(E-{\cal
E})\,,\end{split}$ (5.3)

we can convert each of our answers obtained in the canonical ensemble (or
thermofield double state) by the integral transformation

$\begin{split}F^{\text{micro}}_{(E,{{\delta}})}(t)=\int_{c+i\mathbb{R}}d\beta\
e^{\beta E+\beta^{2}{{\delta}}^{2}}\,F_{\beta}(t)\,.\end{split}$ (5.4)

Then for all three quantities considered so far we get rid of the remaining
$s$ integral:

$\begin{split}\expectationvalue{\ell(t)}_{(E,{{\delta}})}&\propto\text{const}-\Theta(s-s_{*}(\hat{t}))\,{\hat{\rho}^{3}(s)\over
s\,r(s)}\left(1-X(\hat{t},s)\right)^{3}\,,\\\
SFF_{(E,{{\delta}})}(t)&\propto\text{const}-\Theta(s-s_{*}(\hat{t}))\,{\hat{\rho}^{2}(s)\over
s}\left(1-X(\hat{t},s)\right)\,,\\\
\expectationvalue{{{\mathcal{O}}}_{L}(t){{\mathcal{O}}}_{R}(0)}_{(E,{{\delta}})}&\propto\text{const}-\Theta(s-s_{*}(\hat{t}))\,{\hat{\rho}^{2}(s)\over
s}\left(1-X(\hat{t},s)\right){\absolutevalue{{{\Gamma}}\left({{\Delta}}+2is\right)}^{2}\over
2^{2\Delta+1}}\times{{{\Gamma}}\left({{\Delta}}\right)^{2}\over{{\Gamma}}(2{{\Delta}})}\,,\end{split}$
(5.5)

where $E=s^{2}/2$ and the extra $s$ in the denominator comes from the Jacobian
when changing variables from $s$ to $E$ in the integral.232323We note that
$SFF_{(E,{{\delta}})}(t)$ is sometimes denoted by $Y(E,{{\delta}},t)$ in the
literature Saad:2018bqo ; Saad:2019pqd . Both the ER bridge length and the
two-sided correlator are best thought of as being calculated in a
microcanonical thermofield double state (considered also in Stanford:2020wkf )

$\begin{split}\ket{\text{TFD}_{(E,{{\delta}})}}=\int_{c+i\mathbb{R}}d\beta\
e^{\beta E+\beta^{2}{{\delta}}^{2}}\,\ket{\text{TFD}_{\beta}}\,.\end{split}$
(5.6)

Recall that $X(\hat{t},s)$ (defined in (4.6)) is linear in time, hence before
saturation $\expectationvalue{\ell(t)}_{(E,{{\delta}})}$ is a cubic and the
other two quantities are linear functions of time. Up to unimportant $E$ (or
$s$) dependent factors, the microcanonical versions the SFF and the two point
function are the same functions of time, while the SFF is the second time
derivative of $\expectationvalue{\ell(t)}_{(E,{{\delta}})}$ up to the addition
of a constant setting the size of the plateau.242424This can already be
anticipated from (5.1), where in the $\hat{\omega}$ integral the only
difference between the ER bridge length and the SFF is the $1/\omega^{2}$
factor in the former, roughly corresponds to double integration in time. The
integral over $s$ however is nontrivial, so in the canonical ensemble ER
bridge length and the SFF are not related in any simple way. To emphasize this
relationship, we plot these functions in figure 3. Nevertheless, this
relationship does not imply that $\expectationvalue{\ell(t)}_{(E,{{\delta}})}$
exhibits a plateau – instead, the special cancellation discussed above is
necessary for that to occur.

##  6 Variance in the length of Einstein-Rosen bridge

Having computed the time dependence in the length of the ER bridge, we can use
similar techniques to compute its variance. As usual, to compute the variance,
we consider geometries with two asymptotic boundaries, with the geodesics
ending on the two pairs of points located on the two boundaries. However, the
definition of the variance suffers from several ambiguities: (i) in the
connected geometries, it is unclear whether to sum over geodesics which
connect the different asymptotic boundaries, and (ii) since we have restricted
ourselves to geodesics that do not self-intersect, should we also only be
considering geodesics that do not intersect with each other. While we believe
that the qualitative time-dependence of the variance is not heavily affected
by these choices, we will only consider geodesics that go between the same
asymptotic boundaries, and we will restrict to non-intersecting geodesics.

Figure 4: An example (with genus 3) of the type of hyperbolic surfaces which
the path integral (6.2) sums over. The figures represent the cuts
${\mathcal{M}}_{g,2}\to{\mathcal{M}}_{h_{1},1}\oplus{\mathcal{M}}_{h_{2},1}\oplus{\mathcal{M}}_{g-h_{1}-h_{2},2}$
with $h_{1,2}\geq 0$ and $h_{1}+h_{2}\leq g$ (top left),
${\mathcal{M}}_{g,2}\to{\mathcal{M}}_{h,1}\oplus{\mathcal{M}}_{g-h-1,3}$ (top
right),
${\mathcal{M}}_{g,2}\to{\mathcal{M}}_{h,2}\oplus{\mathcal{M}}_{g-h-1,2}$
(bottom left), and ${\mathcal{M}}_{g,2}\to{\mathcal{M}}_{g-2,\,4}$ (bottom
right).

###  6.1 Geometric computation

To probe the variance for the length of the ER bridge we again use a modified
boundary-to-boundary two-point function inserted on each asymptotic boundary,
one for an operator $\chi$, with scaling dimension $\Delta$, and another for
an operator $\chi^{\prime}$, with scaling dimension $\Delta^{\prime}$. We will
denote the pair of boundary-to-boundary two-point functions (with the
assumptions about which geodesics contribute to the correlator as above) by
$\langle\Tr_{\beta}(\chi(x_{1})\chi(x_{2}))\,\Tr_{\beta}(\chi(x_{1}^{\prime})\chi(x_{2}^{\prime}))\rangle_{\text{non-
int.}}$. To compute the variance of the length, we will use

$\begin{split}\sigma_{\langle\ell(t)\rangle}^{2}=\lim_{{\Delta\to
0,\Delta^{\prime}\to
0}}\partial_{\Delta}\partial_{\Delta^{\prime}}\bigg{[}&\langle\Tr_{\beta}\left(\chi(x_{1})\chi(x_{2})\right)\Tr_{\beta}\left(\chi^{\prime}(x_{1})\chi^{\prime}(x_{2})\right)\rangle_{\text{non-
int.}}\\\ &-\,\,\langle\Tr_{\beta}(\chi(x_{1})\chi(x_{2}))\rangle_{\text{non-
int.}}\langle\Tr_{\beta}(\chi^{\prime}(x_{1})\chi^{\prime}(x_{2}))\rangle_{\text{non-
int.}}\bigg{]}\,.\end{split}$ (6.1)

The contribution coming from connected geometries is given by

$\begin{split}\langle\Tr_{\beta}&\left(\chi(x_{1})\chi(x_{2})\right)\Tr_{\beta}\left(\chi^{\prime}(x_{1}^{\prime})\chi^{\prime}(x_{2}^{\prime})\right)\rangle^{\,\text{connected}}_{\,\text{non-
int.}}=\sum_{g}e^{S_{0}(1-2g)}\\\
&\times\int_{\frac{\mathcal{T}_{g,2}}{\text{Mod}({\mathcal{M}}_{g,2})}}\omega\int
D{(\mathcal{W})}D{(\mathcal{W}^{\prime})}e^{-I_{JT,\text{
bdy}}(\mathcal{W})}e^{-I_{JT,\text{
bdy}}(\mathcal{W}^{\prime})}\sum_{\begin{subarray}{c}\gamma\in\mathcal{G}_{x1,x2},\\\
\gamma^{\prime}\in\mathcal{G}_{x1^{\prime},x2^{\prime}},\\\
\gamma\cap\gamma^{\prime}=\emptyset\end{subarray}}e^{-\Delta\ell_{\gamma}-\Delta^{\prime}\ell_{\gamma^{\prime}}}\end{split}$
(6.2)

where the integral over $\mathcal{W}$ and $\mathcal{W}^{\prime}$ are the
integrals over wiggles on the two asymptotic boundaries and the integral over
$\omega$ is over the moduli space of hyperbolic manifolds with two asymptotic
boundaries. Just like we discuss in section 3 for the disconnected case or as
we illustrate through an example in figure 4, cutting along the pair of
geodesics $\gamma$ and $\gamma^{\prime}$ in the sum above, we can either
obtain three or two disconnected manifolds or a single connected component. In
the first case, after cutting,
${\mathcal{M}}_{g,2}\to{\mathcal{M}}_{h_{1},1}\oplus{\mathcal{M}}_{h_{2},1}\oplus{\mathcal{M}}_{g-h_{1}-h_{2},2}$
with $h_{1,2}\geq 0$ and $h_{1}+h_{2}\leq g$, in the second case, we can
either obtain
${\mathcal{M}}_{g,2}\to{\mathcal{M}}_{h,1}\oplus{\mathcal{M}}_{g-h-1,3}$ or
${\mathcal{M}}_{g,2}\to{\mathcal{M}}_{h,2}\oplus{\mathcal{M}}_{g-h-1,2}$, and
finally when obtaining a single connected component
${\mathcal{M}}_{g,2}\to{\mathcal{M}}_{g-2,\,4}$.252525 $g=1$ requires special
consideration, since in the third case we obtain two disconnected manifolds,
thus reducing to the second case.  Once again, the mapping class group acts by
relating all pairs of boundary-to-boundary geodesics within a given cutting
class as well as all closed geodesics (which do not intersect the boundary-to-
boundary geodesics) on the surfaces,
${\mathcal{M}}_{h_{1},1}\oplus{\mathcal{M}}_{h_{2},1}\oplus{\mathcal{M}}_{g-h_{1}-h_{2},2}$
in the first case, ${\mathcal{M}}_{h,1}\oplus{\mathcal{M}}_{g-h-1,3}$ in the
second case, or ${\mathcal{M}}_{g-2,\,4}$ in the third case. Therefore, as
shown in appendix B, we present a generalization of (3.4) for pairs of
boundary-to-boundary geodesics. Putting together the results from that
appendix, we find that

