††institutetext: Kavli Institute for Theoretical Sciences (KITS),
University of Chinese Academy of Sciences (UCAS), Beijing 100190, China

# Half-Wormholes and Ensemble Averages

Cheng Peng, Jia Tian and Yingyu Yang<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study “half-wormhole-like” saddle point contributions to spectral
correlators in a variety of ensemble average models, including various
statistical models, generalized 0d SYK models, 1d Brownian SYK models and an
extension of it. In statistical ensemble models, where more general
distributions of the random variables could be studied in great details, we
find the accuracy of the previously proposed approximation for the half-
wormholes could be improved when the distribution of the random variables
deviate significantly from Gaussian distributions. We propose a modified
approximation scheme of the half-wormhole contributions that also work well in
these more general theories. In various generalized 0d SYK models we identify
new half-wormhole-like saddle point contributions. In the 0d SYK model and 1d
Brownian SYK model, apart from the wormhole and half-wormhole saddles, we find
new non-trivial saddles in the spectral correlators that would potentially
give contributions of the same order as the trivial self-averaging saddles.
However after a careful Lefschetz-thimble analysis we show that these non-
trivial saddles should not be included. We also clarify the difference between
“linked half-wormholes” and “unlinked half-wormholes” in some models.

##  1 Introduction

The AdS/CFT correspondence Maldacena:1997re ; Witten:1998qj ; Gubser:1998bc
provides a non-perturbative definition of quantum gravity. An important lesson
from the recently progress in understanding the black hole information paradox
is that a summation of different configurations in the semi-classical
gravitational path integral is crucial to probe some quantum mechanical
properties of the system, such as the Page curve Penington:2019npb ;
Almheiri:2019psf ; Almheiri:2019hni ; Penington:2019kki , the late-time
behavior of the spectral form factor Saad:2019lba ; Saad:2018bqo , and
correlation functions Saad:2019pqd ; Yan:2022nod , see also a recent review in
Bousso:2022ntt . However, the inclusion of spacetime wormholes leads to an
apparent factorization puzzle Maldacena:2004rf ; a holographic computation of
the correlation functions of field theory partition functions living on
different boundaries gives non-factorized results, i.e. $\langle
Z_{L}Z_{R}\rangle\neq\langle Z_{L}\rangle\times\langle Z_{R}\rangle$, which is
in tension with the general expectation on the field theory side. This
revitalizes the hypothetical connection between wormholes and ensemble
averages Coleman:1988cy ; Giddings:1988wv ; Giddings:1988cx ;
Polchinski:1994zs , and motivates an appealing conjectural duality between a
bulk gravitational theory and (the average of) an ensemble of theories on the
boundary Saad:2019lba ; Stanford:2019vob ; Iliesiu:2019lfc ; Kapec:2019ecr ;
Maxfield:2020ale ; Witten:2020wvy ; Mefford:2020vde ; Altland:2020ccq ;
Eberhardt:2021jvj ; Stanford:2021bhl ; Arefeva:2019buu ; Betzios:2020nry ;
Anninos:2020ccj ; Berkooz:2020uly ; Mertens:2020hbs ; Turiaci:2020fjj ;
Anninos:2020geh ; Gao:2021uro ; Godet:2021cdl ; Johnson:2021owr ;
Blommaert:2021etf ; Okuyama:2019xbv ; Forste:2021roo ; Maloney:2020nni ;
Afkhami-Jeddi:2020ezh ; Cotler:2020ugk ; Benjamin:2021wzr ; Perez:2020klz ;
Cotler:2020hgz ; Ashwinkumar:2021kav ; Afkhami-Jeddi:2021qkf ; Collier:2021rsn
; Benjamin:2021ygh ; Dong:2021wot ; Dymarsky:2020pzc ; Meruliya:2021utr ;
Bousso:2020kmy ; Janssen:2021stl ; Cotler:2021cqa ; Marolf:2020xie ;
Balasubramanian:2020jhl ; Gardiner:2020vjp ; Belin:2020hea ; Belin:2020jxr ;
Altland:2021rqn ; Belin:2021ibv ; Peng:2021vhs ; Banerjee:2022pmw ;
Heckman:2021vzx ; Johnson:2022wsr ; Collier:2022emf ; Chandra:2022bqq ;
Schlenker:2022dyo , whose prototype is the by-now well known duality between
the two-dimensional Jackiw-Teitelboim (JT) gravity Jackiw:1984je ;
Teitelboim:1983ux  and the Schwarzian sector of the Sachdev-Ye-Kitaev (SYK)
model Sachdev:1992fk ; KitaevTalk2 , or more directly the random matrix
theories Saad:2019lba ; Stanford:2019vob . Alternatively, an interesting
question is whether there exist other configurations whose inclusion into the
gravitational path integral would capture properties of a single boundary
theory that are washed out after averaging over the ensemble. This is closely
related to the belief that solving the factorization problem will shed light
on the microscopic structure of quantum gravity such as the microstates or the
states behind the horizon of the black hole; these fine structures are not
universal so they can not be captured by the ensemble averaged quantities
Stanford:2020wkf ; Almheiri:2021jwq . In Saad:2021uzi , the factorization
problem is carefully studied in a toy model introduced in Marolf:2020xie ,
where it is shown that the (approximate) factorization can be restored if
other half-wormhole contributions are included. In the dual field theory
analysis, these half-wormhole contributions are identified with non-self-
averaging saddle points in the ensemble averaged theories. This idea is
explicitly realized in a 0-dimensional “one-time” SYK model in Saad:2021rcu ,
followed by further analyses in different models Mukhametzhanov:2021nea ;
Garcia-Garcia:2021squ ; Choudhury:2021nal ; Mukhametzhanov:2021hdi ;
Okuyama:2021eju ; Goto:2021mbt ; Blommaert:2021fob ; Goto:2021wfs . An
explicit connection between the gravity computation in Saad:2021uzi  and the
field theory computation in Saad:2021rcu  is proposed in Peng:2021vhs .

The construction of half-wormhole in Saad:2021rcu  is based on the $G,\Sigma$
effective action of the model that comes from the Gaussian statistics of the
random coupling. Furthermore, a prescription to identify the half-wormhole
contribution is proposed and verified for the 0-dimensional SYK model and GUE
matrix model in Mukhametzhanov:2021hdi . This raised a question of whether
half-wormhole contributions also exist in different ensemble theories, such as
those with random variables from a Poisson distribution Peng:2020rno  or a
uniform distribution on the moduli space Maloney:2020nni ; Afkhami-
Jeddi:2020ezh ; Cotler:2020ugk ; Perez:2020klz ; Benjamin:2021wzr ;
Dong:2021wot ; Collier:2022emf ; Chandra:2022bqq , and whether these
contributions share the same general properties as those discussed in
Saad:2021rcu  and Mukhametzhanov:2021hdi .

In this paper we study the half-wormhole-like contributions that characterize
the distinct behaviors of each individual theory in an ensemble of theories,
and test the approximation schemes of the half-wormholes in various models.
Our main findings are summarized as follows.

###  1.1 Summary of our main results

  * ✓

To understand the nature of the half-wormhole contributions in the 1-time SYK
model, an approximation scheme is proposed in Mukhametzhanov:2021hdi . Since
the proposal does not rely on specific details of the SYK model, such as the
collective $G$ and $\Sigma$ variables, it is interesting to understand if
there is a similar approximation that applies to more general ensemble
averaged theories. In this paper, we first consider various statistical models
with a single or multiple random variables. We compute a variety of different
quantities, such as simple observables, power-sum observables and product
observables, before and after the statistical average. We propose an
approximation formula for the half-wormhole like contributions in general
statistical models, which generalizes the one in Mukhametzhanov:2021hdi , and
show their validity explicitly. We find the validity of the “wormhole/half-
wormhole” approximation crucially depend on the large-$N$ factorization
property of the observables we consider. The large-$N$ constraints such as
traces and determinants play crucial roles in the validity of this
approximation.

  * ✓

We review the 0-dimensional SYK model introduced in Saad:2021rcu  and fill in
technical details of some calculations. In particular, in the saddle point
analysis of various quantities, such as $\langle\Phi(\sigma)^{2}\rangle$ and
others, we find new non-trivial saddle points whose on-shell values, including
the 1-loop corrections, are of the same order as the the trivial saddle that
is accounted for the half-wormhole. We then carry out explicit Lefschetz-
thimbles analyses to conclude that the contributions from these non-trivial
saddle points should not be included in the path integral, which supports the
previous results in Saad:2021rcu . We also extend some of the computations to
two-loop order and again find our results support previous conclusions in
Saad:2021rcu .

  * ✓

We generalize the 0-dimensional SYK model so that the random coupling
$J_{i_{1}\dots i_{q}}$ can be drawn from more general distributions, with non-
vanishing mean or higher order cumulants.

When $J_{i_{1}\dots i_{q}}$ has a non-vanishing mean value, we find new half-
wormhole saddle of $z$ in additional to the linked half-wormhole saddle of
$z^{2}$. We introduce new collective variables $G,\Sigma$ to compute $\langle
z\rangle$ and identify the contributions from the half-wormhole saddle. We
further consider the half-wormhole proposal in this context. We find that
depending on the relative ratio between the different cumulants, different
“multiple-linked-wormholes” could be dominant. In particular, in very special
limits approximate factorization could hold automatically and no other “half-
wormholes” saddles are needed.

In models with non-vanishing higher cumulants of the random coupling, e.g.
$\langle J_{i_{1},\dots i_{q}}^{4}\rangle\neq 0$, we find a similar conclusion
that the saddle point contributes. Equivalently, the bulk configurations that
dominate the path integral depends crucially on the ratios of the various
cumulants and the result is not universal.

In addition, we do a preliminary analysis of models whose random couplings
$J_{i_{1},\dots i_{q}}$ are drawn from a discrete distribution, the Poisson
distribution, where more complicated saddle points can be found.

  * ✓

We do a similar analysis explicitly to the Brownian SYK model, and identify
the wormhole and half-wormhole saddles at late time. The results are computed
from both an explicit integration and a saddle point analysis, and we find a
perfect agreement between them. We test the approximation of the partition
function by its mean value and the half wormhole saddle, and further show that
this approximation is good by demonstrating that the error of this
approximation is small. Interestingly, like in the 0-dimensional model we also
find non-trivial saddles for $\langle\Phi(\sigma)^{2}\rangle$ and they should
be excluded by a similar Lefschetz thimble analysis.

  * ✓

We further investigate modified 0d and 1d SYK model whose random couplings
have non-vanishing mean values that are written in terms of products of some
background Majorana fermions Goto:2021wfs . We compute explicitly the wormhole
and a new type of saddle point, the “unlinked half-wormholes”, that contribute
to the partition function. We show these unlink half-wormholes are closely
related to the disconnected saddles due to the non-vanishing mean value of the
random coupling.

##  2 Statistical models

In this section we consider statistical models, which can be considered as toy
models of the Random Matrix Theories, to test the idea of half-wormholes in
ensemble theories with random variables drawn from different distributions.

###  2.1 Models of a single random variable

Let $X$ be a random variable with a PDF $P(X)$ that satisfies the inequality

$\displaystyle\langle X^{2}\rangle\geq\langle X\rangle^{2}\,,$ (1)

that is valid for all conventional probability distributions. To identify the
“half-wormhole contributions” in this model, we consider the unaveraged
observable $X$,$X^{2}$ etc., and rewrite

$\displaystyle X^{n}$ $\displaystyle=\int
dx\,\delta(x-X)\frac{x^{n}P(x)}{P(X)}=\int
dx\int\frac{dk}{2\pi}\,e^{\text{i}k(x-X)}\frac{x^{n}P(x)}{P(X)}=\int\frac{dk}{2\pi}\frac{e^{-\text{i}kX}}{P(X)}\langle
x^{n}e^{\text{i}kx}\rangle\,,$ (2)

where as usual the angle bracket denotes the average of $x$ with the
probability distribution $P(x)$

$\displaystyle\langle\mathcal{O}e^{\text{i}kx}\rangle=\int
dx\,\mathcal{O}e^{\text{i}kx}P(x)\ .$ (3)

Such expectation values can further be decomposed into the connected and
disconnected parts, for example

$\displaystyle\langle xe^{\text{i}kx}\rangle=\langle x\rangle\langle
e^{\text{i}kx}\rangle+\langle xe^{\text{i}kx}\rangle_{\text{c}}\,,$ (4)
$\displaystyle\langle x^{2}e^{\text{i}kx}\rangle=\langle x^{2}\rangle\langle
e^{\text{i}kx}\rangle+2\langle x\rangle\langle
xe^{\text{i}kx}\rangle_{c}+\langle x^{2}e^{\text{i}kx}\rangle_{\text{c}},$ (5)
$\displaystyle\langle x^{3}e^{\text{i}kx}\rangle=\langle x^{3}\rangle\langle
e^{\text{i}kx}\rangle+3\langle x^{2}\rangle\langle
xe^{\text{i}kx}\rangle_{c}+3\langle x\rangle\langle
x^{2}e^{\text{i}kx}\rangle_{c}+\langle
x^{3}e^{\text{i}kx}\rangle_{\text{c}}\,,$ (6) $\displaystyle\dots$

where the subscript $c$ denotes “connected” or “cumulant” which can be defined
recursively as

$\displaystyle\langle xe^{\text{i}kx}\rangle_{\text{c}}=\langle
xe^{\text{i}kx}\rangle-\langle x\rangle\langle e^{\text{i}kx}\rangle,$ (7)
$\displaystyle\langle x^{2}e^{\text{i}kx}\rangle_{\text{c}}=\langle
x^{2}e^{\text{i}kx}\rangle-\langle x^{2}\rangle\langle
e^{\text{i}kx}\rangle-2\langle x\rangle\langle
xe^{\text{i}kx}\rangle_{\text{c}},$ (8) $\displaystyle\dots$

There is a diagrammatic way to understand this result that closely resembles
the 2-dimensional topological gravity model which is introduced in
Marolf:2020xie . Formally writing

$\displaystyle\langle\mathcal{O}e^{\text{i}kx}\rangle=\langle\mathcal{O}|e^{\text{i}kx}\rangle,$
(9)

we can interpret the state $|e^{\text{i}kx}\rangle$ as a “spacetime”
D-brane${}_{\text{i}k}$ state that is similar to that introduced in
Marolf:2020xie . Then the relation (5) can be understood as in Figure 1 where
the meaning of the subscript $c$ is transparent.