$\begin{split}\langle\Tr_{\beta}&\left(\chi(x_{1})\chi(x_{2})\right)\Tr_{\beta}\left(\chi^{\prime}(x_{1}^{\prime})\chi^{\prime}(x_{2}^{\prime})\right)\rangle_{\,\text{non-
int.}}={4e^{-S_{0}}\over\langle
Z(\beta)Z(\beta)\rangle}\int_{0}^{\infty}dE_{1}\dots dE_{4}\ \\\
&\times\left\langle\rho(E_{1})\rho(E_{2})\rho(E_{3})\rho(E_{4})\right\rangle
e^{-E_{1}x_{12}-E_{2}(\beta-x_{12})-E_{3}x_{12}^{\prime}-E_{4}(\beta-
x_{12}^{\prime})}\,{{\mathcal{M}}}_{{\Delta}}(E_{1},E_{2}){{\mathcal{M}}}_{{{\Delta}}^{\prime}}(E_{3},E_{4})\,,\end{split}$
(6.3)

where $\left\langle\rho(E_{1})\rho(E_{2})\rho(E_{3})\rho(E_{4})\right\rangle$
is the spectral four-point function in JT or dilaton gravity and $x_{ij}\equiv
x_{i}-x_{j}$. By subtracting the product of two-point functions from (6.3) we
obtain the variance. Of course, in the matrix integral interpretation of the
theories of dilaton gravity, (6.3) is the natural candidate for computing the
square of (3.9) in an ensemble average. More generally, as discussed in
appendix B, using the same geometric techniques we can compute any higher
moments for
$\langle\Tr_{\beta}\left(\chi(x_{1})\chi(x_{2})\right)\rangle_{\text{non-
int.}}$.

###  6.2 Evaluating the variance of the ER bridge length

Given that the dual of JT gravity is an ensemble, we want to quantify the size
of the fluctuations in the volume. We therefore wish to evaluate
$\langle\ell(t)^{2}\rangle$ using (6.3). We take $\Delta\neq\Delta^{\prime}$
in the regularization and therefore geodesics connecting distinct boundaries
do not contribute. We take a $\Delta$ and a $\Delta^{\prime}$ derivative of
(6.3) and set $\Delta=\Delta^{\prime}=0$. This results in the average of the
square of complexity $\langle\ell(t)^{2}\rangle$. To simplify the formulas and
to incorporate the renormalization, we define the renormalized volume
“operator”

$\displaystyle\Tr_{\beta}(\hat{\ell}(t)-\hat{\ell}(0))$ $\displaystyle=\int
dE_{1}dE_{1}\rho(E_{1})\rho(E_{2})G(E_{1},E_{2})$ (6.4)

where the time dependence is in the function $G(E,E^{\prime})$ and it reads as

$\begin{split}G(E_{1},E_{2})=-{e^{-S_{0}}\over 4\pi^{2}Z(\beta)}\ {1\over
r(\bar{s})r\left({\omega\over
2}\right)}\exp\left[-\beta\left({\bar{s}^{2}\over 2}+{\omega^{2}\over
8}\right)\right]\left[\cos(\bar{s}\,\omega t)-1\right].\end{split}$ (6.5)

This kernel is obtained from formula (4.3), after we have symmetrized the
integrand in $\omega=s_{1}-s_{2}$ (the antisymmetric part does not
contribute). Finally, we made a time independent subtraction so that the
operator is $0$ at $t=0$. This takes care of all the divergences, which we
have seen to be time independent. Physically, it means that we set the volume
of the ER bridge to be zero at zero time in all members of the matrix
ensemble.

Now we want to evaluate
$\langle[\Tr_{\beta}(\hat{\ell}(t)-\hat{\ell}(0))]^{2}\rangle$. We need the
four point density correlation in order to do this. In general, for a one
matrix model, the $n$ eigenvalue marginals are given in terms of the
Christoffel-Darboux kernel $K(E_{1},E_{2})$ (see millerRMT  for a concise
pedagogical review):

$\displaystyle R_{k}(E_{1},...,E_{k})\equiv\frac{N!}{(N-k)!}\int
dE_{k+1}...dE_{N}P(E_{1},...,E_{N})=\det K(E_{i},E_{j})|_{i,j=1,...,k},$ (6.6)

for instance $\langle\rho(E)\rangle=R_{1}(E)=K(E,E)$ and

$\displaystyle\langle\rho(E_{1})\rho(E_{2})\rangle$
$\displaystyle=R_{2}(E_{1},E_{2})+\delta(E_{1}-E_{2})R_{1}(E_{1})$ (6.7)
$\displaystyle=\langle\rho(E_{1})\rangle\langle\rho(E_{2})\rangle+\langle\rho(E_{1})\rangle\delta(E_{1}-E_{2})-K(E_{1},E_{2})^{2}.$

From this and (4.4), we read the kernel in the scaling regime $E_{1}\to
E_{2}$, relevant for the sine kernel:

$\displaystyle
K(s_{1},s_{2})\approx{1\over\pi\omega}\,\sin\left(e^{S_{0}}\pi\hat{\rho}(\bar{s})\,\omega\right),\quad\quad\omega=s_{1}-s_{2}.$
(6.8)

The easiest way to deal with contact terms in density correlators is to write
(6.4) as a sum over the discrete spectrum in a single member of the matrix
ensemble

$\displaystyle\Tr_{\beta}(\hat{\ell}(t)-\hat{\ell}(0))$
$\displaystyle=\sum_{i,j|E_{i}\neq E_{j}}G(E_{i},E_{j})$ (6.9)

and directly integrate this expression against the joint distribution
$P(E_{1},...,E_{N})$. Then we can use that the function $G$ is symmetric for
exchange of its variables, and the $P$ and the $R_{k}$ are also totally
symmetric. This way, we get that

$\displaystyle\sigma_{\ell}^{2}$ $\displaystyle=\int
dE_{1}...dE_{4}[R_{4}(E_{1},...,E_{4})-R_{2}(E_{1},E_{2})R_{2}(E_{3},E_{4})]G(E_{1},E_{2})G(E_{3},E4)$
(6.10) $\displaystyle+4\int
dE_{1}dE_{2}dE_{3}R_{3}(E_{1},E_{2},E_{3})G(E_{1},E_{2})G(E_{2},E_{3})$
$\displaystyle+2\int dE_{1}dE_{2}R_{2}(E_{1},E_{2})G(E_{1},E_{2})^{2}.$

where the second and third lines yield the contribution of contact terms in
$\left\langle\rho(s_{1})\rho(s_{2})\rho(s_{3})\rho(s_{4})\right\rangle$, and
we have already subtracted the disconnected contribution
$\langle\Tr_{\beta}(\hat{\ell}(t)-\hat{\ell}(0))\rangle^{2}$. The dominant
contribution to all these integrals come from the region where all energies
are nearby. Therefore, we can use (6.8) along with (6.6). Evaluating the
integrals at large times results in262626The limit of the last integral is not
a misprint, we changed it from $s_{*}(t)$ to $s_{*}(2t)$ so that the integrand
of the middle term simplifies.

$\displaystyle\sigma_{\ell}^{2}$
$\displaystyle={e^{3S_{0}}\over\pi^{2}Z(\beta)^{2}}\Big{[}\int_{s_{*}(2\hat{t})}^{\infty}ds\frac{e^{-\beta
s^{2}}}{r(s)^{2}}\frac{s^{5}{\hat{t}}^{5}}{60\pi}$ (6.11)
$\displaystyle+\int_{s_{*}(\hat{t})}^{s_{*}(2\hat{t})}ds\frac{e^{-\beta
s^{2}}}{r(s)^{2}}\frac{(2\pi\hat{\rho}-\hat{t}s)^{3}(7s^{2}{\hat{t}}^{2}+2\pi
s\hat{t}\hat{\rho}+8\pi^{2}{\hat{\rho}}^{2})}{60\pi}+\int_{0}^{s_{*}(2\hat{t})}ds\frac{e^{-\beta
s^{2}}}{r(s)^{2}}\frac{2}{15}\pi^{3}{\hat{\rho}}^{4}(5s\hat{t}-7\pi\hat{\rho})\Big{]},$

with $s_{*}(\hat{t})$ as in (4.6). The most important property of this
expression is that for $\hat{t}=e^{-S_{0}}t=O(1)$, it is $O(e^{S_{0}})$. Since
this is the square of the noise, we find that the signal to noise ratio is

$\frac{\sigma_{\ell}}{\ell}\propto e^{-\frac{S_{0}}{2}}\,.$ (6.12)

This is to be contrasted with the spectral form factor, for which this ratio
is order one. At early times $t=O(1)$, the first integral dominates and the
expression starts as $\propto e^{S_{0}}{\hat{t}}^{5}=e^{-4S_{0}}t^{5}$, so at
early times, the volume is reliably calculated by semi-classical gravity, as
expected.272727This is a non-trivial check of our prescription to calculate a
non-perturbative volume by summing over all non-intersecting geodesics. At
late times, the last integral dominates, and we get a linear growth $\propto
e^{S_{0}}\hat{t}$. Somewhat surprisingly, this growth never stops, and the
signal to noise ratio becomes $O(1)$ at $\hat{t}\propto e^{S_{0}}$, or
$t\propto e^{2S_{0}}$. We show this growing noise on the late parts of the
plateau for JT gravity in figure 5.282828Note that we can understand the
source of this linear growth quite explicitly. Is is coming from the last term
in (6.10) involving $R_{2}$. At small $\omega$ relevant for late time,
$R_{2}$, and its contribution to $\sigma_{\ell}^{2}$, read as
$\displaystyle\frac{1}{\omega^{4}}\left(\rho^{2}-\frac{\sin^{2}\pi\rho\omega}{\pi^{2}\omega^{2}}\right)\approx\frac{\pi^{2}\rho^{4}}{3\omega^{2}},\qquad\int
d\omega\frac{\pi^{2}\rho^{4}}{3\omega^{2}}[\cos(s\omega
t)-1]^{2}=\frac{\pi^{3}\rho^{4}}{3}|st|\,,$ (6.13) respectively. This might be
because of the ambiguity in the definition of the variance since in our
calculation we have not included the contribution of intersecting (but not
self-intersecting) geodesics. Such intersecting configurations start existing
for the genus one two-boundary geometry so we might expect them to affect the
very late-time value of the variance.