Figure 1:  Each $x$ denotes a circular boundary and the bracket
$\langle\cdot\rangle$ denotes a bulk amplitude. The first two diagrams denote
$\langle x^{2}\rangle\langle e^{\text{i}kx}\rangle$ and the last two diagrams
denote the “connected” parts of the correlation function $2\langle
x\rangle\langle xe^{\text{i}kx}\rangle_{c}+\langle
x^{2}e^{\text{i}kx}\rangle_{c}$.

We would like to get an estimation of the difference between any quantity
$X^{n}$ and its ensemble average $\langle X^{n}\rangle$, which requires a
simple evaluation of $\langle x^{n}e^{ikx}\rangle$. Motivated by the diagrams
in Figure 1 and a similar proposal in Mukhametzhanov:2021hdi , we propose the
following approximation

$\displaystyle{\langle x^{2}e^{\text{i}kx}\rangle_{c}}\approx\langle
xe^{\text{i}kx}\rangle_{\text{c}}\frac{1}{\langle
e^{\text{i}kx}\rangle}\langle xe^{\text{i}kx}\rangle_{\text{c}}\,,$ (10)

which has a diagrammatic interpretation as a recursive computation of
configurations with a higher number of contractions to the spacetime brane
from gluing the fundamental building blocks $\langle
xe^{\text{i}kx}\rangle_{\text{c}}$ with the “propagator” $\langle
e^{\text{i}kx}\rangle^{-1}$.

Equivalently, this relation can be presented as

$\displaystyle\frac{\langle x^{2}e^{\text{i}kx}\rangle}{\langle
e^{\text{i}kx}\rangle}$ $\displaystyle\approx$ $\displaystyle\langle
x^{2}\rangle-\langle x\rangle^{2}+\frac{\langle xe^{\text{i}kx}\rangle\langle
xe^{\text{i}kx}\rangle}{\langle e^{\text{i}kx}\rangle\langle
e^{\text{i}kx}\rangle}$ (11) $\displaystyle=$ $\displaystyle\langle
x^{2}\rangle+2\langle x\rangle\frac{\langle
xe^{\text{i}kx}\rangle_{\text{c}}}{\langle
e^{\text{i}kx}\rangle}+\frac{\langle xe^{\text{i}kx}\rangle_{\text{c}}\langle
xe^{\text{i}kx}\rangle_{\text{c}}}{\langle e^{\text{i}kx}\rangle\langle
e^{\text{i}kx}\rangle}\ .$ (12)

Making use of the fact that the quantity $\langle
e^{\text{i}kx}\rangle\equiv\varphi(k)$ is the characteristic function of the
probability distribution whose inverse Fourier transformation is the PDF

$\displaystyle\frac{1}{2\pi}\int\varphi(k)e^{-\text{i}kX}dk=P(X)\,,$ (13)

the relation (10) is equivalent to

$\displaystyle X^{2}$ $\displaystyle\approx$ $\displaystyle\langle
X^{2}\rangle-\langle X\rangle^{2}+\Phi\,,\quad\Phi=\frac{1}{2\pi}\int
dk\frac{e^{-\text{i}kX}}{P(X)}\langle e^{\text{i}kx}\rangle\left(\frac{\langle
xe^{\text{i}kx}\rangle}{\langle e^{\text{i}kx}\rangle}\right)^{2}.$ (14)

A more instructive form of this approximation is

$\displaystyle X^{2}\approx\langle
X^{2}\rangle+\tilde{\Phi}\,,\quad\tilde{\Phi}=\frac{1}{2\pi}\int
dk\frac{e^{-\text{i}kX}}{P(X)}\langle e^{\text{i}kx}\rangle\left(\frac{\langle
xe^{\text{i}kx}\rangle^{2}_{c}}{\langle
e^{\text{i}kx}\rangle^{2}}+\frac{2\langle x\rangle\langle
xe^{\text{i}kx}\rangle_{c}}{\langle e^{\text{i}kx}\rangle}\right)\,,$ (15)

where $\langle\tilde{\Phi}\rangle=0$. We will call the connected piece
$\langle X^{2}\rangle_{c}\equiv\langle X^{2}\rangle-\langle X\rangle^{2}$ the
“wormhole” contribution and $\Phi$ the “half-wormhole” contribution although
it’s mean value is non-vanishing.

As a simple example, the Gaussian distribution
$\mathcal{N}(\mu,t^{2}+\mu^{2})$ has the non-vanishing cumulants

$\displaystyle c_{1}=\mu,\quad c_{2}=t^{2},$ (16)

such that

$\displaystyle\frac{\langle xe^{\text{i}kx}\rangle}{\langle
e^{\text{i}kx}\rangle}=\mu+\text{i}kt^{2},\quad\left(\frac{\langle
xe^{\text{i}kx}\rangle}{\langle
e^{\text{i}kx}\rangle}\right)^{2}=\mu^{2}-k^{2}t^{4}+2\text{i}k\mu t^{2}.$
(17)

Substituting the above into (14) gives

$\displaystyle\Phi=X^{2}-t^{2},$ (18)

which means that for Gaussian distribution the approximation (14) is actually
exact. Clearly, this approximation cannot be exact for an arbitrarily general
probability distribution. For example, for exponential distribution
$\mathcal{E}(\lambda)$ the half-wormhole part is given by

$\displaystyle\Phi=\frac{X^{2}}{2}\ ,\qquad x\geq 0\,,$ (19)

and we quantify the error by its ratio to the variance of $X^{2}$

$\displaystyle\text{Error}=X^{2}-\langle X^{2}\rangle+\langle
X\rangle^{2}-\Phi\,,\qquad\rho=\frac{\langle\text{Error}^{2}\rangle}{\langle
X^{4}\rangle}=\frac{5}{24}\ .$ (20)

In fact, the error of the approximation (10) or (14) can be derived explicitly
for any general distribution. Denoting the cumulants of the probability
distribution as $c_{n}$, namely

$\displaystyle\log\langle
e^{\text{i}kx}\rangle\equiv\log\varphi(k)=\sum_{n=0}^{\infty}c_{n}\frac{(ik)^{n}}{n!}\,,$
(21)

we find111Notice that $\langle\cdot\rangle_{c}$ is not a linear functional, so
we don’t expect similar relations for $\langle x^{n}e^{ikx}\rangle$.

$\displaystyle(-\text{i}\partial_{k})\log\langle
e^{\text{i}kx}\rangle=\frac{\langle xe^{\text{i}kx}\rangle}{\langle
e^{\text{i}kx}\rangle}=\frac{\langle xe^{\text{i}kx}\rangle_{c}}{\langle
e^{\text{i}kx}\rangle}+\langle
x\rangle=\sum_{n=0}^{\infty}c_{n+1}\frac{(\text{i}k)^{n}}{n!}\,,$ (22)

which means

$\displaystyle\frac{\langle xe^{\text{i}kx}\rangle_{c}}{\langle
e^{\text{i}kx}\rangle}=\sum_{n=1}^{\infty}c_{n+1}\frac{(\text{i}k)^{n}}{n!}\
.$ (23)

Similarly,

$\displaystyle(-\text{i}\partial_{k})^{2}\log\langle
e^{\text{i}kx}\rangle=\frac{\langle x^{2}e^{\text{i}kx}\rangle}{\langle
e^{\text{i}kx}\rangle}-\frac{\langle xe^{\text{i}kx}\rangle\langle
xe^{\text{i}kx}\rangle}{\langle e^{\text{i}kx}\rangle\langle
e^{\text{i}kx}\rangle}=\sum_{n=0}^{\infty}c_{n+2}\frac{(\text{i}k)^{n}}{n!}\,,$
(24)

which means

$\displaystyle\frac{\langle x^{2}e^{\text{i}kx}\rangle_{c}}{\langle
e^{\text{i}kx}\rangle}-\frac{\langle xe^{\text{i}kx}\rangle_{\text{c}}\langle
xe^{\text{i}kx}\rangle_{\text{c}}}{\langle e^{\text{i}kx}\rangle\langle
e^{\text{i}kx}\rangle}=\sum_{n=1}^{\infty}c_{n+2}\frac{(\text{i}k)^{n}}{n!}\
.$ (25)

The approximation (10) is thus originated from neglecting all higher $c_{k}$
with $k>2$.

This implies that indeed the approximation (10) or (14) is exact when the
distribution is Gaussian, namely $c_{n}=0$ for $n>2$.

Similarly we can consider the approximation of $X^{n}$. We first derive the
approximation of the connected correlators in the presence of spacetime brane.
Taking the higher order derivative of the cumulant generating functions, for
example when $n=3$, we get

$\displaystyle(-\text{i}\partial_{k})^{3}\log\langle
e^{\text{i}kx}\rangle=\frac{\langle x^{3}e^{\text{i}kx}\rangle}{\langle
e^{\text{i}kx}\rangle}-3\frac{\langle x^{2}e^{\text{i}kx}\rangle\langle
xe^{\text{i}kx}\rangle}{\langle e^{\text{i}kx}\rangle\langle
e^{\text{i}kx}\rangle}+2\left(\frac{\langle xe^{\text{i}kx}\rangle}{\langle
e^{\text{i}kx}\rangle}\right)^{3}\ .$ (26)

Separating out connected and disconnected parts, we get

$\displaystyle(-\text{i}\partial_{k})^{3}\log\langle
e^{\text{i}kx}\rangle=\frac{\langle x^{3}e^{\text{i}kx}\rangle_{c}}{\langle
e^{\text{i}kx}\rangle}-3\frac{\langle x^{2}e^{\text{i}kx}\rangle_{c}\langle
xe^{\text{i}kx}\rangle_{c}}{\langle e^{\text{i}kx}\rangle\langle
e^{\text{i}kx}\rangle}+2\left(\frac{\langle
xe^{\text{i}kx}\rangle_{c}}{\langle e^{\text{i}kx}\rangle}\right)^{3}+\langle
x^{3}\rangle_{c}\,,$ (27)

where

$\displaystyle\langle x^{3}\rangle_{c}=\langle x^{3}\rangle-3\langle
x^{2}\rangle\langle x\rangle+2\langle x\rangle^{3}\,,$ (28)

is the connected correlator that equals to $c_{3}$. Therefore we arrive at

$\displaystyle\frac{\langle x^{3}e^{\text{i}kx}\rangle_{c}}{\langle
e^{\text{i}kx}\rangle}-3\frac{\langle x^{2}e^{\text{i}kx}\rangle_{c}\langle
xe^{\text{i}kx}\rangle_{c}}{\langle e^{\text{i}kx}\rangle\langle
e^{\text{i}kx}\rangle}+2\left(\frac{\langle
xe^{\text{i}kx}\rangle_{c}}{\langle
e^{\text{i}kx}\rangle}\right)^{3}=\sum_{n=1}^{\infty}c_{n+3}\frac{(\text{i}k)^{n}}{n!}\
.$ (29)

This means up to the third cumulant we have approximately

$\displaystyle\frac{\langle x^{3}e^{\text{i}kx}\rangle_{c}}{\langle
e^{\text{i}kx}\rangle}\approx 3\frac{\langle
x^{2}e^{\text{i}kx}\rangle_{c}\langle xe^{\text{i}kx}\rangle_{c}}{\langle
e^{\text{i}kx}\rangle\langle e^{\text{i}kx}\rangle}-2\left(\frac{\langle
xe^{\text{i}kx}\rangle_{c}}{\langle e^{\text{i}kx}\rangle}\right)^{3}\,,$ (30)

and the error of this approximation is due to neglecting all $c_{k}$ with
$k>3$. It is clear from this computation that the error of this approximation
can be determined by (14). If the accuracy requirement is only up to the
second moment, it up to quadratic fluctuations, we can use the approximation
(10) again to get

$\displaystyle\frac{\langle x^{3}e^{\text{i}kx}\rangle_{c}}{\langle
e^{\text{i}kx}\rangle}\approx\left(\frac{\langle
xe^{\text{i}kx}\rangle_{c}}{\langle e^{\text{i}kx}\rangle}\right)^{3}\ ,$ (31)

which becomes exact when the distribution is Gaussian. In fact, we can derive
similar relations by taking higher order derivatives in (26) to get relations
among higher order $\langle x^{i}e^{\text{i}kx}\rangle_{c}$’s. If again we
need accuracy up to quadratic order one can prove by induction

$\displaystyle\frac{\langle x^{n}e^{\text{i}kx}\rangle_{c}}{\langle
e^{\text{i}kx}\rangle}\approx\left(\frac{\langle
xe^{\text{i}kx}\rangle_{c}}{\langle e^{\text{i}kx}\rangle}\right)^{n}\ .$ (32)

We can then approximate the un-average $X^{3}$ to a required accuracy. In
practice, we rewrite the definition of $X^{n}$ according to (2), then expand
the $\langle x^{n}e^{\text{i}kx}\rangle$ in (2) in terms of the connected
correlators $\langle x^{i}e^{\text{i}kx}\rangle_{c}$ according to e.g.
(4)-(6). Then depending on the accuracy requirement, we use relations
analogous to either (30) or (61), (32), to write down the approximation and
the error of the final approximation is the composition of the errors the
different approximations of $\langle x^{n}e^{ikx}\rangle$. The general
expression of the approximation of $X^{n}$ and the corresponding errors are
complicated. But we will present some general procedures that work for any
distribution once an accuracy goal is given.

####  2.1.1 Recursion relations for approximations to arbitrary accuracy

Define $\Phi_{n}=\frac{1}{2\pi}\int\frac{e^{\text{i}kX}}{P(X)}\langle
e^{\text{i}kx}\rangle^{1-n}\langle xe^{\text{i}kx}\rangle^{n}$, we have

$\displaystyle X^{m}\Phi_{n}$
$\displaystyle=\frac{1}{2\pi}\int\frac{X^{m}e^{\text{i}kX}}{P(X)}\langle
e^{\text{i}kx}\rangle^{1-n}\langle
xe^{\text{i}kx}\rangle^{n}=\frac{1}{2\pi}\int\frac{e^{\text{i}kX}}{P(X)}\left(-i\partial_{k}\right)^{m}\left(\langle
e^{\text{i}kx}\rangle^{1-n}\langle xe^{\text{i}kx}\rangle^{n}\right)\ .$ (33)

Evaluating the derivative gives a result involving $\langle
x^{i}e^{\text{i}kx}\rangle$ with $1\leq i\leq m+1$. Rewriting them in terms of
$\langle x^{i}e^{\text{i}kx}\rangle_{c}$ with the help of e.g. (4)-(6). Then
use the approximation either (30) or (61), (32) according to the required
accuracy. Then rewrite the $\langle x^{i}e^{\text{i}kx}\rangle_{c}$ in the
approximated results back in terms of $\langle x^{i}e^{\text{i}kx}\rangle$,
and the result will be a relation among $\Phi_{i}$ with $1\leq i\leq m+1$.
Making use of the fact that $\Phi_{1}=X$ and recursively carrying out the
above procedure to evaluate $X^{n-1}\Phi_{1}$, we get the approximation of
$X^{n}$ to the desired accuracy.

For example, if we require accuracy to the second order, we simply consider

$\displaystyle X\Phi_{n}$
$\displaystyle=\frac{1}{2\pi}\int\frac{e^{\text{i}kX}}{P(X)}\langle
e^{\text{i}kx}\rangle^{-n}\left(n\langle x^{2}e^{\text{i}kx}\rangle\langle
xe^{\text{i}kx}\rangle^{n-1}\langle e^{\text{i}kx}\rangle+(1-n)\langle
xe^{\text{i}kx}\rangle^{n+1}\right)\ .$ (34)

Following the above procedure to rewrite $\langle x^{2}e^{\text{i}kx}\rangle$,
we arrive at

$\displaystyle X\Phi_{n}=n\left(\langle x^{2}\rangle-\langle
x\rangle^{2}\right)\Phi_{n-1}+\Phi_{n+1}\ .$ (35)

For example, we can evaluate

$\displaystyle
X^{3}=X^{2}\Phi_{1}=3\left(\mu_{2}-\mu_{1}^{2}\right)\Phi_{1}+\Phi_{3}\,,$
(36)

where we keep only accuracy up to the quadratic order, so $\mu_{3}$ does not
appear independently; it is simply replaced by

$\displaystyle\mu_{3}=3\mu_{1}\mu_{2}-2\mu_{1}^{3}\ .$ (37)

####  2.1.2 Explicit relations for Gaussian approximation

If we only want Gaussian approximations of $X^{n}$, we can get an explicit
approximation formula. First let introduce some convenient notations

$\displaystyle\phi_{n}=\frac{\langle x^{n}e^{\text{i}kx}\rangle}{\langle
e^{\text{i}kx}\rangle},\quad\phi^{c}_{n}=\frac{\langle
x^{n}e^{\text{i}kx}\rangle_{c}}{\langle e^{\text{i}kx}\rangle},$ (38)
$\displaystyle\langle x^{n}\rangle=\mu_{n},\quad\langle
x^{n}\rangle_{\text{cumulant}}=c_{n}.$ (39)

The cumulant $c_{m}$ can be expressed as a polynomial of moments

$\displaystyle c_{m}=P_{m}(\mu_{m},\mu_{m-1},\dots,\mu_{1}).$ (40)

Some examples are

$\displaystyle c_{1}=\mu_{1},\quad c_{2}=\mu_{2}-\mu_{1}^{2},\quad
c_{3}=\mu_{3}-3\mu_{1}\mu_{2}+2\mu_{1}^{3},\dots$ (41)

Note that the coefficient of $\mu_{m}$ is 1. Of course the relations can be
inverted

$\displaystyle\mu_{m}=Q_{m}(c_{m},c_{m-1},\dots c_{1}).$ (42)

Similar to (4),(5) and (6), $\phi_{n}$ can be decomposed as

$\displaystyle\phi_{m}=\tilde{P}_{m}(\phi_{m}^{c},\dots,\phi_{0}^{c}),$ (43)

for example

$\displaystyle\phi_{1}=\phi_{1}^{c}+\mu_{1}\phi_{0}^{c},\quad\phi_{2}=\phi_{2}^{c}+2\mu_{1}\phi_{1}^{c}+\mu_{2},\dots.$
(44)

Since $\log\langle e^{\text{i}kx}\rangle$ is the generating function of
$c_{n}$ we have 222The simplest way to see this is to set $k=0$, then it
reduces to (40) and to notice that the coefficients of the polynomial $P_{m}$
do not depend on $k$.

$\displaystyle(-\text{i}\partial_{k})^{m}\log\langle
e^{\text{i}kx}\rangle=P_{m}(\phi_{m},\phi_{m-1},\dots,\phi_{1})=\sum_{n=0}c_{n+m}\frac{(\text{i}k)^{n}}{n!}.$
(45)

Using (43) and (42) the left-hand side can be expanded as a polynomial of
$c_{i}$ with coefficients to be functions of $\phi_{i}^{c}$:

$\displaystyle
P_{m}(\phi_{m},\phi_{m-1},\dots,\phi_{1})=P_{m}(\tilde{P}_{m}(\phi_{i}^{c}),\tilde{P}_{m-1}(\phi_{i}^{c}),\dots,\tilde{P}_{1}(\phi_{i}^{c}))\
.$ (46)

For example

$\displaystyle P_{2}$ $\displaystyle=$
$\displaystyle\phi_{2}-\phi_{1}^{2}=\tilde{P}_{2}-2{\tilde{P}_{1}}^{2}$ (47)
$\displaystyle=$
$\displaystyle\phi_{2}^{c}+2\mu_{1}\phi_{1}^{c}+\mu_{2}-{\phi_{1}^{c}}^{2}-\mu_{1}^{2}{\phi_{0}^{c}}^{2}-2\mu_{1}\phi_{1}^{c}\phi_{0}^{c}$
(48) $\displaystyle=$
$\displaystyle\phi_{2}^{c}-{\phi_{1}^{c}}^{2}+c_{1}(2\phi_{1}^{c}-2\phi_{1}^{c}\phi_{0}^{c})+c_{2}.$
(49)

Therefore we end up with

$\displaystyle
P_{m}=M_{m}+c_{1}M_{m-1}^{(1)}+(c_{1}^{2}M_{m-2}^{(1)}+c_{2}M_{m-2}^{(2)})+\dots+c_{m}=\sum_{n=0}c_{n+m}\frac{(\text{i}k)^{n}}{n!}\,,$
(50)

where each $M_{i}^{(k)}$ is a function of the $\phi^{c}_{i}$’s. Since the
subscript $i$ of $\phi^{c}_{i}$ and $M_{i}$ both indicate the power of $x$, it
is clear that

$\displaystyle\sum_{a}i_{a}=m\,,\qquad\forall\left(\prod_{a}\phi_{i_{a}}^{c}\right)\in
M_{m}\,,$ (51)

where $\prod_{a}\phi_{i_{a}}^{c}$ is any term in $M_{m}$. Notice that these
relations are true for arbitrary $k$, $m$ and distributions, then the non-
trivial solution is only

$\displaystyle\quad M_{n}^{(p)}=0\,,\qquad
M_{m}=P_{m}(\phi^{c}_{m},\phi^{c}_{m-1},\dots,\phi^{c}_{1})=\sum_{n=1}c_{n+m}\frac{(\text{i}k)^{n}}{n!}\
.$ (52)

The Gaussian approximation means $c_{m}=0$ for all $m>2$. This requires

$\displaystyle P_{m}(\phi^{c}_{m},\phi^{c}_{m-1},\dots,\phi^{c}_{1})\approx
0\,,\quad\forall m>1\ .$ (53)

At $m=2$ this relation means

$\displaystyle P_{2}(\phi^{c}_{2},\phi^{c}_{1})\approx 0\,,$ (54)

which combines with (51) means
$\phi^{c}_{2}=\alpha\left(\phi^{c}_{1}\right)^{2}$ and

$\displaystyle P_{2}(\alpha\phi^{c}_{2},\phi^{c}_{1})=0\ .$ (55)

To fix the normalization $\alpha$, we notice that since the above relations
(40) -(53), in particular the functional form of $P$, are true for arbitrary
distribution, we can choose the delta function distribution such that
$c_{n}=0,\forall n\geq 2$ and $\mu_{m}=\mu_{1}^{m}$, we can get the identity

$\displaystyle P_{m}(\mu^{m},\mu^{m-1},\dots\mu)=0,\quad m\geq 2,$ (56)

thus combining this with (55) we conclude $\alpha=1$ and

$\displaystyle\phi^{c}_{2}\approx\left(\phi^{c}_{1}\right)^{2}\,,$ (57)

where $\approx$ is due to the Gaussian approximation. This is nothing but the
approximation (10). Iterating this procedure successively for different $m$,
we reach to

$\displaystyle\phi_{m}^{c}\approx{(\phi_{1}^{c})}^{m}\,,$ (58)

in the Gaussian approximation. Then we can approximate $X^{m}$ as

$\displaystyle X^{m}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int\frac{e^{\text{i}kX}}{P(X)}\langle
e^{\text{i}kx}\rangle\phi_{m}=\frac{1}{2\pi}\int\frac{e^{\text{i}kX}}{P(X)}\langle
e^{\text{i}kx}\rangle\tilde{P}_{m}(\phi_{m}^{c},\dots,\phi_{1}^{c},1)$ (59)
$\displaystyle\approx$
$\displaystyle\frac{1}{2\pi}\int\frac{e^{\text{i}kX}}{P(X)}\langle
e^{\text{i}kx}\rangle\tilde{P}_{m}((\phi_{1}^{c})^{m},(\phi_{1}^{c})^{m-1},\dots,\phi_{1}^{c},1)$
(60) $\displaystyle=$ $\displaystyle\sum_{i=0}^{m}{m\choose
i}\mu_{i}\Phi_{m-i}\,,$ (61)

where $\Phi_{i}=\frac{1}{2\pi}\int\text{d}k\frac{e^{\text{i}kX}}{P(X)}\langle
e^{\text{i}kx}\rangle(\phi_{1}^{c})^{i}$, and it may be understood as
generalized wormholes which we will report somewhere else.