The findings above present some tension with the interpretation of the volume
as complexity. First, we expect complexity to have time independent
fluctuations until the Poincaré recurrence time, which is
$\propto\exp[\exp[S_{0}]]$. Second, for times $t\propto e^{S_{0}}$, we expect
complexity to have O(1) fluctuations, instead of $e^{S_{0}/2}$ that we have
found here.292929We thank Adam Brown and Leonard Susskind for explaining this
to us. However, we should emphasize that the distribution of $\ell(t)$ seems
to be highly non-Gaussian, since it is dominated by the region where all
energies are nearby. This is opposed to the spectral form factor, where the
four boundary correlator $\langle\rho\rho\rho\rho\rangle$ is well approximated
by Wick contractions (two boundary wormhole contributions), indicating
Gaussian noise. Because the noise is highly non-Gaussian, one should be
careful in interpreting $\sigma_{\ell}$. For example it could come from a
distribution which has fixed noise near the origin, but has a growing bump for
certain very large fluctuations. Resolving this requires one to study higher
moments of $\ell$. This is in principle possible, but we leave it to future
work. In any case, to resolve this tension between the variance of the ER
length and the expected variance of complexity, it would be interesting to
explore all alternate definitions of the variance which we have described
above.

Figure 5: We show $\langle\ell\rangle$ and the band
$\langle\ell\rangle\pm\sigma_{\ell}$, which at long times goes as
$e^{S_{0}}\pm\sqrt{t}$ giving the graph above on a log-log plot. We put
$\beta=15$ and $S_{0}=10$ on this plot. The opening time of the band in
$\hat{t}=e^{-S_{0}}t$ can be made arbitrarily late by increasing $S_{0}$.

##  7 Discussion

###  7.1 Summary

To recapitulate, we have computed the length of the ER bridge at all times. At
early times, this length grows linearly with time, as expected from a semi-
classical analysis.303030Even when accounting for the backreaction of the
operator insertion on the metric Yang:2018gdb . At late times, it saturates at
a time and value both proportional to $e^{S_{0}}$. This saturation is due to a
universal cancellation between the classical contribution and the non-
perturbative corrections to the spectral two-point function. This type of
saturation is distinct from the cancellation between the leading wormhole
geometry and the non-perturbative corrections that is responsible for the
plateau in the spectral form factor and in the probe matter two-point
function. Also, as opposed to these latter quantities, we have found that for
the length of the ER bridge, the “noise” on the plateau is much smaller than
the signal, i.e. $\sigma_{\ell}\ll\ell$ for $t\propto e^{S_{0}}$. However, we
have also found that $\sigma_{\ell}$ grows forever and that it becomes the
same size as the signal at $t\propto e^{2S_{0}}$. This raises some challenges
in the interpretation of the volume as complexity. A possible resolution is
that our geometric prescription to compute the variance breaks down at very
late times. Indeed, in summing over pairs geodesics in two boundary
geometries, we made a choice to only include geodesic pairs that do not
intersect. It is possible that the contribution of these intersecting
geodesics would change the picture at very late times and it would be
interesting to investigate this in the future.

###  7.2 A new spectral quantity

Regarding the relation between the length of the ER bridge and complexity, one
can take several different perspectives. The first perspective, which we
present in this subsection, is that we can use the results in this paper to
define the dual for the volume of the ER bridge in the boundary theory. Since
our final formula solely depends on the spectral data in the matrix ensemble
dual to dilaton gravity, one can imagine defining a similar quantity to (4.3)
for any quantum system. Explicitly, for systems with discrete energy levels,
one can define313131Note that we replaced the function $r(s_{1}-s_{2})$ in the
denominator of (4.3) simply by $(E_{1}-E_{2})^{2}$ since, at sufficiently late
times both (4.3) and (7.1) are dominated by closely spaced energy pairs.
Clearly, the spectral complexity can be defined for any ensemble of theories
by averaging (7.1).

$\begin{split}\mathcal{C}(t)\sim\sum_{E_{1}\neq
E_{2}}{1\over(E_{1}-E_{2})^{2}}\exp\left[-\frac{\beta}{2}\left(E_{1}+E_{2}\right)\right]\left(\cos\left[{(E_{1}-E_{2})t}\right]-1\right)\,,\end{split}$
(7.1)

where we changed from the $s$ parametrization of energy to energy itself,
which also changes the argument of the cosine compared to previous formulas
(e.g. (6.5)). We shall call this quantity the spectral complexity of the
thermofield double state. We believe that most chaotic systems have spectral
statistics which closely follow that of random matrices, and hence, we expect
$\mathcal{C}(t)$ to have a similar behavior to the ER length computed in this
paper. Once again, we expect it to start with a linear growth and, at a late
time (order $e^{S}$), way past the thermalization time, asymptotes to a
constant. Thus, just like the complexity defined in Susskind:2014rva ;
Susskind:2014moa , $\mathcal{C}(t)$ grows past when conventional probes reach
their thermal equilibrium.

While $\mathcal{C}(t)$ is defined when preparing the $t=0$ state by the
Hartle-Hawking prescription and is given by the thermofield double state in
the boundary theory, one can also consider generalizing the formula for the
length of the ER bridge for different states that are prepared by the
Euclidean path integral with different boundary conditions. In turn, this
would lead to a generalization of (7.1) for the spectral complexity of various
states in the boundary theory. For instance, one generalization already
discussed in this paper is described in section 5 where we have defined the
length of the ER bridge in the microcanonical ensemble, i.e., when in gravity
one fixes $\partial_{n}\phi$, instead of its canonical conjugate
$\sqrt{g_{uu}}$ on the boundary of the dilaton gravity theory Goel:2020yxl .

###  7.3 Other Dyson ensembles

Figure 6: We show $\mathcal{C}_{\text{micro}}(t)$ and
$\text{SFF}_{\text{micro}}(t)$ for GOE (blue), GUE (orange) and GSE (green).

We can generalize the results in this paper to compute the spectral complexity
in any random matrix ensemble. The spectral correlators of dilaton gravity
theories investigated above are universally governed by the Gaussian unitary
ensemble (GUE) for small energy separations (or late times). It is an
interesting question to ask, what is the behavior of $\mathcal{C}(t)$ in the
other orthogonal (GOE) and symplectic (GSE) ensembles. While these ensembles
are related to versions of JT gravity that includes a sum over unoriented
surfaces, one encounters divergences for higher genus surfaces
Stanford:2019vob . At the level of the matrix integral this appears due to the
double-scaling limit necessary in order to compute observables in two-
dimensional gravity. Unless this divergence can somehow be resolved, this
means that the computation below will not have the obvious interpretation of
an ER bridge length in a theory of gravity. Nevertheless, it is interesting to
understand the behavior of this quantity in matrix models.

Towards that end, it is sufficient to work with the microcanonical version of
the spectral complexity or ER bridge length discussed around (5.5). Up to
inessential overall factors this is given by

$\begin{split}\mathcal{C}_{\text{micro}}(t)=-\int_{-\infty}^{\infty}&d\omega\
{\expectationvalue{\rho(E+\omega/2)\rho(E-\omega/2)}\over{\omega}^{2}}\left[\cos({\omega}t)-1\right]\,.\end{split}$
(7.2)

Comparing with the microcanonical SFF

$\begin{split}\text{SFF}_{\text{micro}}(t)=\int_{-\infty}^{\infty}&d\omega\
{\expectationvalue{\rho(E+\omega/2)\rho(E-\omega/2)}}\left[\cos({\omega}t)-1\right]\,,\end{split}$
(7.3)

it is clear that

$\begin{split}\text{SFF}_{\text{micro}}(t)=\text{const}+{d^{2}\over
dt^{2}}\mathcal{C}_{\text{micro}}(t)\,,\end{split}$ (7.4)

as was already discussed in section 5. We give the density two point functions
and the explicit formulas for both of these quantities in appendix C, and plot
them on figure 6. Curiously, $\mathcal{C}_{\text{micro}}$ does not lead to a
plateau in GOE, but grows logarithmically forever as
$\expectationvalue{\rho(E)}\pi^{2}\log(t)/3$. This can be anticipated from the
fact that, while the SFF in GUE and GSE reaches a constant at finite $t$, in
GOE its asymptotics is
$\text{SFF}_{\text{micro}}(t)=\expectationvalue{\rho(E)}\left[1-\pi^{2}/(3t^{2})+\dots\right]$
and combining with the relation (7.4). Note however, that for all ensembles
there is still a significant cancellation between the disconnected piece of
the density two point function giving linear growth and the connected piece
cancelling this at late times with the very slow asymptotic logarithmic growth
remaining. Nevertheless, it is important to better understand the significance
of this logarithmic growth in UV complete gravitational theories that can be
related to individual members in the GOE ensemble.323232It would also be
interesting to study the behavior of the spectral complexity in supersymmetric
ensembles altland1997nonstandard , which should capture the behavior of the ER
birdge in theories of supergravity Stanford:2019vob ; Heydeman:2020hhw ;
Fan:2021wsb . A first step in that direction was taken in Fan:2021wsb  who
showed that the length of the ER birdge also grows linearly with time in
$\mathcal{N}=1$ JT super-gravity.