It is easy to check that the result (61) agrees with (36) once the relation
(37) is used.

###  2.2 Models with multiple independent identical random variables

In statistical models with a single random variable, the various moments are
all observables that we can compute. On the other hand, we would like to
consider other interesting observables. We therefore proceed to consider
operators in statistical models with multiple independent identical random
variables.

One class of operators in these models is the light operators that are simply
linear combinations of the random variables $X_{i}$. We conjecture that if
$Y(X_{i})$ is some function of a large number $N$ independent random variables
$X_{i}$ such that $Y$ is approximately Gaussian, then the approximation

$\displaystyle Y^{2}\approx\langle Y^{2}\rangle-\langle Y\rangle^{2}+\Phi,$
(62)
$\displaystyle\Phi(X)=\frac{1}{(2\pi)^{N}}\int\prod_{i}\left(dk_{i}\frac{e^{-\text{i}k_{i}X_{i}}}{P(X_{i})}\right)\langle
e^{\text{i}\sum_{i}k_{i}x_{i}}\rangle\left(\frac{\langle
Y(x)e^{\text{i}\sum_{i}k_{i}x_{i}}\rangle}{\langle
e^{\text{i}\sum_{i}k_{i}x_{i}}\rangle}\right)^{2}.$ (63)

is good in the sense that

$\displaystyle\rho\equiv\frac{\langle\text{Error}^{2}\rangle}{\langle
Y^{2}\rangle^{2}}\,,$ (64)

is suppressed by $1/N$.

Like (15) we can rewrite it into

$\displaystyle Y^{2}\approx\langle Y^{2}\rangle+\tilde{\Phi},$ (65)
$\displaystyle\tilde{\Phi}(X)=\frac{1}{(2\pi)^{N}}\int\prod_{i}\left(dk_{i}\frac{e^{-\text{i}k_{i}X_{i}}}{P(X_{i})}\right)\langle
e^{\text{i}\sum_{i}k_{i}x_{i}}\rangle\left(\frac{\langle
Y(x)e^{\text{i}\sum_{i}k_{i}x_{i}}\rangle_{c}^{2}}{\langle
e^{\text{i}\sum_{i}k_{i}x_{i}}\rangle^{2}}+\frac{2\langle Y\rangle\langle
Y(x)e^{\text{i}\sum_{i}k_{i}x_{i}}\rangle_{c}}{\langle
e^{\text{i}\sum_{i}k_{i}x_{i}}\rangle}\right).$ (66)

####  2.2.1 Simple observables

The fundamental logic in this section is that by the central limit theorem
(CLT), summing over a large number of i.i.d random variables gives a random
variable that approximately obey a Gaussian distribution. Explicitly, if
$X_{i}$ is from a normal distribution ${\cal N}(\mu,\sigma^{2})$, then the
mean of $N$ such i.i.d’s

$\displaystyle\tilde{Y}=\frac{1}{N}\sum_{i=1}^{N}X_{i}\,,$ (67)

is approximately a Gaussian random variable from ${\cal N}(\mu,\sigma^{2}/N)$
when $N$ is large enough.

In this paper, it turns out that it is more convenient to define

$\displaystyle Y=\sum_{i=1}^{N}X_{i}\,,$ (68)

so that the connection to the SYK model is more transparent. Then $Y$ is a
Gaussian random variable with probability distribution ${\cal
N}(N\mu,N\sigma^{2})$ when $N$ is large. In particular, we expect

$\displaystyle\langle Y^{4}\rangle\approx 3\langle Y^{2}\rangle^{2}-2\langle
Y\rangle^{4}\,,\quad\langle Y^{2}\rangle\approx N\left(\langle
X^{2}\rangle-\langle X\rangle^{2}\right)+N^{2}\langle
X\rangle^{2}\,,\quad\langle Y\rangle\approx N\langle X\rangle\ .$ (69)

They can be checked by a direct calculation

$\displaystyle\langle Y^{2}\rangle=N\langle X^{2}\rangle+N(N-1)\langle
X\rangle^{2},$ (70) $\displaystyle\langle Y^{4}\rangle=N\langle
X^{4}\rangle+N(N-1)\left(4\langle X^{3}\rangle\langle X\rangle+3\langle
X^{2}\rangle^{2}\right)$ $\displaystyle\quad+6N(N-1)(N-2)\langle
X^{2}\rangle\langle X\rangle^{2}+N(N-1)(N-2)(N-3)\langle X\rangle^{4}\ .$ (71)

Because all the $X_{i}$ are independent so that it is straightforward to
obtain

$\displaystyle\frac{\langle Ye^{\text{i}k_{i}x_{i}}\rangle}{\langle
e^{\text{i}k_{i}x_{i}}\rangle}=\sum_{i}\frac{\langle
x_{i}e^{\text{i}k_{i}x_{i}}\rangle}{\langle
e^{\text{i}k_{i}x_{i}}\rangle}\equiv\sum_{i}k_{i}[1].$ (72)

Next we can rewrite the square of (72) into the diagonal terms and off-
diagonal terms

$\displaystyle\left(\frac{\langle Ye^{\text{i}k_{i}x_{i}}\rangle}{\langle
e^{\text{i}k_{i}x_{i}}\rangle}\right)^{2}=\sum_{i}k_{i}[1]^{2}+\sum_{i\neq
j}k_{i}[1]k_{j}[1].$ (73)

To compute the off-diagonal contributions to the half-wormhole, we observe
that

$\displaystyle\frac{1}{(2\pi)^{N}}\int\prod_{i}\left(dk_{i}\frac{e^{-\text{i}k_{i}X_{i}}}{P(X_{i})}\right)\langle
e^{\text{i}\sum_{i}k_{i}x_{i}}\rangle k_{i}[1]k_{j}[1]$ (74)
$\displaystyle=\frac{1}{(2\pi)^{2}}\int
dk_{i}dk_{j}\frac{e^{-\text{i}k_{i}X_{i}-\text{i}k_{j}X_{j}}}{P(X_{i})P(X_{j})}\langle
x_{i}e^{\text{i}k_{i}x_{i}}\rangle\langle
x_{j}e^{\text{i}k_{j}x_{j}}\rangle=X_{i}X_{j}\ .$ (75)

In terms of $\widehat{k_{i}[n]^{m}}$ which are defined in (585) the half-
wormhole can be written as

$\displaystyle\Phi=\sum_{i}\widehat{k_{i}[1]^{2}}+\sum_{i\neq j}X_{i}X_{j},$
(76)

and the error is given by

Error
$\displaystyle=\sum_{i}\left(X_{i}^{2}-\widehat{k_{i}[1]^{2}}-t^{2}\right),\quad
t^{2}=\langle X_{i}^{2}\rangle-\langle X_{i}\rangle^{2}\,,$ (77)
$\displaystyle\langle\text{Error}^{2}\rangle$
$\displaystyle=\sum_{i,j}\langle(X_{i}^{2}-\widehat{k_{i}[1]^{2}})(X_{j}^{2}-\widehat{k_{j}[1]^{2}})\rangle+N^{2}t^{4}-2Nt^{2}\sum_{i}(X_{i}^{2}-\widehat{k_{i}[1]^{2}})\
.$ (78)

Recalling that $\langle Y^{2}\rangle\sim Nt^{2}$ so to prove the conjecture
(62) we need to show that the ${\cal O}(N^{2})$ term in (78) vanish. A direct
calculation gives

$\displaystyle\langle\widehat{k_{i}[1]^{2}}\rangle_{X_{i}}$ $\displaystyle=$
$\displaystyle\int dX_{i}P(X_{i})k_{i}[1]^{2}\langle
e^{\text{i}k_{i}x_{i}}\rangle_{x_{i}}\frac{e^{-\text{i}k_{i}X_{i}}}{P(X_{i})}dk_{i}$
(79) $\displaystyle=$ $\displaystyle\int dk_{i}\delta(-k_{i})\langle
e^{\text{i}k_{i}x_{i}}\rangle_{x_{i}}k_{i}[1]^{2}=\langle X_{i}\rangle^{2}\ .$
(80)

This means

$\displaystyle\langle(X_{i}^{2}-\widehat{k_{i}[1]^{2}}\rangle=\langle
X^{2}_{i}\rangle-\langle X_{i}\rangle^{2}=t^{2}\
.\quad\Leftrightarrow\quad\langle\widehat{k_{i}[1]^{2}}\rangle_{X_{i}}\approx\langle
X_{i}\rangle^{2}\ .$ (81)

In particular, a consequence of this relation is that although all the 3 terms
in (78) are of order ${\cal O}(N^{2})$, the sum of them cancelled exactly
since (81) does not depend on $i$. This then shows that
$\langle\text{Error}^{2}\rangle\ll\langle Y^{2}\rangle$ and hence the
approximation (62) is valid.

We can derive this result in a more illuminating fashion. First using (23)
$k_{i}[1]$ can be expressed as

$\displaystyle k_{i}[1]=\sum_{n=0}\frac{(-\text{i}k)^{n}}{n!}c_{n+1}.$ (82)

Then using the fact that the inverse Fourier transformation of the
characteristic function is the PDF we find

$\displaystyle\langle\widehat{k_{i}[1]^{2}}\rangle_{X_{i}}=\int
dX_{i}P(X_{i})\sum_{n,m=0}\frac{c_{n+1}c_{m+1}}{n!m!}(-\text{i}k_{i})^{n+m}\langle
e^{\text{i}k_{i}x_{i}}\rangle_{x_{i}}\frac{e^{-\text{i}k_{i}X_{i}}}{P(X_{i})}dk_{i}$
(83) $\displaystyle\quad=\int
dX_{i}\sum_{n,m=0}\frac{c_{n+1}c_{m+1}}{n!m!}(\partial_{X_{i}})^{n+m}P(X_{i})=c_{1}^{2}=\langle
X_{i}\rangle^{2}\ .$ (84)

####  2.2.2 Power-sum observables

In this section, we consider another class of more general observables

$\displaystyle Y=\sum_{i}f(X_{i}),\quad Y^{2}=\sum_{i,j}f(X_{i})f(X_{j}),$
(85)

where $X_{i}$ are still independent identical random variables with PDF
$P_{X_{i}}$ and $f$ is some smooth function so that $f(X_{i})$ are also
independent and identical random variables with a new PDF $P_{f}$:

$\displaystyle\int dXF[f(X)]P_{X}=\int dfF(f)P_{f}.$ (86)

The CLT is still valid but the proposal may not because naively it depends on
the function $f$. By smooth function we mean $f(X_{i})$ is not singular
anywhere such that it can be Taylor expanded

$\displaystyle f(X_{i})=\sum_{n}a_{n}X_{i}^{n}\,,$ (87)

whose expansion coefficients satisfy

$\displaystyle a_{n}\approx 0\,,\qquad\forall n>n_{0}\,,\quad n_{0}\ll N\ .$
(88)

Accordingly (72) and (73) become

$\displaystyle\frac{\langle Ye^{\text{i}k_{i}x_{i}}\rangle}{\langle
e^{\text{i}k_{i}x_{i}}\rangle}=\sum_{i}\sum_{n}a_{n}k_{i}[n].$ (89)
$\displaystyle\left(\frac{\langle Ye^{\text{i}k_{i}x_{i}}\rangle}{\langle
e^{\text{i}k_{i}x_{i}}\rangle}\right)^{2}=\sum_{i,j}\left(\sum_{n}a_{n}k_{i}[n]\sum_{m}a_{m}k_{j}[m]\right).$
(90)

So the error is given by

Error
$\displaystyle=\sum_{i}\left(f^{2}(X_{i})-t^{2}-\sum_{n,m}a_{n}a_{m}\widehat{k_{i}[n]k_{i}[m]}\right)\,,$
(91) $\displaystyle\langle\text{Error}^{2}\rangle$
$\displaystyle=\langle\sum_{i,j}(f^{2}(X_{i})-\sum_{n,m}a_{n}a_{m}\widehat{k_{i}[n]k_{i}[m]})(f^{2}(X_{j})-\sum_{n,m}a_{n}a_{m}\widehat{k_{j}[n]k_{j}[m]})\rangle$
$\displaystyle\quad+N^{2}t^{4}-2Nt^{2}\sum_{i}(f^{2}(X_{i})-\sum_{n,m}a_{n}a_{m}\widehat{k_{i}[n]k_{i}[m]})\,,$
(92)

where $t^{2}=\langle f^{2}(X_{i})\rangle-\langle f(X_{i})\rangle^{2}$. Similar
to the calculation of (80), one can find

$\displaystyle\langle\sum_{n,m}a_{n}a_{m}\widehat{k_{i}[n]k_{i}[m]}\rangle=\langle
f(X_{i})\rangle^{2},$ (93)

which means the leading order terms, ie of order $N^{2}$, in (92) is

$\displaystyle
2\left(\langle(f^{2}(X_{j})-\sum_{n,m}a_{n}a_{m}\widehat{k_{j}[n]k_{j}[m]})\rangle^{2}-t^{4}\right)N^{2}=0\,,$
(94)

As a result, the error is small and indeed the approximation (62) is
reasonable in this case too. We also show some explicit examples in the
Appendix (B). More generally, following the same procedure one can show that
the half wormhole proposal is correct for the following family of functions

$\displaystyle
Y_{k}=\sum_{i}^{N}\left(f(X_{i_{1}},X_{i_{2}},\dots,X_{i_{k}})\right),$ (95)

where $X_{i_{p}}$ are independent and identical random variables.

####  2.2.3 Product observables

Previously the function $Y$ we considered are a summation of (polynomials of)
independent random variables. The proposal works very well for all the
probability distributions. However in the original construction of half
wormhole introduced in Saad:2021rcu , the function $Y$ is a determinant
observables which are “heavy” in the traditional field theory language

$\displaystyle
Y=\text{PF}(J)=\sum^{\prime}_{A_{1}<A_{2}<\dots<A_{p}}\text{sgn}(A){J}_{A_{1}}{J}_{A_{2}}\dots
J_{A_{p}},$ (96)

where the function $\text{PF}(J)$ is called the hyperpfaffian Barvinok  which
is a tensorial generalization of pfaffian and $J_{A_{i}}$ are random
variables. To mimic this construction let us consider a similar model:

$\displaystyle Y=\sum_{i_{1}\neq i_{2}\neq\dots\neq
i_{q}}^{N}X_{i_{1}}X_{i_{2}}\dots X_{i_{q}}.$ (97)

$\bullet$ $q=2$ Gaussian distribution
The simplest case is $q=2$:

$\displaystyle Y=\sum_{i\neq j}X_{i}X_{j},$ (98) $\displaystyle
Y^{2}=\sum_{i\neq j\neq p\neq q}X_{i}X_{j}X_{p}X_{q}+4\sum_{i\neq j\neq
p}X_{i}^{2}X_{j}X_{p}+2\sum_{i\neq j}X_{i}^{2}X_{j}^{2}\ .$ (99)

It is straightforward to get

$\displaystyle\langle
Y^{2}\rangle=N(N-1)\left(2t^{4}+4(N-1)\mu^{2}t^{2}+N(N-1)\mu^{4}\right),$
(100) $\displaystyle\langle X_{i}\rangle=\mu,\quad\langle X^{2}\rangle-\langle
X\rangle^{2}=t^{2}.$ (101)

So in general $\langle Y^{2}\rangle$ will scale as $N^{4}$ if $\mu\neq 0$,
while if $\mu=0$ it scales as $N^{2}$.

One example of the $\mu=0$ case is the Gaussian distribution
$\mathcal{N}(\mu=0,t^{2})$. We then verifies

$\displaystyle\langle Y\rangle=0,\quad\langle Y^{2}\rangle=2t^{4}N(N-1)$ (102)

and

$\displaystyle\Phi=\sum_{i\neq j\neq p\neq q}X_{i}X_{j}X_{p}X_{q}+4\sum_{i\neq
j\neq p}(X_{i}^{2}-t^{2})X_{j}X_{p}+2\sum_{i\neq
j}(X_{i}^{2}-t^{2})(X_{j}^{2}-t^{2})\ .$ (103)

Therefore we obtain

$\displaystyle\text{Error}=-2t^{4}N(N-1)+4t^{2}(N-2)\sum_{i\neq
j}X_{i}X_{j}+4(N-1)t^{2}\sum_{i}X_{i}^{2}-2t^{4}N(N-1),$
$\displaystyle\langle(\text{Error}/4)^{2}\rangle=(2+1+1-2)N^{4}t^{8}+\\#N^{3}+\dots$
(104)

the leading term does not vanish so the approximation

$\displaystyle Y^{2}\approx\langle Y^{2}\rangle-\langle Y\rangle^{2}+\Phi\,,$
(105)

is not good.

However, for more general Gaussian distributions $\mathcal{N}(\mu,t^{2})$
similar calculation gives

$\displaystyle\langle Y\rangle=N(N-1)\mu^{2},\quad\langle Y^{2}\rangle-\langle
Y\rangle^{2}\equiv\tilde{t}^{2}=2t^{2}N(N-1)(t^{2}+2(N-1)\mu^{2}),$ (106)

and

$\displaystyle\text{Error}=-\tilde{t}^{2}+4t^{2}(N-2)\sum_{i\neq
j}X_{i}X_{j}+4(N-1)t^{2}\sum_{i}X_{i}^{2}-2t^{4}N(N-1),$ (107)

now we find that

$\displaystyle\langle\text{Error}^{2}\rangle=32(3t^{2}\mu^{2}+\mu^{4})N^{5}+32(t^{4}-12t^{2}\mu^{2}-4\mu^{4})N^{4}+\dots$
(108)

and

$\displaystyle\frac{\langle\text{Error}^{2}\rangle}{\langle
Y^{2}\rangle^{2}}=\frac{2(\mu^{4}+3t^{2}\mu^{2}-3t^{4})}{(2t^{4}-4t^{2}\mu^{2})^{2}N}+\dots.$
(109)

Notice that the error is always small, even when $\mu\rightarrow 0$, and the
proposal is valid. This is because when $\mu\neq 0$, the moments of $Y$ behave
as

$\displaystyle\langle Y\rangle\approx N^{2}\mu,\quad\langle
Y^{2}\rangle\approx N^{4}\mu^{2},\quad\langle Y^{4}\rangle\approx
N^{8}\mu^{4},$ (110)

as expected from (69). It is thus clear that the $\mu\to 0$ limit is not
smooth.

It seems $\mu\neq 0$ is fundamentally better than the $\mu=0$ case in the
sense that the approximation (62) is good. But as we will discuss shortly in
section 2.3.1 this is not the case and the crucial point is that it is more
appropriate to compare the error with the connected contributions and left out
the disconnected contributions.

$\bullet$ General $q$
Next we consider general distributions. We show some details of the
computation for exponential distribution and Poisson distribution in the
Appendix (C). Here we only give a more abstract derivation. In terms of (585)
the half wormhole (63) can be written as

$\displaystyle\Phi$ $\displaystyle=$ $\displaystyle\sum_{i\neq j\neq p\neq
q}\widehat{k_{i}[1]}\widehat{k_{j}[1]}\widehat{k_{p}[1]}\widehat{k_{q}[1]}+4\sum_{i\neq
j\neq
p}\widehat{k_{i}[1]^{2}}\widehat{k_{j}[1]}\widehat{k_{p}[1]}+2\sum_{i\neq
j}\widehat{k_{i}[1]^{2}}\widehat{k_{j}[1]^{2}}$ (111) $\displaystyle=$
$\displaystyle\sum_{i\neq j\neq p\neq q}X_{i}X_{j}X_{p}X_{q}+4\sum_{i\neq
j\neq p}\widehat{k_{i}[1]^{2}}X_{j}X_{p}+2\sum_{i\neq
j}\widehat{k_{i}[1]^{2}}\widehat{k_{j}[1]^{2}}.$ (112)

Therefore the error of the proposal is

$\displaystyle\text{Error}=4\sum_{i\neq j\neq
q}(X_{i}^{2}-\widehat{k_{i}[1]^{2}})X_{j}X_{p}+2\sum_{i\neq
j}(X_{i}^{2}X_{j}^{2}-\widehat{k_{i}[1]^{2}}\widehat{k_{j}[1]^{2}})-\tilde{t}^{2}.$
(113)

The maximal power of $N$ in $\langle\text{Error}^{2}\rangle$ will be $6$.

When $\mu\neq 0$, $\langle Y^{2}\rangle^{2}\sim N^{8}$. So in this case the
error is small and the approximation is good.

When $\mu=0$, $\langle Y^{2}\rangle^{2}\sim N^{4}$. The terms of $N^{4}$ in
$\langle\text{Error}^{2}\rangle$ come from

$\displaystyle\langle\text{Error}^{2}\rangle$ $\displaystyle=$
$\displaystyle\langle\sum_{i\neq j\neq p\neq q}\\{16\times
2(X_{i}^{2}-\widehat{k_{i}[1]^{2}})(X_{j}^{2}-\widehat{k_{j}[1]^{2}})X_{p}^{2}X_{q}^{2}$
(114) $\displaystyle+$ $\displaystyle
4(X_{i}^{2}X_{j}^{2}-\widehat{k_{i}[1]^{2}}\widehat{k_{j}[1]^{2}})(X_{p}^{2}X_{q}^{2}-\widehat{k_{p}[1]^{2}}\widehat{k_{q}[1]^{2}})\\}$
$\displaystyle+$ $\displaystyle 4t^{16}N^{4}-8t^{4}N^{2}\sum_{i\neq
j}(X_{i}^{2}X_{j}^{2}-\widehat{k_{i}[1]^{2}}\widehat{k_{j}[1]^{2}})\rangle+\dots$
$\displaystyle=$ $\displaystyle
N^{4}t^{16}\left(32+4+4-8\right)+\\#N^{3}\dots=32N^{4}t^{16}+\\#N^{3}\dots$
(115)

which is not vanishing so the error is large and we cannot approximate $Y^{2}$
by $\langle Y^{2}\rangle+\Phi$ probably for the same reason as the $q=2$ case.
One could ask that when $\langle X_{i}^{2}-\widehat{k_{i}[1]^{2}}\rangle=0$,
the approximation might be fine, but it requires $t^{2}=0$ which we do not
consider at the moment.

$\bullet$ General distributions

Now we consider the general case (97):

$\displaystyle Y=\sum_{i_{1}\neq i_{2}\neq\dots\neq
i_{q}}^{N}X_{i_{1}}X_{i_{2}}\dots X_{i_{q}},$ (116) $\displaystyle
Y^{2}=\sum_{k=0}^{q}\frac{(q!/(q-k)!)^{2}}{k!}\sum_{j_{1}\neq j_{2}\dots
j_{k}\neq i_{1}\dots\neq i_{2q-2k}}X_{j_{1}}^{2}\dots
X_{j_{k}}^{2}X_{i_{1}}\dots X_{i_{2q-2k}}.$ (117)

If $N\gg q$ then the average $\langle Y^{2}\rangle$ will have the following
scaling behavior in the large $N$ limit

$\displaystyle\langle Y^{2}\rangle\sim\begin{cases}N^{2q}\mu^{2q}&\quad\mu\neq
0\\\ N^{q}q!t^{2q}&\quad\mu=0\end{cases}$ (118)

Similar to (112), one can find that the half wormhole contribution $\Phi$ can
be written as

$\displaystyle\Phi$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{q}\frac{(q!/(q-k)!)^{2}}{k!}\sum_{j_{1}\neq
j_{2}\dots j_{k}\neq i_{1}\dots\neq
i_{2q-2k}}\widehat{k_{j_{1}}[1]^{2}}\dots\widehat{k_{j_{k}}[1]^{2}}X_{i_{1}}\dots
X_{i_{2q-2k}},$ (119)

so that the error is

Error $\displaystyle=$
$\displaystyle\sum_{k=1}^{q}\frac{(q!/(q-k)!)^{2}}{k!}\sum_{\begin{subarray}{c}j_{1}\neq
j_{2}\dots j_{k}\neq\\\ i_{1}\dots\neq
i_{2q-2k}\end{subarray}}(X_{j_{1}}^{2}\dots
X_{j_{k}}^{2}-\widehat{k_{j_{1}}[1]^{2}}\dots\widehat{k_{j_{k}}[1]^{2}})X_{i_{1}}\dots
X_{i_{2q-2k}}$ (120) $\displaystyle-$ $\displaystyle\langle
Y^{2}\rangle+\langle Y\rangle^{2}.$

When $\mu\neq 0$, the leading contribution to $\langle\text{Error}^{2}\rangle$
scales as $N^{2q-2}$ so the approximation (62) is correct.

However when $\mu=0$, the leading contributions to
$\langle\text{Error}^{2}\rangle$ are

$\displaystyle\langle\text{Error}^{2}\rangle$
$\displaystyle=E_{1}+E_{2}+\\#N^{2q-1},$ (121) $\displaystyle E_{1}$
$\displaystyle=\langle\sum_{k=1}^{q}\left(\frac{(q!/(q-k)!)^{2}}{k!}\right)^{2}(2q-2k)!$
(122) $\displaystyle\quad\times\sum_{\begin{subarray}{c}j_{1}\neq
j_{2}\dots\neq j_{2k}\\\ \neq i_{1}\neq\dots\neq
i_{2q-2k}\end{subarray}}(X_{j_{1}}^{2}-\widehat{k_{j_{1}}[1]^{2}})\dots(X_{j_{k}}^{2}-\widehat{k_{j_{2k}}[1]^{2}})X_{i_{1}}^{2}\dots
X_{i_{2q-2k}}^{2}\rangle$
$\displaystyle=N^{2q}t^{4q}(2q)!\left(\,{}_{3}F_{2}\left(-q,-q,-q;1,\frac{1}{2}-q;\frac{1}{4}\right)-1\right)\neq
0,$ (123) $\displaystyle E_{2}$ $\displaystyle=\langle\left(q!\sum_{i_{1}\neq
i_{2}\neq\dots\neq i_{q}}^{N}(X_{i_{1}}^{2}\dots
X_{i_{q}}^{2}-\widehat{k_{i_{1}}[1]^{2}}\dots\widehat{k_{i_{q}}[1]^{2}})-q!N^{q}\right)^{2}\rangle=0\
.$ (124)

So the error is large as in the previous case (115) and the approximation (62)
is not good.