###  7.4 The quantity in SYK

Figure 7: The spectral complexity in the SYK model with $N=18$, $q=4$ and
$\beta J=23$, averaged over $90$ members of the ensemble. Like in the random
matrix theory example, the spectral complexity grows at early times as
$\\#\log\cosh(\\#t/\beta)$ and plateaus at late times. The inset shows the
linear scaled plot around $t=0$ to better emphasize the validity of the
logarithmic fit.

Yet another concrete example in which we can compute the spectral complexity
(this time numerically) that is different from the random matrix examples
discussed throughout this paper, is the SYK model, a chaotic system with
$q$-local interacting fermions sachdev1993gapless ; kitaevTalks ;
Maldacena:2016hyu ; Cotler:2016fpe . In Figure 7 we show the spectral
complexity for the $N=18$ and $q=4$ SYK model at inverse temperature $\beta
J=23$. For this value of $N$ the corresponding ensemble is GUE Cotler:2016fpe
. Before thermalization time $\sim\beta$ the gravitational results suggest
that the spectral complexity grows quadratically with time, after which it has
a linear period of growth, followed by the plateau at later times.

In accordance with expectations, our results for $\mathcal{C}(t)$ in SYK show
an early time growth period followed by a plateau period. Due to the lower
value of $N$ (due to computational constraints) as well as the higher value of
$\beta$ (because we wish to see the plateau appear at a time which is
computationally reachable) that we have simulated, the time difference between
the thermalization time and plateau time is not parametrically large. Inspired
by the semi-classical result for the ER bridge length we therefore fit the
simulated values of $\mathcal{C}(t)$ at early times to
$\ell(t)\propto\log\cosh(\\#t/\beta)$.

###  7.5 Relation to circuit complexity

The second perspective on the relation between the ER bridge length and the
complexity of the state is that $\mathcal{C}(t)$ could be a good proxy for the
circuit complexity of the thermofield double state in a variety of quantum
systems.333333Some previous efforts for defining complexity or a proxy for it
in field theory with an eye towards holography include Miyaji:2015woj ;
Jefferson:2017sdb ; Caputa:2017yrh ; Chapman:2017rqy ; Caputa:2018kdj ;
Belin:2018bpg ; Erdmenger:2020sup ; Kar:2021nbm . If this is indeed the case,
$\mathcal{C}(t)$ would be a much easier quantity to compute since it only
depends on the spectral statistics of the system. This brings up two
challenges. The first is to check, at least in some systems where complexity
is computable, whether the temperature dependence of $\mathcal{C}(t)$ matches
that of complexity. The second challenge is that in the models of dilaton
gravity discussed in this paper, there is no obvious notion of $k$-locality in
the matrix integral duals. In the context of complexity, $k$-locality means
that the Hamiltonian is a sum of terms that are products of at most $k$
operators that are considered to be simple (or local). In a theory without
$k$-locality, one expects complexity to grow much faster than in $k$-local
systems and saturate at its maximal value at much earlier times (perhaps not
exponential in the entropy). Since $\mathcal{C}(t)$ does not have such a
behavior, it is difficult to see how it would be a proxy for complexity in
such systems.343434The definition of complexity in such matrix models is
unclear. To define complexity, one needs to define a set of simple gates that
can be used to produce the time-evolved state. In the ensemble average over
random matrices, it is unclear whether to start with a unique set of gates for
all members of the ensemble or whether to use a different set of gates for
each member.  Nevertheless, it would be interesting to understand if in
$k$-local chaotic systems the spectral complexity indeed approximates the
circuit complexity of the state.

###  7.6 Back to gravity

Finally, it would be interesting to understand the geometric origin of the
plateau. In the case of the SFF Saad:2019lba  and the two-point function
Saad:2019pqd , a possible understanding of the plateau came from studying the
effects of spacetime D-branes on the non-perturbative corrections to these
observables in the matrix integral. In the case of the volume of the interior,
we would need to understand the origin of the cancellation between the
classical contribution and the non-perturbative contributions that come from
D-branes.

It would also be important to understand if $\expectationvalue{\ell(t)}$ can
be understood as the expectation value of a linear operator acting on some
gravitational Hilbert space. This is a subtle question in gravity
Jafferis:2017tiu ; Marolf:2020xie , and the circuit complexity in the dual
boundary theory is certainly not a linear operator.

An interesting extension of our work would be to consider single sided black
holes that could be modeled by considering JT gravity with end of the world
branes Kourkoulou:2017zaj ; Penington:2019kki ; Gao:2021uro . Finally, given
that the models of two-dimensional gravity that we have studied are closely
related to higher dimensional near-extremal black holes Sachdev:2015efa ;
Almheiri:2016fws ; Sarosi:2017ykf ; Nayak:2018qej ; Moitra:2018jqs ;
Moitra:2019bub ; Sachdev:2019bjn ; Iliesiu:2020qvm , we expect that our
conclusions extend to the volume of such bridges for these more realistic
black holes. However, it would be important to make this precise, since there
are a plethora of geometries that the dimensionally reduced two-dimensional
gravity theory does not take into account. Since black holes are chaotic
systems, we believe that their spectral complexity defined in (7.1) will
behave in the same way as analyzed in this paper.

## Acknowledgements

We thank Adam Brown, Daniel Jafferis, Douglas Stanford, Leonard Susskind, and
Gustavo Joaqíun Turiaci for very useful discussions, Zhenbin Yang and Stephen
Shenker for thoughtful comments on the manuscript, as well as Jordan Cotler
and Nicholas Hunter-Jones for correspondence. MM is supported by the Simons
Center for Geometry and Physics. LVI is supported by the Simons Collaboration
on Ultra-Quantum Matter, a Simons Foundation Grant with No. 651440. The paper
was finalized while LVI and MM were participating in the 2021 Simons Summer
Workshop.

##  Appendix A The modified two-point function for dilaton potential $V(\phi)$

To emphasize the universality of our JT gravity results, we can consider the
two-point function as probe for the length of the ER bridge in more general
models of dilaton gravity.353535Such toy models of gravity also have two-sided
black hole solutions with an ER bridge Witten:2020ert .  Our analysis will
follow closely that of Turiaci:2020fjj ; Witten:2020wvy , combined with some
of the basic building blocks discussed in section 3. We will start with models
of dilaton gravity where the JT action is modified,363636We follow the
convention for $\alpha$ in Turiaci:2020fjj , which is related to the
convention for $\alpha$ in Witten:2020wvy  by $2\pi(1-\alpha)\to\alpha$.

$\begin{split}I_{\text{dilaton}}=I_{JT}-\lambda\int
d^{2}x\sqrt{g}e^{-2\pi(1-\alpha)\phi}\,,\end{split}$ (A.1)

and as in Turiaci:2020fjj ; Witten:2020wvy , for reasons that we shall review
shortly, our techniques will solely apply when $0<\alpha\leq
1/2$.373737However, some of the results of Turiaci:2020fjj ; Witten:2020wvy
were extended to $0<\alpha<1$ by using the equivalence between the theory of
dilaton gravity and deformation of the $(2,p)$ minimal string theory
Turiaci:2020fjj . It would be interesting to see if the two-point function
which we shall discuss below could also be understood as an observable in the
deformed $(2,p)$ minimal string theory. For the disk with a defect, this point
was addressed in Mertens:2019tcm .

Expanding the exponential, $\exp(-{I_{\text{dilaton}}})$, that appears in the
path integral for the theory (A.1) in powers of $\lambda$, we have that

$\begin{split}e^{-I_{\text{dilaton}}}\supset
e^{-I_{JT}}\frac{\lambda^{k}}{k!}\int d^{2}x_{1}\sqrt{g}\dots\int
d^{2}x_{k}\sqrt{g}\left(e^{-2\pi(1-\alpha)\left[\phi(x_{1})+\dots\phi(x_{k})\right]}\right)\,.\end{split}$
(A.2)

Integrating out $\phi(x)$ in the path integral of this dilaton gravity theory
and pulling the integrals over coordinates out of the path integral, we find
that

$\begin{split}R(x)+2=2\pi(1-\alpha)\sum_{i=1}^{k}\delta^{2}(x-x_{i})\,.\end{split}$
(A.3)

Thus, the term in (A.2) can be understood as the insertion of $k$ conical-
defects in a theory of JT gravity, where we then integrate the defects over
the entire manifold Mertens:2019tcm .

More generally, we can deform JT gravity by adding the potential

$\begin{split}U(\phi)=\sum_{i=1}^{r}\lambda_{i}e^{-2\pi(1-\alpha_{i})\phi}\,.\end{split}$
(A.4)

By expanding $e^{-I_{\text{dilaton}}}$ in powers of $\lambda_{i}$ and once
again integrating-out the dilaton $\phi$, the term of order $k$ are equivalent
to inserting $k$ defects in JT gravity whose deficit angles are among
$\\{\alpha_{1},\dots,\alpha_{r}\\}$. To simplify some of the formulas in this
appendix, we will however focus on the theory (A.1) from which the
generalization to any $U(\phi)$ in (A.4) (once again, with $0<\alpha_{i}<1/2$)
will immediately follow.

Figure 8: An example (with genus 1 and 1 defect) of the type of hyperbolic
surfaces which the path integral with defect insertions sums over. Once again,
while there are an infinite number of non-self-intersecting boundary-to-
boundary geodesics on such surfaces, above we have drawn an example of all
possible surface topology that can result by cutting a genus 1 and 1 defect
surface along the geodesics. Once again, the purple curves represent the
closed geodesics which we use to glue the trumpet wavefunctions in (3.7) to
different bordered Riemann surfaces. Compared to our computation in section 3,
in the center figure we encounter a new ingredient as the decomposition of the
manifold includes a “defect wavefunction”.

We can now study the insertion of the two-point function in the path integral
of (A.1). Once again expanding in powers of $\lambda$ we find that to evaluate
the term $\sim\lambda^{k}$ we have to compute the two-point function in JT
gravity in the presence of $k$ defects. This term is schematically given by

$\begin{split}\langle\Tr_{\beta}\left(\chi(x_{1})\chi(x_{2})\right)\rangle_{k}=\frac{\lambda^{k}}{k!}\sum_{g}e^{S_{0}(1-2g)}\int_{\frac{\mathcal{T}_{g,1,k}}{\text{Mod}({\mathcal{M}}_{g,1,k})}}\omega\int
D{(\mathcal{W})}e^{-I_{JT,\text{
bdy}}(\mathcal{W})}\sum_{\gamma\in\mathcal{G}_{x1,x2}^{k}}e^{-\Delta\ell_{\gamma}}\,,\end{split}$
(A.5)

where the first sum is over surfaces of genus $g$, with $k$ defects, the first
integral is over the moduli space of such surfaces
$\int_{{\mathcal{T}_{g,n,k}}}\omega=\int_{{\mathcal{T}_{g,n,k}}}\sum_{j=1}^{3g-3+n+k}db_{j}\wedge
d\tau_{j}$. We once again have to quotient this space by the mapping class
group of each manifold, and, as before, the integral over $D{(\mathcal{W})}$
gives the integral over the boundary wiggles in the presence of these defects.
Finally, since we are interested in probing the volume of the ER bridge, we
will only sum over geodesics that do not self-intersect, whose set we denote
by $\mathcal{G}_{x1,x2}^{k}$. As in section 3, cutting along the boundary-to-
boundary geodesic separates the surface into two disconnected components or
yields a single connected component. In the former case,
${\mathcal{M}}_{g,1,k}\to{\mathcal{M}}_{h,1,p}\oplus{\mathcal{M}}_{g-h,1,k-p}$
where the geodesic can separate the set of defects to $p$ on one side (with
$0\leq p\leq k$), and $k-p$ on the other side. In the latter case,
${\mathcal{M}}_{g,1,k}\to{\mathcal{M}}_{g-1,2,k}$ (see figure 8 for an example
of both cases).383838We do not expect the case, when the geodesic passes
through a defect (or a set of defects) to significantly contribute to the path
integral in (A.6). This is because we are already summing over configurations
in which the geodesics pass arbitrarily close to the defect and the geodesic
length is continuous as the geodesic passes through the defect (since it is
the length and not its derivative that appears in the action it is unimportant
that the latter is discontinuous).  Once again, we have a trade-off between
the integral over the moduli space and the sum over geodesics and can rewrite

$\begin{split}\int_{\frac{\mathcal{T}_{g,1,k}}{\text{Mod}({\mathcal{M}}_{g,1,k})}}\omega\sum_{\gamma\in\mathcal{G}_{x1,x2}}e^{-\Delta\ell_{\gamma}}&=e^{-\Delta\ell}\int_{\frac{\mathcal{T}_{g-1,2,k}}{\text{Mod}({\mathcal{M}}_{g-1,2,k})}}\omega\\\
&+\sum_{p\leq k}\sum_{h\geq
0}e^{-\Delta\ell}\int_{\frac{\mathcal{T}_{h,1,p}}{\text{Mod}({\mathcal{M}}_{h,1,p})}}\omega\int_{\frac{\mathcal{T}_{g-h,1,k-p}}{\text{Mod}({\mathcal{M}}_{g-h,1,k-p})}}\omega\,,\end{split}$
(A.6)

where the geodesic of length $\ell$ is now on all the boundaries appearing for
surfaces in (A.6). Having performed (A.6) we now want to break down all
surfaces into further building blocks. For $0<\alpha<1/2$, surfaces of the
type ${\mathcal{M}}_{h,1,p}$ (with $h\neq 0$ or $p>1$) or
${\mathcal{M}}_{g-1,2,k}$ appearing in (A.6) always have a closed geodesic (or
two closed geodesics in the latter case) which separates all the defects and
holes from the asymptotic boundary and the boundary-boundary
geodesic.393939This follows from a straightforward use of the Gauss-Bonnet
theorem. Using the same theorem we can prove that this closed geodesic never
intersects the boundary-to-boundary geodesics; if that is not the case, the
resulting surface would include a bigon which would be disallowed in
hyperbolic space farb2011primer .  In such a case, we can again separate our
surface into trumpet wavefunctions and bordered Riemann surfaces of some genus
$g$, with some number of boundaries $n$, and some number of defects $p$. The
integral over the moduli space of such bordered Riemann surfaces with defects
gives the Weil-Peterson volume

$\begin{split}{{\mathrm{Vol}}}_{g,n,k}(b_{1},\,\dots,\,b_{n};\alpha_{1},\,\dots,\,\alpha_{k})={{\mathrm{Vol}}}_{g,n+k}(b_{1},\,\dots,\,b_{n},b_{n+1}=2\pi
i\alpha_{1},\,\dots,\,b_{n+k}=2\pi i\alpha_{k})\,,\end{split}$ (A.7)

which can be viewed as an analytic continuation of the usual Weil-Peterson
volumes tan2006generalizations .404040 Hyperbolic surfaces have a natural
$PSL(2,\mathbb{R})$ connection. The holonomy $g$ around a cycle homotopic to a
closed geodesic is given by hyperbolic $PSL(2,\mathbb{R})$ elements.
Similarly, one finds that the holonomy $g^{\prime}$ around a cycle homotopic
to a defect is given by elliptic $PSL(2,\mathbb{R})$ elements. Since the Weil-
Peterson volumes only depend on the conjugacy classes of the holonomies and
the analytic continuation of the eigenvalues for an element in the hyperbolic
class can yield an element in the elliptic class, (A.7) follows.  There is
however one exception to the decomposition above, when it involves a surface
of the type ${\mathcal{M}}_{0,1,1}$; in such a case, there is no closed
geodesic separating the defect from the asymptotic boundary. We call this
object, that has one defect and an asymptotic boundary glued to a boundary-to-
boundary geodesic, a “defect wavefunction”. The integral over the wiggles in
(A.6) for such an object can be evaluated in precisely the same way as for the
trumpet wavefunction. One finds,414141As in footnote 40, this follows from the
fact that the result for the wavefunction can only depend on the conjugacy
class of the holonomy around the closed geodesic of the trumpet or around the
defect. Since the latter can be viewed as an analytic continuation of the
former (A.8) follows.

$\begin{split}\psi_{\text{Defect},x}(\ell,\alpha)&=\leavevmode\hbox
to77.67pt{\vbox to41.31pt{\pgfpicture\makeatletter\hbox{\hskip
63.72223pt\lower-35.82727pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.6}{0.0}{0.0}{0.6}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.5}]{defectwave.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (0,0);
{}{{}}{}{{}}{}\pgfsys@moveto{-57.19072pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-60.38922pt}{-2.15277pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\alpha$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}\pgfsys@moveto{8.5359pt}{-29.02205pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{6.45256pt}{-32.49426pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\ell$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\int_{0}^{\infty}dE\rho_{\text{Defect},\,b}(E)\psi_{\text{Disk,
E}}(\ell)e^{-xE}\\\ &=\psi_{\text{Trumpet},x}(\ell,b=2\pi
i\alpha)\,,\end{split}$ (A.8)

with $\rho_{\text{Defect},\,\alpha}(E)=\rho_{\text{Trumpet},\,b=2\pi
i\alpha}(E)$. Putting all these building blocks together we find that

$\begin{split}\langle&\Tr_{\beta}(\chi(x_{1})\chi(x_{2}))\rangle_{g,k}\sim
e^{S_{0}(1-2g)}\int d\ell\,e^{\ell}\int
db_{1}b_{1}\,db_{2}b_{2}\,\psi_{\text{Trumpet},x}(\ell,b_{1})\psi_{\text{Trumpet},\beta-x}(\ell,b_{2})e^{-\Delta\ell}\\\
&\times\lambda^{k}\bigg{[}\frac{1}{k!}{{\mathrm{Vol}}}_{g-1,2,k}(b_{1},b_{2};\underbrace{\alpha,\,\dots,\,\alpha}_{k\text{
times}})\\\ &\qquad\qquad+\sum_{p\leq k}\sum_{h\geq
0}\frac{1}{p!(k-p)!}{{\mathrm{Vol}}}_{g-h,1,p}(b_{1};\underbrace{\alpha,\dots,\alpha}_{p\text{
times}}){{\mathrm{Vol}}}_{h,1,p-k}(b_{2};\underbrace{\pi i\alpha,\dots,2\pi
i\alpha}_{k-p\text{ times}})\bigg{]}\,,\end{split}$ (A.9)

where, in order to not separately consider the decomposition into surfaces
which involve a defect wavefunction, we will formally define
${{\mathrm{Vol}}}_{0,1,1}(b;\alpha)$ such that