In our toy model (97) we did not include the “diagonal” terms while from our
analysis above we have shown in the large $N$ limit it is the “off-diagonal”
term that dominates. So our conclusions for (97) are also valid for the
following general function

$\displaystyle Y=\sum_{i_{1},i_{2},\dots,i_{q}=1}^{N}X_{i_{1}}X_{i_{2}}\dots
X_{i_{q}}..$ (125)

As a simple demonstration, let us still consider the simplest case with $q=2$:

$\displaystyle Y=\sum_{i,j}X_{i}X_{j},$ (126) $\displaystyle
Y^{2}=\sum_{i}X_{i}^{4}+4\sum_{i\neq j}X_{i}^{3}X_{j}+3\sum_{i\neq
j}X_{i}^{2}X_{j}^{2}$ $\displaystyle\qquad+6\sum_{i\neq j\neq
k}X_{i}X_{j}X_{k}^{2}+\sum_{i\neq j\neq m\neq n}X_{i}X_{j}X_{m}X_{n}.$ (127)

Comparing

$\displaystyle\langle Y^{2}\rangle=$ $\displaystyle
N^{4}\kappa_{1}^{4}+4N^{2}\kappa_{3}\kappa_{1}+3N^{2}\kappa_{2}^{2}+6N^{3}\kappa_{2}\kappa_{1}^{2}+N\kappa_{4},$
(131) $\displaystyle\kappa_{1}=\langle X\rangle=\mu,\quad\kappa_{2}=\langle
X^{2}\rangle-\langle X\rangle^{2}=t^{2},$ $\displaystyle\kappa_{3}=\langle
X^{3}\rangle-3\langle X\rangle\langle X^{2}\rangle+2\langle X^{3}\rangle,$
$\displaystyle\kappa_{4}=\langle X^{4}\rangle-3\langle
X^{2}\rangle^{2}-4\langle X\rangle\langle X^{3}\rangle+12\langle
X\rangle^{2}\langle X^{2}\rangle-6\langle X\rangle^{4},$

with (100) one find that if $t\neq 0$, the scaling behavior of $\langle
Y^{2}\rangle$ is same as before. The half wormhole contribution $\Phi$ can be
work out similarly:

$\displaystyle\Phi$ $\displaystyle=$
$\displaystyle\sum_{i}\widehat{k_{i}[2]^{2}}+\sum_{i\neq
j}\widehat{k_{i}[2]}\widehat{k_{j}[2]}+\sum_{i\neq j\neq m\neq
n}X_{i}X_{j}X_{m}X_{n}+4\sum_{i\neq j\neq m}\widehat{k_{i}[1]^{2}}X_{j}X_{m}$
(132) $\displaystyle+$ $\displaystyle 2\sum_{i\neq
j}\widehat{k_{i}[1]^{2}}\widehat{k_{j}[1]^{2}}+2\sum_{i\neq j\neq
k}\widehat{k_{i}[2]}X_{j}X_{k}+4\sum_{i\neq j}\widehat{k_{i}[2]k_{i}[1]}X_{j}$

Then the error is given by

Error $\displaystyle=$ $\displaystyle 4\sum_{i\neq j\neq
k}(X_{i}^{2}-\widehat{k_{i}[1]^{2}})X_{j}X_{k}+2\sum_{i\neq j\neq
k}(X_{i}^{2}-\widehat{k_{i}[2]})X_{j}X_{k}+2\sum_{i\neq
j}(X_{i}^{2}X_{j}^{2}-\widehat{k_{i}[1]^{2}}\widehat{k_{j}[1]^{2}})$ (133)
$\displaystyle+$ $\displaystyle\sum_{i\neq
j}(X_{i}^{2}X_{j}^{2}-\widehat{k_{i}[2]}\widehat{k_{j}[2]})+4\sum_{i\neq
j}(X_{i}^{3}-\widehat{k_{i}[2]k_{i}[1]})X_{j}+\sum_{i}(X_{i}^{4}-\widehat{k_{i}[2]^{2}})\rangle-\tilde{t}^{2}.$
$\displaystyle=$ $\displaystyle 4\sum_{i\neq j\neq
k}(X_{i}^{2}-\widehat{k_{i}[1]^{2}})X_{j}X_{k}+2\sum_{i\neq
j}(X_{i}^{2}X_{j}^{2}-\widehat{k_{i}[1]^{2}}\widehat{k_{j}[1]^{2}})$
$\displaystyle+$ $\displaystyle 4\sum_{i\neq
j}(X_{i}^{3}-\widehat{k_{i}[2]k_{i}[1]})X_{j}+\sum_{i}(X_{i}^{4}-\widehat{k_{i}[2]^{2}})\rangle-\tilde{t}^{2},$

where we have used the identity

$\displaystyle\widehat{k[2]}=\int\text{d}k\frac{e^{-\text{i}kX}}{P(X)}\langle
x^{2}e^{\text{i}kx}\rangle=X^{2}.$ (134)

Comparing with (113), there are two extra terms in (133), but they will never
contribute333If $\mu\neq 0$ they maximally contribute to $N^{5}$ and when
$\mu=0$ they maximally contribute to $N^{3}$. to the leading power of $N$ when
$t\neq 0$. So again it seems the approximation (62) is good when $\mu\neq 0$
but not good when $\mu=0$. We will explain in the next section how to
understand these results and modify the proposal (62).

###  2.3 Large-$N$ constraints and half-wormhole approximation

In the previous sections we consider a few different examples. To summarize,
the half-wormhole conjecture (62) and (63) is valid for a large families of
statistical models. However, for some examples discussed in section 2.2.3 this
approximation is not good.

####  2.3.1 Why and how to modify the approximation proposal

The failed examples indicate that the proposed $\Phi$ does not capture all
semi-classical components in the observable $Y^{2}$ to be approximated.

As discussed previously, the approximation (62) should come from the
approximation (10). The relation (10) indeed fails for the case where the
approximation (62) is not good in section (2.2.3). To see this explicitly, we
consider the simplest example (98) where

$\displaystyle Y^{2}$ $\displaystyle=\sum_{i\neq j\neq p\neq
q}X_{i}X_{j}X_{p}X_{q}+4\sum_{i\neq p\neq q}X_{i}^{2}X_{p}X_{q}+2\sum_{i\neq
j}X_{i}^{2}X_{j}^{2}\,,$ (135)

which means we need to consider the following terms in the approximation
$\Phi$

$\displaystyle\langle
X_{i}X_{j}X_{p}X_{q}e^{i\sum_{a}k_{a}X_{a}}\rangle\,,\qquad\langle
X_{j}^{2}X_{p}X_{q}e^{i\sum_{a}k_{a}X_{a}}\rangle,,\qquad\langle
X_{i}^{2}X_{j}^{2}e^{i\sum_{a}k_{a}X_{a}}\rangle\ .$ (136)

However, in the proposal (62) the $\Phi$ term contains only $\langle
Ye^{ik_{a}x_{a}}\rangle^{2}$, which means only terms like

$\displaystyle\langle X_{i}X_{j}e^{i\sum_{a}k_{a}X_{a}}\rangle\langle
X_{p}X_{q}e^{i\sum_{a}k_{a}X_{a}}\rangle\,,\qquad i\neq j\,,p\neq q\,,$ (137)

contribute. Therefore to check why the proposal (62) that fails, we want to
understand what is “missing” in (137) comparing with the correct answer
involving (136).

Because the $x_{i}$’s are identical independent random variables, the cumulant
$c_{n}$ for each $x_{i}$ are the same and the moment generating function is
just a product of the moment generating functions of each $x_{i}$. Therefore
we can reduce the problem of finding a good approximation of the above product
terms to each flavor of $x_{i}$ and find the approximation for each of them.
This should give a good approximation for each term. 444Although this would
obscure the interpretation of $Y$ as an independent function, we still choose
to proceed this way in order to check how the approximation (62) fails.

Recall the approximation is to replace $\frac{\langle
X_{i}^{n}e^{ikx}\rangle_{c}}{\langle e^{ikx}\rangle}$ by $\left(\frac{\langle
X_{i}e^{ikx}\rangle_{c}}{\langle e^{ikx}\rangle}\right)^{n}$ for $n>1$, ie
(32), therefore only the last two terms in (136) are affected by the
approximation. In particular, the first term in (136) gives the same
contribution as the term (137) that leads to the inaccurate approximation
(62). So the non-vanishing contributions from the last two terms in the
leading order of $1/N$ should then be responsible for the failure of the
approximation (62) in this example. As discussed above, a good approximation
to the $x_{j}$ factor of $\langle
X_{j}^{2}X_{p}X_{q}e^{i\sum_{a}k_{a}X_{a}}\rangle$ should be

$\displaystyle\frac{\langle X_{j}^{2}e^{ik_{j}X_{j}}\rangle}{\langle
e^{ik_{j}X_{j}}\rangle}\approx\langle X_{j}^{2}\rangle+2\langle
X_{j}\rangle\frac{\langle X_{j}e^{ik_{j}X_{j}}\rangle_{c}}{\langle
e^{ik_{j}X_{j}}\rangle}+\left(\frac{\langle
X_{j}e^{ik_{j}X_{j}}\rangle_{c}}{\langle e^{ik_{j}X_{j}}\rangle}\right)^{2}\
.$ (138)

The contribution to the half-wormhole $\Phi$ from this term $\langle
X_{j}^{2}X_{p}X_{q}e^{i\sum_{a}k_{a}X_{a}}\rangle$ is thus

$\displaystyle(t^{2}+\mu^{2})X_{p}X_{q}+2\mu
X_{j}X_{p}X_{q}+\Phi_{2}^{j}X_{p}X_{q}\ .$ (139)

Similarly, the $\langle X_{p}^{2}X_{q}^{2}e^{i\sum_{a}k_{a}X_{a}}\rangle$ type
terms gives a contribution

$\displaystyle(t^{2}+\mu^{2})^{2}+4\mu^{2}X_{p}X_{q}+\Phi_{2}^{p}\Phi_{2}^{q}+4\mu(t^{2}+\mu^{2})\left(X_{p}+X_{q}\right)$
$\displaystyle\quad+(t^{2}+\mu^{2})\left(\Phi_{2}^{p}+\Phi_{2}^{q}\right)+4\mu\left(X_{p}\Phi_{2}^{q}+X_{q}\Phi_{2}^{p}\right)\
.$ (140)

Now we should sum over $j,p,q$ to get all the contributions to the computation
of $\langle Y^{2}\rangle$ and further to Error2.

To understand the structure of the contribution to Error2, we denote

$\displaystyle\text{Error}=(Y^{2}-\Phi+\langle Y\rangle^{2})-\langle
Y^{2}\rangle=M-\langle Y^{2}\rangle\,,\qquad\langle M\rangle=\langle
Y^{2}\rangle\,,$ (141)

then if we switch the notation of $\langle\text{Error}^{2}\rangle$ to a
slightly more indicative one $\langle\text{Error}_{1}\text{Error}_{2}\rangle$,
we have

$\displaystyle\langle\text{Error}_{1}\text{Error}_{2}\rangle=\langle
M_{1}M_{2}\rangle+\langle Y^{2}\rangle^{2}-2\langle Y^{2}\rangle\langle
M\rangle=\langle M_{1}M_{2}\rangle-\langle Y^{2}\rangle^{2}\ .$ (142)

Therefore we find if $\langle M_{1}M_{2}\rangle\approx\langle
M_{1}\rangle\langle M_{2}\rangle$ to the leading order, then the error is
small and the approximation (62) is good.

This is precisely how the previous proposal (62) failed. For example, in the
Error (104), it is precisely the contraction among the two factors
$\sum_{i\neq j}X_{i}X_{j}$ that gives another factor of $2N^{4}t^{8}$ in
$\langle(\text{Error}/4)^{2}\rangle$ and prevent it from vanishing. On the
other hand, if we check the results (139) and (140) we find, to the leading
order of $N$, the term that have non-trivial contribution to Error2 is

$\displaystyle 4(N-2)(t^{2}+\mu^{2})X_{p}X_{q}\,,$ (143)

that comes from summing the first terms in (139) over $j$; the other terms are
either suppressed by $1/N$ or do not give nontrivial contraction between the
two copies of Error as discussed above. Then we immediately notice that this
is precisely the term, with $\mu=0$ in this case, that is missing in $\Phi$ to
remove the “problematic” term in the Error that we just discussed. Therefore,
once we use the correct approximation with all terms in (136), the error
should be small and the approximation should be good. The other examples in
section (2.2.3) could also be modified in a similar way so that the errors
become small.