$\int
dbb{{\mathrm{Vol}}}_{0,1,1}(b;\alpha)\psi_{\text{Trumpet},x}(\ell,b)=\psi_{\text{Defect},x}(\ell,\alpha)\,.$
(A.10)

(A.9) can then be compactly rewritten as

$\begin{split}\langle\Tr_{\beta}(\chi(x_{1})&\chi(x_{2}))\rangle=\int_{0}^{\infty}dE_{1}dE_{2}\
\left\langle\rho(E_{1})\rho(E_{2})\right\rangle^{\text{dilaton}}e^{-E_{1}(x_{1}-x_{2})-E_{2}(\beta-
x_{1}+x_{2})}\,{{\mathcal{M}}}_{{\Delta}}(E_{1},E_{2})\,,\end{split}$ (A.11)

where $\left\langle\rho(E_{1})\rho(E_{2})\right\rangle^{\text{dilaton}}$ is
the spectral two-point function for the dilaton gravity theory (A.1), which at
order $\lambda^{k}$ has a genus $g$ contribution given by

$\begin{split}\left\langle\rho(E_{1})\rho(E_{2})\right\rangle^{\text{dilaton}}_{g,k}&\sim\lambda^{k}\int
db_{1}b_{1}db_{2}b_{2}\rho_{\text{Trumpet},\,b_{1}}(E_{1})\rho_{\text{Trumpet},\,b_{2}}(E_{2})\bigg{[}\frac{1}{k!}{{\mathrm{Vol}}}_{g-1,2,k}(b_{1},b_{2};\underbrace{\alpha,\,\dots,\,\alpha}_{k\text{
times}})\\\ &+\sum_{p\leq k}\sum_{h\geq
0}\frac{1}{p!(k-p)!}{{\mathrm{Vol}}}_{g-h,1,p}(b_{1};\underbrace{\alpha,\dots,\alpha}_{p\text{
times}}){{\mathrm{Vol}}}_{h,1,p-k}(b_{2};\underbrace{\pi i\alpha,\dots,2\pi
i\alpha}_{k-p\text{ times}})\bigg{]}\end{split}$ (A.12)

Following precisely the same cutting and gluing algorithm, the formula (A.11)
follows for any theory of dilaton gravity with a potential $U(\phi)$ as in
(A.4).

##  Appendix B Details about the geometric computation of the variance

In this section we will provide more details about the derivation of (6.3).
The possible surfaces obtained from cutting along the two non-intersecting
geodesics and the action of the mapping class group on geodesics on such
surfaces in discussed in section 6. Just like in (3.4), we can rewrite

$\begin{split}\int_{\frac{\mathcal{T}_{g,2}}{\text{Mod}({\mathcal{M}}_{g,2})}}\omega&\sum_{\begin{subarray}{c}\gamma\in\mathcal{G}_{x1,x2},\\\
\gamma^{\prime}\in\mathcal{G}_{x1^{\prime},x2^{\prime}},\\\
\gamma\cap\gamma^{\prime}=\emptyset\end{subarray}}e^{-\Delta\ell_{\gamma}-\Delta^{\prime}\ell_{\gamma^{\prime}}}=e^{-\Delta\ell-\Delta^{\prime}\ell^{\prime}}\int_{\frac{\mathcal{T}_{g-2,4}}{\text{Mod}({\mathcal{M}}_{g-2,4})}}\omega\\\
&+\sum_{h\geq
0}e^{-\Delta\ell-\Delta^{\prime}\ell^{\prime}}\left(\int_{\frac{\mathcal{T}_{h,1}}{\text{Mod}({\mathcal{M}}_{h,1})}}\omega\int_{\frac{\mathcal{T}_{g-h-1,3}}{\text{Mod}({\mathcal{M}}_{g-h-1,3})}}\omega+\int_{\frac{\mathcal{T}_{h,2}}{\text{Mod}({\mathcal{M}}_{h,2})}}\omega\int_{\frac{\mathcal{T}_{g-h-1,2}}{\text{Mod}({\mathcal{M}}_{g-h-1,2})}}\omega\right)\\\
&+\sum_{h_{1},h_{2}\geq
0}e^{-\Delta\ell-\Delta^{\prime}\ell^{\prime}}\int_{\frac{\mathcal{T}_{h_{1},1}}{\text{Mod}({\mathcal{M}}_{h_{1},1})}}\omega\int_{\frac{\mathcal{T}_{h_{2},1}}{\text{Mod}({\mathcal{M}}_{h_{2},1})}}\omega\int_{\frac{\mathcal{T}_{g-h_{1}-h_{2},2}}{\text{Mod}({\mathcal{M}}_{g-h_{1}-h_{2},2})}}\omega\end{split}$
(B.1)

where the geodesics $\ell$ and $\ell^{\prime}$ are once again always on the
boundaries of the manifolds resulting from the cut and where the sum over $h$
is again bounded such that disconnected surfaces of genus $h$ and $g-h$ are
not counted twice and the sum over $h_{1,2}$ are also bounded such that all
three disconnected surfaces within the same genus configuration are not
counted multiple times. As before, we can once again then decompose our
manifold into trumpet (or disk, when a component has $g=0$) wavefunctions and
bordered Riemann surfaces. Peforming the integrals in (B.1) explicitly to
obtain the volumes of these bordered Riemann surfaces, we find that the genus
$g$ contribution to
$\langle\Tr_{\beta}\left(\chi(x_{1})\chi(x_{2})\right)\Tr_{\beta}\left(\chi(x_{1}^{\prime})\chi(x_{2}^{\prime})\right)\rangle$,
coming from connected geometries, is given by

$\begin{split}\langle\Tr_{\beta}&\left(\chi(x_{1})\chi(x_{2})\right)\Tr_{\beta}\left(\chi(x_{1}^{\prime})\chi(x_{2}^{\prime})\right)\rangle_{\text{connected,
}g}\,\,\,\sim\,\,\,e^{-2S_{0}g}\int
db_{1}b_{1}\,db_{2}b_{2}db_{3}b_{3}\,db_{4}b_{4}\,\\\ &\times\int
d\ell\,e^{\ell}\int
d\ell^{\prime}\,e^{\ell^{\prime}}\psi_{\text{Trumpet},x}(\ell,b_{1})\psi_{\text{Trumpet},\beta-x}(\ell,b_{2})\psi_{\text{Trumpet},x}(\ell^{\prime},b_{3})\psi_{\text{Trumpet},\beta-x}(\ell^{\prime},b_{4})\\\
&\times
e^{-\Delta\ell-\Delta^{\prime}\ell^{\prime}}V_{\text{connected}}(b_{1},b_{2},b_{3},b_{4})\end{split}$
(B.2)

where, we define $V_{\text{connected}}(b_{1},b_{2},b_{3},b_{4})$ as,424242We
once again have to treat the components that have genus $g=0$ separately.

$\begin{split}V_{\text{connected}}&(b_{1},b_{2},b_{3},b_{4})={{\mathrm{Vol}}}_{g-2,4}(b_{1},b_{2},b_{3},b_{4})\\\
&+\sum_{h\geq
0}\left[{{\mathrm{Vol}}}_{h,1}(b_{1}){{\mathrm{Vol}}}_{g-h,1}(b_{2},b_{3},b_{4})+{{\mathrm{Vol}}}_{h,1}(b_{1}){{\mathrm{Vol}}}_{g-h,1}(b_{2},b_{3},b_{4})\right]\\\
&+\sum_{h_{1},h_{2}\geq
0}\left[{{\mathrm{Vol}}}_{h_{1},1}(b_{1}){{\mathrm{Vol}}}_{h_{2}}(b_{2}){{\mathrm{Vol}}}_{g-h_{1}-h_{2},2}(b_{3},b_{4})\right]\end{split}$
(B.3)

Putting together the connected and disconnected contributions, writing the
trumpet wavefunctions as in (3.5), and integrating over the $b$’s and $\ell$’s
we can compactly rewrite the total two-boundary correlator
$\langle\Tr_{\beta}\left(\chi(x_{1})\chi(x_{2})\right)\Tr_{\beta}\left(\chi(x_{1}^{\prime})\chi(x_{2}^{\prime})\right)\rangle_{\text{non-
int.}}$ as

$\begin{split}\langle\Tr_{\beta}&\left(\chi(x_{1})\chi(x_{2})\right)\Tr_{\beta}\left(\chi(x_{1}^{\prime})\chi(x_{2}^{\prime})\right)\rangle_{\text{non-
int.}}={4e^{-S_{0}}\over\langle
Z(\beta)Z(\beta)\rangle}\int_{0}^{\infty}dE_{1}\dots dE_{4}\
\left\langle\rho(E_{1})\dots\rho(E_{4})\right\rangle\\\ &\times
e^{-E_{1}x_{12}-E_{2}(\beta-x_{12})-E_{3}x_{12}^{\prime}-E_{4}(\beta-
x_{12}^{\prime})}\,{{\mathcal{M}}}_{{\Delta}}(E_{1},E_{2}){{\mathcal{M}}}_{{{\Delta}}^{\prime}}(E_{3},E_{4})\,.\end{split}$
(B.4)

More generally, using the same techniques to address the issues related to the
mapping class group, we can compute even higher “moments” of the length of the
ER bridge by using434343The same ambiguities that exist for the variance, also
exist for the higher moments and for the result (B.5) we make the same choices
as for the variance.

$\begin{split}\bigg{\langle}\Tr_{\beta}&\left(\chi^{(1)}(x_{1}^{(1)})\chi^{(1)}(x_{2}^{(1)})\right)\dots\Tr_{\beta}\left(\chi^{(n)}(x_{1}^{(n)})\chi^{(n)}(x_{2}^{(n)})\right)\bigg{\rangle}={4e^{-S_{0}}\over\langle\underbrace{Z(\beta)Z(\beta)\dots
Z(\beta)\rangle}_{n\text{ copies}}}\\\ &\times\int_{0}^{\infty}dE_{1}\dots
dE_{2n}\
\left\langle\rho(E_{1})\dots\rho(E_{2n})\right\rangle{{\mathcal{M}}}_{{{\Delta}}^{(1)}}(E_{1},E_{2})\dots{\mathcal{M}}_{{{\Delta}}^{(n)}}(E_{2n-1},E_{2n})\\\
&\times e^{-E_{1}(x_{1}^{(1)}-x_{2}^{(1)})-E_{2}(\beta-
x_{1}^{(1)}+x_{2}^{(1)})}\dots
e^{-E_{2n-1}(x_{1}^{(n)}-x_{2}^{(n)})-E_{2n}(\beta-
x_{1}^{(n)}+x_{2}^{(n)})}\,\,,\end{split}$ (B.5)

and by taking $\lim_{\Delta^{(1)},\,\dots\,,\Delta^{(n)}\to
0}\partial_{\Delta^{(1)}}\dots\partial_{\Delta^{(n)}}$ for the above
expression.

##  Appendix C Explicit formulas in Dyson ensembles

We largely follow the notation of Guhr:1997ve . The density two point
functions in Dyson ensembles are given by

$\begin{split}\expectationvalue{\rho\left(E+{\Omega\over
2\expectationvalue{\rho(E)}}\right)\rho\left(E-{\Omega\over
2\expectationvalue{\rho(E)}}\right)}&=\expectationvalue{\rho(E)}^{2}\left(1-Y(\Omega)\right)\,,\\\
Y^{\text{GOE}}(\Omega)&=s^{2}(\Omega)+s^{\prime}(\Omega)\int_{\Omega}^{\infty}d\nu\
s(\nu)\,,\\\ Y^{\text{GUE}}(\Omega)&=s^{2}(\Omega)\,,\\\
Y^{\text{GSE}}(\Omega)&=s^{2}(2\Omega)-s^{\prime}(2\Omega)\int_{0}^{\Omega}d\nu\
s(2\nu)\,,\\\ s(\Omega)&\equiv{\sin(\pi\Omega)\over\pi\Omega}\,.\end{split}$
(C.1)

The SFF is given by formula (7.3) and evaluates to

$\begin{split}\text{SFF}^{\text{GOE}}_{\text{micro}}(2\pi\expectationvalue{\rho(E)}T)&=\expectationvalue{\rho(E)}\begin{cases}2T-T\log\left(2T+1\right)\qquad&(T\leq
1)\,,\\\ -T\log\left(2T+1\over 2T-1\right)\qquad&(T>1)\,.\end{cases}\\\
\text{SFF}^{\text{GUE}}_{\text{micro}}(2\pi\expectationvalue{\rho(E)}T)&=\expectationvalue{\rho(E)}\begin{cases}T\qquad&(T\leq
1)\,,\\\ 1\qquad&(T>1)\,.\end{cases}\\\
\text{SFF}^{\text{GSE}}_{\text{micro}}(2\pi\expectationvalue{\rho(E)}T)&=\expectationvalue{\rho(E)}\begin{cases}\frac{1}{2}T-\frac{1}{4}T\log\absolutevalue{1-T}\qquad&(T\leq
2)\,,\\\ 1\qquad&(T>2)\,.\end{cases}\end{split}$ (C.2)

where we used a rescaled time variable $t=2\pi\expectationvalue{\rho(E)}T$. We
can similarly compute $\mathcal{C}(t)$ using (7.2) with the result

$\begin{split}\mathcal{C}^{\text{GOE}}_{\text{micro}}(2\pi\expectationvalue{\rho(E)}T)&=\langle{\rho(E)}\rangle{\pi^{2}\over
18}\begin{cases}(30T-48T^{2}+34T^{3})+(3+9T-12T^{3})\log\left(2T+1\right)\qquad&(T\leq
1)\,,\\\ (4+12T^{2})+(3-9T+12T^{3})\log\left(2T-1\right)\\\
+(3+9T-12T^{3})\log\left(2T+1\right)\qquad&(T>1)\,.\end{cases}\\\
\mathcal{C}^{\text{GUE}}_{\text{micro}}(2\pi\expectationvalue{\rho(E)}T)&=\langle{\rho(E)}\rangle{2\pi^{2}\over
3}\begin{cases}3T-3T^{2}+T^{3}\qquad&(T\leq 1)\,,\\\
1\qquad&(T>1)\,.\end{cases}\\\
\mathcal{C}^{\text{GSE}}_{\text{micro}}(2\pi\expectationvalue{\rho(E)}T)&=\langle{\rho(E)}\rangle{4\pi^{2}\over
9}\begin{cases}{1\over
16}\left[(60T-60T^{2}+17T^{3})-(12-18T+6T^{3})\log\absolutevalue{1-T}\right]\qquad&(T\leq
2)\,,\\\ 1\qquad&(T>2)\,.\end{cases}\end{split}$ (C.3)

The relation (7.4) is now easily verified.