Further notice that one of the upshot of the approximation (62) is, as pointed
out in Mukhametzhanov:2021hdi , that we can safely ignore the direct
correlation between the two $Y$’s (or $z$’s in the context of
Mukhametzhanov:2021hdi ) and the two terms are “linked” through the
correlation with $e^{ikx}$. What we found in the previous section are however
cases where these direct correlations cannot be ignored. The new ingredient of
the approximation (145) we will present shortly is precisely a partial
correlation between the $Y$’s directly, not just through the $e^{ikx}$
factors. In this sense the saddles in the general models discussed in section
2.2.3 are hyper-linked half-wormholes with extra partially direct connections.

With this we propose a modified approximation

$\displaystyle Y^{2}\approx\langle Y^{2}\rangle+\tilde{\Phi},$ (144)
$\displaystyle\tilde{\Phi}(X)=\frac{1}{(2\pi)^{N}}\int\prod_{i}\left(dk_{i}\frac{e^{-\text{i}k_{i}X_{i}}}{P(X_{i})}\right)\langle
e^{\text{i}\sum_{i}k_{i}x_{i}}\rangle\frac{\left[Y(x)^{2}e^{\text{i}\sum_{i}k_{i}x_{i}}\right]}{\langle
e^{\text{i}\sum_{i}k_{i}x_{i}}\rangle}\,,$ (145)

where $\left[Y(x)^{2}e^{\text{i}\sum_{i}k_{i}x_{i}}\right]$ denotes all
possible terms contains at least one contraction between $Y^{2}$ and the
spacetime brane $e^{ikx}$.

In the example (98), each term in $Y$ contains two $X_{i}$ legs, therefore we
have

$\displaystyle\frac{\left[Y(x)^{2}e^{\text{i}\sum_{i}k_{i}x_{i}}\right]}{\langle
e^{\text{i}\sum_{i}k_{i}x_{i}}\rangle}=\frac{2\langle Y\rangle\langle
Y(x)e^{\text{i}\sum_{i}k_{i}x_{i}}\rangle_{c}}{\langle
e^{\text{i}\sum_{i}k_{i}x_{i}}\rangle}+\frac{\langle
Y(x)e^{\text{i}\sum_{i}k_{i}x_{i}}\rangle_{c}^{2}}{\langle
e^{\text{i}\sum_{i}k_{i}x_{i}}\rangle^{2}}+\frac{\langle
Y(x)^{2}e^{\text{i}\sum_{i}k_{i}x_{i}}\rangle_{c}}{\langle
e^{\text{i}\sum_{i}k_{i}x_{i}}\rangle}\,,$ (146)

where the different terms correspond to one contraction to the brane, two
separate contractions to the brane and a pair of connected contractions to the
brane. The c here means the contribution cannot be made disconnected if we
only cut on the brane. Among these terms the last one is precisely the one
missed in the previous proposal (63). A demonstration of these terms are shown
in Figure 2. We notice that this approximation is closely related to the
relation between $\hat{Z}$ and $\hat{W}$ discussed in Peng:2021vhs , see e.g.
Figure. 9 there.

Figure 2: A pictorial illustration of the 3 terms in (146) respectively. Each
vertex on the left is a factor of $Y(x)$, brane on the right denotes
$e^{ikx}$, and each bracket $\langle\cdot e^{ikx}\rangle_{c}$ corresponds to a
component of the bulk amplitude that connects the brane with a set of
vertices. Notice that the first diagrams should be considered as two diagrams
each has a vertex connected to the brane.

From this analysis, it is more obvious to understand why the Errors are all
small when $\mu\neq 0$ in section 2.2.3. When disconnected contributions
exist, the leading order contributions of the Error2 always come from the
disconnected component and hence the Error is guaranteed to be small. However,
this is not very meaningful since as in most of the large-$N$ theories studied
in the literature, we isolate away the disconnected contributions and always
focus on the connected contributions.

####  2.3.2 Why the proposal works for the Pfaffian in the SYK model

From the above discussion, it seems that for a generic operators with
complicated product structure, the original proposal (62) almost surely fails.
However, we know from explicit computations in Saad:2021rcu ;
Mukhametzhanov:2021nea  that the approximation works well for the
hyperpfaffian of the random couplings which is also related to the partition
function of the SYK model.

We believe the reason for this is the large-$N$ factorizations properties due
to large-$N$ constraints. By this we mean when the operators are defined to
have extra structures, for example as a trace or a determinant over the $N$
flavors, such extra structure remains to affect the computation of the Error.
When this is true, which indeed is our case, then the contractions between the
two copies of Error are necessarily suppressed by the large-$N$ factors;
either $1/N$ when the structure is trace as in (77) or higher powers of $1/N$
when the structure is a determinant. Therefore all contractions between the
two copies of Errors are suppressed and at the leading order the result
factorizes and hence the original proposal (62) works. 555A related fact is
that when the approximation is no longer good the relation between the
4${}^{\text{th}}$ moment $\langle Y^{4}\rangle$ of the observable (98) and the
second moment $\langle Y^{2}\rangle$ deviates significantly from the Gaussian
distribution. In Gaussian distribution, this contribution is $3\langle
Y^{2}\rangle\subset\langle Y^{4}\rangle$, on the other hand, for the
observable $Y$ in (98) we get  $\displaystyle\langle Y^{4}\rangle$
$\displaystyle=$ $\displaystyle 8\sum_{i\neq j}\langle
X_{i}^{4}X_{j}^{4}\rangle+60\sum_{i\neq j\neq p\neq q}\langle
X_{i}^{2}X_{j}^{2}X_{p}^{2}X_{q}^{2}\rangle+48\sum_{i\neq j\neq p}\langle
X_{i}^{4}X_{j}^{2}X_{p}^{2}\rangle$ (147) $\displaystyle\approx$
$\displaystyle 60N^{4}t^{8}\neq 3\langle Y^{2}\rangle^{2}-2\langle
Y\rangle^{4}\approx 12N^{4}t^{8}\ .$ (148) But at the moment we have not
succeeded in making a causal relation between this fact and the fact that the
Error is small. The explanation in the main text does better in doing so.

A somewhat ad hoc reason for the need of traces or determinant in the
definition of the operator to make the discussion about (half-)wormhole
meaningful is the following. There is no “spacetime” in our statistical
models, so we cannot use any locality property to identify a function of the
random variables as a single operator; the most we can do is to use a trace or
determinant structure to identify a group of random variables as an operator.
If there is no such trace/determinant constraints, it is equally legitimate to
regard the result as computing correlations of a large number of the
fundamental random variables and the (half-)wormhole interpretation is not
necessarily relevant.

A different interpretation of the importance of the existence of such trace or
determinant structure could be considered as some emergent global symmetry
among the random variables (probably when appropriately analytically
continued). By this we simply mean if we treat the random variables $X_{i}$ as
“fields”, then the action, ie the probability distribution, and the operators
we considered in the computation all have $SO(N)$ symmetry among them. Then
the invariant tensors of $SO(N)$ directly lead to the trace or determinant
structures we just described. It is interesting to make this point more clear,
and we plan to come back to this question somewhere else.

We did not find a general proof of the above assertion (145) or (146), but as
a check we can, according to our assertion, modify the definition of the
function $Y$ and put in by hand some constraints, mimicking a trace structure.
Then we find with this constraints the approximation (62) is indeed valid. For
instance we could introduce a restriction in the sum

$\displaystyle Y=\sum_{i+j=M}X_{i}X_{j},\quad N<M<2N,\quad i\neq j\,,$ (149)

where $N$ is the total number of $X$’s and $M$ is an integer. Without loss of
generality we assume $M$ is even in the following, and the computation for odd
$M$ is the same. Following the previous computations, we get

$\displaystyle Y^{2}$
$\displaystyle=2\sum_{i+j=M}X_{i}^{2}X_{j}^{2}+\sum_{i\neq j\neq m\neq
n}X_{i}X_{j}X_{m}X_{n}\,,$ (150)

and

$\displaystyle\langle Y\rangle=K\mu^{2},\quad\langle
Y^{2}\rangle=2K(t^{2}+\mu^{2})^{2}+K(K-2)\mu^{4},\quad K=2N-M\ .$ (151)

Taking $X_{i}$ from the same Gaussian distribution in the previous cases we
get the expression for the error

$\displaystyle\text{Error}=4t^{2}\sum_{i+j=M}X_{i}^{2}-4Kt^{4}-4Kt^{2}\mu^{2}.$
(152)

It is straightforwardly to show that the expectation values

$\displaystyle\langle\text{Error}\rangle=0\,,\qquad\langle(\text{Error}/4)^{2}\rangle=2Kt^{8}+4Kt^{6}\mu^{2}\
.$ (153)

Clearly in this case $\langle(\text{Error}/4)^{2}\rangle$ is $1/N$ suppressed
compared to $\langle Y^{2}\rangle^{2}$ independent on the value of $\mu$.
Hence the approximation (62) is always valid in the presence of this extra
constraint. Similar restrictions could be imposed to models with general $q$.
It turns out that again the computation is quite similar and we expect the
approximation to be valid in these cases too.

##  3 SYK at one time point: $\langle J_{a}\rangle=0$

In this section, we study the half-wormhole contributions in some 0d SYK model
that can be considered as the usual 0+1d SYK model on a single instant of
time. This section is largely a review of previous results in Saad:2021rcu ;
Mukhametzhanov:2021nea ; Mukhametzhanov:2021hdi ; we provide more details of
various saddle point results and carry out Lefschetz thimble analysis of some
computations when needed.

###  3.1 SYK model with one time point

Let us first revisit the analysis of the 0-dimensional SYK model introduced in
Saad:2021rcu . We are interested in the following Grassmann integral

$\displaystyle z=\int d^{N}\psi\exp(\text{i}^{q/2}\sum J_{i_{1}\dots
i_{q}}\psi_{i_{1}\dots i_{q}})\,,$ (154)

where $\psi_{i_{1}\dots i_{q}}=\psi_{a_{1}}\psi_{a_{2}}\dots\psi_{a_{q}}$ and
$\psi_{i}$ are Grassmann numbers. The number $z$ can be understood as the
partition function of $0+0$ dimensional analogue of SYK model. The random
couplings $J_{i_{1}\dots i_{q}}$ is drawn from a Gaussian distribution

$\displaystyle\langle J_{i_{1}\dots i_{q}}\rangle=0,\quad\langle J_{i_{1}\dots
i_{q}}J_{j_{1}\dots
j_{q}}\rangle=t^{2}\delta_{i_{1}j_{1}}\dots\delta_{i_{q}j_{q}},\quad
t^{2}=\frac{(q-1)!}{N^{q-1}}\ .$ (155)

We sometimes use the collective indies $A,B$ to simplify the notation

$\displaystyle A=\\{a_{1}<\dots<a_{q}\\}\,,\qquad J_{A}\psi_{A}\equiv
J_{a_{1}\dots a_{q}}\psi_{a_{1}\dots a_{q}}\ .$ (156)

Integrating out the Grassmann numbers directly gives (96)666Here we choose the
measure of Grassmann integral to be $\int d^{N}\psi\psi_{1\dots
N}=\text{i}^{-N/2}$.:

$\displaystyle z=\int
d^{N}\psi\exp(\text{i}^{q/2}J_{A}\psi_{A})=\sum^{\prime}_{A_{1}<\dots<A_{p}}\text{sgn}(A)J_{A_{1}}\dots
J_{A_{p}}\,,\quad p=N/q\,,$ (157)

where the expression (157) is nothing but the hyperpfaffian $\text{Pf}(J)$.
Since $\langle z\rangle=0$ due to (155), we focus on $z^{2}$ and $\langle
z^{2}\rangle$

$\displaystyle
z^{2}=z_{L}z_{R}=\int\text{d}^{N}\psi^{L}\text{d}^{N}\psi^{R}\exp\left\\{\text{i}^{q/2}\sum_{A}J_{A}\left(\psi_{A}^{L}+\psi_{A}^{R}\right)\right\\}\,,$
(158) $\displaystyle\langle
z^{2}\rangle=\int\text{d}^{2N}\psi\exp\left\\{\frac{N}{q}\left(\frac{1}{N}\sum_{i=1}^{N}\psi_{i}^{L}\psi_{i}^{R}\right)^{q}\right\\}\,,$
(159)

where we have assumed that $q$ and $N$ are even. The exact values of (159) can
be computed by introducing the standard $G,\Sigma$ variables

$\displaystyle\langle z^{2}\rangle$ $\displaystyle=$
$\displaystyle\int\text{d}^{2N}\psi\int_{\mathbb{R}}\text{d}G\delta\left(G-\frac{1}{N}\sum_{i=1}^{N}\psi_{i}^{L}\psi_{i}^{R}\right)\exp\left(\frac{N}{q}G^{q}\right)$
(160) $\displaystyle=$
$\displaystyle\int_{\mathbb{R}}\text{d}G\int_{\text{i}\mathbb{R}}\frac{\text{d}\Sigma}{2\pi\text{i}/N}\exp\left\\{N\left(\log(\Sigma)-\Sigma
G+\frac{1}{q}G^{q}\right)\right\\}$ (161) $\displaystyle=$ $\displaystyle
N^{-N}\int_{\mathbb{R}}\text{d}G\exp\left(\frac{N}{q}G^{q}\right)(-\partial_{G})^{N}\delta(G)$
(162) $\displaystyle=$
$\displaystyle\frac{N!(N/q)^{N/q}}{N^{N}(N/q)!}=e^{-(1-\frac{1}{q})N}\sqrt{q}\left(1+\frac{1-q}{12N}+\mathcal{O}(\frac{1}{N^{2}})\right)\,,$
(163)

where in the last step we expand around $N\to\infty$ to the next-to-leading
order.