## References

  * (1) J. M. Maldacena, _The Large N limit of superconformal field theories and supergravity_ , _Adv. Theor. Math. Phys._ 2 (1998) 231 [hep-th/9711200].
  * (2) E. Witten, _Anti-de Sitter space and holography_ , _Adv. Theor. Math. Phys._ 2 (1998) 253 [hep-th/9802150].
  * (3) S. S. Gubser, I. R. Klebanov and A. M. Polyakov, _Gauge theory correlators from noncritical string theory_ , _Phys. Lett. B_ 428 (1998) 105 [hep-th/9802109].
  * (4) L. Susskind, _Computational Complexity and Black Hole Horizons_ , _Fortsch. Phys._ 64 (2016) 24 [1403.5695].
  * (5) L. Susskind, _Entanglement is not enough_ , _Fortsch. Phys._ 64 (2016) 49 [1411.0690].
  * (6) D. Stanford and L. Susskind, _Complexity and Shock Wave Geometries_ , _Phys. Rev. D_ 90 (2014) 126007 [1406.2678].
  * (7) A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle and Y. Zhao, _Holographic Complexity Equals Bulk Action?_ , _Phys. Rev. Lett._ 116 (2016) 191301 [1509.07876].
  * (8) A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle and Y. Zhao, _Complexity, action, and black holes_ , _Phys. Rev. D_ 93 (2016) 086006 [1512.04993].
  * (9) J. Couch, W. Fischler and P. H. Nguyen, _Noether charge, black hole volume, and complexity_ , _JHEP_ 03 (2017) 119 [1610.02038].
  * (10) A. Belin, R. C. Myers, S.-M. Ruang, G. Sarosi and A. J. Speranza, _To appear_ , .
  * (11) L. Susskind, _The Typical-State Paradox: Diagnosing Horizons with Complexity_ , _Fortsch. Phys._ 64 (2016) 84 [1507.02287].
  * (12) A. R. Brown, L. Susskind and Y. Zhao, _Quantum Complexity and Negative Curvature_ , _Phys. Rev. D_ 95 (2017) 045010 [1608.02612].
  * (13) V. Balasubramanian, M. Decross, A. Kar and O. Parrikar, _Quantum Complexity of Time Evolution with Chaotic Hamiltonians_ , _JHEP_ 01 (2020) 134 [1905.05765].
  * (14) L. Susskind, _Black Holes at Exp-time_ , 2006.01280.
  * (15) V. Balasubramanian, M. Decross, A. Kar, C. Li and O. Parrikar, _Complexity Growth in Integrable and Chaotic Models_ , 2101.02209.
  * (16) J. Haferkamp, P. Faist, N. B. T. Kothakonda, J. Eisert and N. Y. Halpern, _Linear growth of quantum circuit complexity_ , 2106.05305.
  * (17) P. Saad, S. H. Shenker and D. Stanford, _JT gravity as a matrix integral_ , 1903.11115.
  * (18) J. Cotler and K. Jensen, _AdS 3 gravity and random CFT_, _JHEP_ 04 (2021) 033 [2006.08648].
  * (19) A. Belin and J. de Boer, _Random Statistics of OPE Coefficients and Euclidean Wormholes_ , 2006.05499.
  * (20) A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, _Replica Wormholes and the Entropy of Hawking Radiation_ , _JHEP_ 05 (2020) 013 [1911.12333].
  * (21) G. Penington, S. H. Shenker, D. Stanford and Z. Yang, _Replica wormholes and the black hole interior_ , 1911.11977.
  * (22) C. Teitelboim, _Gravitation and Hamiltonian Structure in Two Space-Time Dimensions_ , _Phys. Lett. B_ 126 (1983) 41.
  * (23) R. Jackiw, _Lower Dimensional Gravity_ , _Nucl. Phys. B_ 252 (1985) 343.
  * (24) J. Maldacena, D. Stanford and Z. Yang, _Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space_ , _PTEP_ 2016 (2016) 12C104 [1606.01857].
  * (25) E. Witten, _Matrix Models and Deformations of JT Gravity_ , _Proc. Roy. Soc. Lond. A_ 476 (2020) 20200582 [2006.13414].
  * (26) H. Maxfield and G. J. Turiaci, _The path integral of 3D gravity near extremality; or, JT gravity with defects as a matrix integral_ , _JHEP_ 01 (2021) 118 [2006.11317].
  * (27) A. Blommaert, _Dissecting the ensemble in JT gravity_ , 2006.13971.
  * (28) Z. Yang, _The Quantum Gravity Dynamics of Near Extremal Black Holes_ , _JHEP_ 05 (2019) 205 [1809.08647].
  * (29) A. Lewkowycz and J. Maldacena, _Generalized gravitational entropy_ , _JHEP_ 08 (2013) 090 [1304.4926].
  * (30) P. Saad, _Late Time Correlation Functions, Baby Universes, and ETH in JT Gravity_ , 1910.10311.
  * (31) T. G. Mertens, G. J. Turiaci and H. L. Verlinde, _Solving the Schwarzian via the Conformal Bootstrap_ , _JHEP_ 08 (2017) 136 [1705.08408].
  * (32) M. Mirzakhani, _Simple geodesics and weil-petersson volumes of moduli spaces of bordered riemann surfaces_ , _Inventiones mathematicae_ 167 (2007) 179.
  * (33) D. Stanford and E. Witten, _Fermionic Localization of the Schwarzian Theory_ , _JHEP_ 10 (2017) 008 [1703.04612].
  * (34) D. Stanford and E. Witten, _JT Gravity and the Ensembles of Random Matrix Theory_ , 1907.03363.
  * (35) A. Blommaert, T. G. Mertens and H. Verschelde, _The Schwarzian Theory - A Wilson Line Perspective_ , _JHEP_ 12 (2018) 022 [1806.07765].
  * (36) A. Kitaev and S. J. Suh, _Statistical mechanics of a two-dimensional black hole_ , _JHEP_ 05 (2019) 198 [1808.07032].
  * (37) L. V. Iliesiu, S. S. Pufu, H. Verlinde and Y. Wang, _An exact quantization of Jackiw-Teitelboim gravity_ , _JHEP_ 11 (2019) 091 [1905.02726].
  * (38) D. Kolchmeyer, D. Jafferis, B. Mukhametzhanov and J. Sonner, “to appear.”.
  * (39) A. Blommaert, T. G. Mertens and H. Verschelde, _Eigenbranes in Jackiw-Teitelboim gravity_ , _JHEP_ 02 (2021) 168 [1911.11603].
  * (40) M. L. Mehta, _Random matrices_. Elsevier, 2004.
  * (41) T. Guhr, A. Muller-Groeling and H. A. Weidenmuller, _Random matrix theories in quantum physics: Common concepts_ , _Phys. Rept._ 299 (1998) 189 [cond-mat/9707301].
  * (42) P. D. Miller, “Lectures on random matrix theory for course at samsi.” November, 2006.
  * (43) D. Harlow and D. Jafferis, _The Factorization Problem in Jackiw-Teitelboim Gravity_ , _JHEP_ 02 (2020) 177 [1804.01081].
  * (44) J. Cotler and N. Hunter-Jones, _Spectral decoupling in many-body quantum chaos_ , _JHEP_ 12 (2020) 205 [1911.02026].
  * (45) J. S. Cotler, G. Gur-Ari, M. Hanada, J. Polchinski, P. Saad, S. H. Shenker et al., _Black Holes and Random Matrices_ , _JHEP_ 05 (2017) 118 [1611.04650].
  * (46) P. Saad, S. H. Shenker and D. Stanford, _A semiclassical ramp in SYK and in gravity_ , 1806.06840.
  * (47) D. Stanford, _More quantum noise from wormholes_ , 2008.08570.
  * (48) A. Goel, L. V. Iliesiu, J. Kruthoff and Z. Yang, _Classifying boundary conditions in JT gravity: from energy-branes to $\alpha$-branes_, _JHEP_ 04 (2021) 069 [2010.12592].
  * (49) A. Altland and M. R. Zirnbauer, _Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures_ , _Physical Review B_ 55 (1997) 1142.
  * (50) M. Heydeman, L. V. Iliesiu, G. J. Turiaci and W. Zhao, _The statistical mechanics of near-BPS black holes_ , 2011.01953.
  * (51) Y. Fan and T. G. Mertens, _Supergroup Structure of Jackiw-Teitelboim Supergravity_ , 2106.09353.
  * (52) S. Sachdev and J. Ye, _Gapless spin-fluid ground state in a random quantum heisenberg magnet_ , _Physical review letters_ 70 (1993) 3339\.
  * (53) A. Kitaev, _Talks given at the Fundamental Physics Prize Symposium and KITP seminars_ , .
  * (54) J. Maldacena and D. Stanford, _Remarks on the Sachdev-Ye-Kitaev model_ , _Phys. Rev._ D94 (2016) 106002 [1604.07818].
  * (55) M. Miyaji, T. Numasawa, N. Shiba, T. Takayanagi and K. Watanabe, _Distance between Quantum States and Gauge-Gravity Duality_ , _Phys. Rev. Lett._ 115 (2015) 261602 [1507.07555].
  * (56) R. Jefferson and R. C. Myers, _Circuit complexity in quantum field theory_ , _JHEP_ 10 (2017) 107 [1707.08570].
  * (57) P. Caputa, N. Kundu, M. Miyaji, T. Takayanagi and K. Watanabe, _Liouville Action as Path-Integral Complexity: From Continuous Tensor Networks to AdS/CFT_ , _JHEP_ 11 (2017) 097 [1706.07056].
  * (58) S. Chapman, M. P. Heller, H. Marrochio and F. Pastawski, _Toward a Definition of Complexity for Quantum Field Theory States_ , _Phys. Rev. Lett._ 120 (2018) 121602 [1707.08582].
  * (59) P. Caputa and J. M. Magan, _Quantum Computation as Gravity_ , _Phys. Rev. Lett._ 122 (2019) 231302 [1807.04422].
  * (60) A. Belin, A. Lewkowycz and G. Sárosi, _Complexity and the bulk volume, a new York time story_ , _JHEP_ 03 (2019) 044 [1811.03097].
  * (61) J. Erdmenger, M. Gerbershagen and A.-L. Weigel, _Complexity measures from geometric actions on Virasoro and Kac-Moody orbits_ , _JHEP_ 11 (2020) 003 [2004.03619].
  * (62) A. Kar, L. Lamprou, M. Rozali and J. Sully, _Random Matrix Theory for Complexity Growth and Black Hole Interiors_ , 2106.02046.
  * (63) D. L. Jafferis, _Bulk reconstruction and the Hartle-Hawking wavefunction_ , 1703.01519.
  * (64) D. Marolf and H. Maxfield, _Transcending the ensemble: baby universes, spacetime wormholes, and the order and disorder of black hole information_ , _JHEP_ 08 (2020) 044 [2002.08950].
  * (65) I. Kourkoulou and J. Maldacena, _Pure states in the SYK model and nearly- $AdS_{2}$ gravity_, 1707.02325.
  * (66) P. Gao, D. L. Jafferis and D. K. Kolchmeyer, _An effective matrix model for dynamical end of the world branes in Jackiw-Teitelboim gravity_ , 2104.01184.
  * (67) S. Sachdev, _Bekenstein-Hawking Entropy and Strange Metals_ , _Phys. Rev._ X5 (2015) 041025 [1506.05111].
  * (68) A. Almheiri and B. Kang, _Conformal Symmetry Breaking and Thermodynamics of Near-Extremal Black Holes_ , _JHEP_ 10 (2016) 052 [1606.04108].
  * (69) G. Sárosi, _AdS 2 holography and the SYK model_, _PoS_ Modave2017 (2018) 001 [1711.08482].
  * (70) P. Nayak, A. Shukla, R. M. Soni, S. P. Trivedi and V. Vishal, _On the Dynamics of Near-Extremal Black Holes_ , _JHEP_ 09 (2018) 048 [1802.09547].
  * (71) U. Moitra, S. P. Trivedi and V. Vishal, _Extremal and near-extremal black holes and near-CFT 1_, _JHEP_ 07 (2019) 055 [1808.08239].
  * (72) U. Moitra, S. K. Sake, S. P. Trivedi and V. Vishal, _Jackiw-Teitelboim Gravity and Rotating Black Holes_ , 1905.10378.
  * (73) S. Sachdev, _Universal low temperature theory of charged black holes with AdS 2 horizons_, _J. Math. Phys._ 60 (2019) 052303 [1902.04078].
  * (74) L. V. Iliesiu and G. J. Turiaci, _The statistical mechanics of near-extremal black holes_ , 2003.02860.
  * (75) E. Witten, _Deformations of JT Gravity and Phase Transitions_ , 2006.03494.
  * (76) G. J. Turiaci, M. Usatyuk and W. W. Weng, _Dilaton-gravity, deformations of the minimal string, and matrix models_ , 2011.06038.
  * (77) T. G. Mertens and G. J. Turiaci, _Defects in Jackiw-Teitelboim Quantum Gravity_ , _JHEP_ 08 (2019) 127 [1904.05228].
  * (78) B. Farb and D. Margalit, _A primer on mapping class groups (pms-49)_. Princeton University Press, 2011.
  * (79) S. P. Tan, Y. L. Wong, Y. Zhang et al., _Generalizations of mcshane’s identity to hyperbolic cone-surfaces_ , _Journal of Differential Geometry_ 72 (2006) 73.