Next we consider the non-averaged quantity (158). Following Saad:2021rcu , we
rewrite

$\displaystyle
z^{2}=\int_{R}\text{d}\sigma\Psi(\sigma)\Phi(\sigma)\,,\quad\Psi(\sigma)=\int\frac{dg}{2\pi/N}\exp[N(-\text{i}\sigma
g-1/qg^{q})]\,,$ (164)

where the coupling dependent piece $\Phi$ is

$\displaystyle\Phi(\sigma)=\int\text{d}^{2N}\psi\exp\left\\{\text{i}e^{-\frac{\text{i}\pi}{q}}\sigma\psi_{i}^{L}\psi_{i}^{R}+\text{i}^{q/2}J_{A}(\psi_{A}^{L}+\psi_{A}^{R})-\frac{N}{q}\left(\frac{1}{N}\psi_{i}^{L}\psi_{i}^{R}\right)^{q}\right\\}\,.$
(165)

Its averaged value is

$\displaystyle\langle\Phi(\sigma)\rangle=(\text{i}e^{-\frac{\text{i}\pi}{q}}\sigma)^{N}\
.$ (166)

As suggested in Saad:2021rcu , to understand the relation between each
individual result and the averaged result, we could figure out in what region
of the $\sigma$-plane $\Phi$ is self-averaging. This is reflected in the
quantity
$\langle\left(\Phi(\sigma)-\langle\Phi(\sigma)\rangle\right)^{2}\rangle$.
Therefore we compare $\langle\Phi(\sigma)\rangle^{2}$ with
$\langle\Phi(\sigma)^{2}\rangle$

$\displaystyle\langle\Phi(\sigma)^{2}\rangle=\int_{R}\frac{\text{d}^{4}\sigma_{AB}\text{d}^{4}g_{AB}}{(2\pi/N)^{4}}e^{N\left[\log(-e^{-\frac{2\text{i}\pi}{q}}(\sigma^{2}+\sigma_{14}\sigma_{23}-\sigma_{13}\sigma_{24}))-\text{i}\sigma_{AB}g_{AB}-\frac{1}{q}g_{AB}^{q}\right]}\,,$
(167)

where we relabel $L=1,L^{\prime}=3,R=2,R^{\prime}=4$ and
$(AB)=(13),(14),(23),(24)$. The integral can be done exactly Saad:2021rcu
following a similar computation we used to get (163)

$\displaystyle\langle\Phi(\sigma)^{2}\rangle=(-e^{-\frac{2\text{i}\pi}{q}})^{N}\sum_{n_{1}+n_{2}+n_{3}=\frac{N}{q},n_{i}\geq
0}\frac{N!}{N^{2q(n_{2}+n_{3})}}\left(\frac{N}{q}\right)^{2(n_{2}+n_{3})}\frac{\sigma^{2qn_{1}}(qn_{2})!(qn_{3})!}{(qn_{1})!(n_{2}!)^{2}(n_{3}!)^{2}}\,,$
(168)

which can be organized into a polynomial in $\sigma$

$\displaystyle\langle\Phi(\sigma)^{2}\rangle$ $\displaystyle=$
$\displaystyle(-e^{-\frac{2\text{i}\pi}{q}})^{N}\left(\sigma^{2N}+\frac{2N!q!}{(N-q)!q^{2}N^{2q-2}}\sigma^{2N-2q}+\dots+e^{2N\frac{1-q}{q}}2q\right)$
(169) $\displaystyle\sim$
$\displaystyle(-e^{-\frac{2\text{i}\pi}{q}})^{N}\left(\sigma^{2N}+\frac{2(q-1)!}{qN^{q-2}}\sigma^{2N-2q}+\dots+e^{2N\frac{1-q}{q}}2q\right)\,,$
(170)

where the phase factor is trivial whenever $q$ divides $N$.

###  3.2 The saddle points analysis

The above results can be reproduced by saddle point approximation in large $N$
limit.

####  3.2.1 The averaged $\langle z^{2}\rangle$

To obtain the same result (163) from saddle point approximation, we first we
rotate the contour

$\displaystyle\Sigma=\text{i}e^{-\text{i}\frac{\pi}{q}}\sigma,\quad
G=e^{\text{i}\frac{\pi}{q}}g\,,$ (171)

to get

$\displaystyle\langle
z^{2}\rangle=\int_{R}\text{d}g\int_{R}\frac{\text{d}\sigma}{2\pi/N}\exp\left\\{N\left(\log(\text{i}e^{-\frac{\text{i}\pi}{q}}\sigma)-\text{i}\sigma
g-\frac{1}{q}g^{q}\right)\right\\}\equiv\int_{R}\text{d}g\int_{R}\frac{\text{d}\sigma}{2\pi/N}e^{NS}\,,$
(172)

so that the integral converges. The saddle point equations are

$\displaystyle-\text{i}\sigma-g^{q-1}=0\,,\quad
g^{q}=-1\,,\quad\rightarrow\quad g=e^{\frac{(2m+1)\text{i}\pi}{q}}\,,\quad
m=0,\dots,q-1\ .$ (173)

All of them give the same on-shell action

$\displaystyle\langle z^{2}\rangle_{s}=\frac{N}{2\pi}e^{-(1-\frac{1}{q})N}\ .$
(174)

To match with the exact result (163) we need to consider fluctuations around
the saddle points. For simplicity let us take $q=4$ and focus on one of the
saddle points

$\displaystyle\sigma_{s}=g_{s}=-(-1)^{\frac{3}{4}},\quad\langle
z^{2}\rangle_{s}=\frac{N}{2\pi}e^{-\frac{3}{4}N}.$ (175)

Expanding the exponent around this saddle

$\displaystyle\sigma=\sigma_{s}+x,\quad g=g_{s}+y$ (176)

to the second order

$\displaystyle
S_{2}\sim-\frac{3}{4}+\frac{3\text{i}x^{2}}{2}-\text{i}xy-\frac{\text{i}y^{2}}{2}+[(-1)^{3/4}x^{3}+\frac{(-1)^{3/4}}{3}y^{3}]+\frac{y^{4}-x^{4}}{4}\,,$
(177)

and evaluating the integral directly gives the fluctuation that combines with
the saddle contribution to

$\displaystyle\langle
z^{2}\rangle_{\text{saddle}+\text{loop}}=e^{-\frac{3}{4}N}\frac{1}{2}\left(1-\frac{1}{4N}\right)\
.$ (178)

Adding contributions from all 4 saddles we arrive at

$\displaystyle\langle
z^{2}\rangle_{\text{saddle}+\text{loop}}=2e^{-\frac{3}{4}N}\left(1-\frac{1}{4N}\right)\,,$
(179)

that agrees with (163) at the two-loop order.

####  3.2.2 The unaveraged $z^{2}$: the wormhole saddle

The result (170) can be reproduced from a saddle point analysis in the
large-$N$ limit. The saddle point equations are

$\displaystyle
g_{AB}^{q-1}=-\text{i}\sigma_{AB}\,,\quad-\text{i}g_{13}=\frac{\sigma_{24}}{f},\quad\text{i}g_{14}=\frac{\sigma_{23}}{f},\quad\text{i}g_{23}=\frac{\sigma_{14}}{f},\quad-\text{i}g_{24}=\frac{\sigma_{13}}{f}\,,$
(180)

where $f\equiv\sigma_{14}\sigma_{23}-\sigma_{13}\sigma_{24}+\sigma^{2}$. The
trivial solution $\sigma_{AB}=g_{AB}=0$ leads to

$\displaystyle\langle\Phi(\sigma)^{2}\rangle_{\text{trivial}+1\text{loop}}=\langle\Phi(\sigma)\rangle^{2}\,,$
(181)

which says the trivial saddle always agrees with the first term in (170).

Next let us consider non-trivial solutions with $\sigma_{AB}\neq 0$. From the
equations of motion we obtain

$\displaystyle
x^{q-2}=y^{q-2},\quad(x^{q-1}-y^{q-1}+\sigma^{2})^{2}=x^{q-2}=y^{q-2}\,,$
(182) $\displaystyle g_{13}^{q}=g_{24}^{q},\quad g_{23}^{q}=g_{14}^{q}$ (183)

where

$\displaystyle x=g_{13}g_{24},\quad y=g_{14}g_{23}\ .$ (184)

It is easy to check that solutions of the above equation satisfies
$x=ye^{\frac{2m\pi\text{i}}{q-2}}$, and for each choice of $m$ there are
$2q^{2}$ solutions of $g_{ab}$. For simplicity let us again focus on the $q=4$
case such that there are only two classes $x=\pm y$.

$\bullet$ When $x=y$ we find another 32 non-trivial saddles. The on-shell
action of all of them are the same

$\displaystyle\langle\Phi(\sigma)^{2}\rangle_{\text{non-
trivial}}^{+}=N^{4}\langle\Phi(\sigma)\rangle^{2}=\langle\Phi(\sigma)^{2}\rangle_{\text{trivial}}\,,$
(185)

where the factor $N^{4}$ comes from the measure of (167). However the 1-loop
fluctuations around them are different

$\displaystyle\text{trivial saddle}:\frac{1}{N^{4}}\,,\quad\text{non-trivial
saddles}:\frac{1}{8N^{4}}\ .$ (186)

We notice that including the 1-loop effect, the trivial saddle is larger and
it reproduces the large $N$ behavior of the exact result. On the other hand,
the non-trivial saddle contributions are also comparable; so it is possible
that we should also take into account of their contributions as well. However,
if we add all the trivial and non-trivial saddle-point values, the result will
obviously exceed the exact value (170). In fact, by a simple Lefschetz-thimble
analysis, see e.g. Witten:2010cx , which is reviewed In Appendix E, we
conclude that these non-trivial saddles should not be included.

Figure 3: Anti-thimble on the $\sigma_{13}$ plane (left) and the $\sigma_{24}$
plane (right).

In particular, we choose a Morse function to be the real part of the action
(167)

$\displaystyle h\equiv\Re(S)=$
$\displaystyle\sum_{abj}\left(-\frac{g_{abj}^{4}}{4}+\frac{3g_{ab1}^{2}g_{ab2}^{2}}{2}+g_{ab1}\sigma_{ab2}+g_{ab2}\sigma_{ab1}\right)$
$\displaystyle\quad+\frac{1}{2}\log\left((\sigma_{142}\sigma_{231}+\sigma_{141}\sigma_{232}-\sigma_{132}\sigma_{241}-\sigma_{131}\sigma_{242})^{2}\right.$
$\displaystyle\left.\quad+(1+\sigma_{141}\sigma_{231}-\sigma_{142}\sigma_{232}-\sigma_{131}\sigma_{241}+\sigma_{132}\sigma_{242})^{2}\right)\,,$
(187)

where we have chosen $q=4$ for simplicity and $\sigma=1$ since we are
interested in the case $\sigma\neq 0$777The $\sigma=0$ case is analyzed in
Saad:2021rcu  . The $g_{abi}$ and $\sigma_{abj}$ are the real and imaginary
parts of the field $g_{ab}$ and $\sigma_{ab}$

$\displaystyle
g_{ab}=g_{ab1}+\text{i}g_{ab2},\quad\sigma_{ab}=\sigma_{ab1}+\text{i}\sigma_{ab2}\
.$ (188)

The downward flow equations of the Morse function are

$\displaystyle\frac{dg_{abj}}{dt}=-\frac{\partial h}{\partial
g_{abj}},\quad\frac{d\sigma_{abj}}{dt}=-\frac{\partial
h}{\partial\sigma_{abj}}\ .$ (189)

The end point of each anti-thimble is one of the saddles at $g_{abj}^{c}$ and
$g_{abj}^{c}$, which leads to the following boundary conditions of the flow
equation

$\displaystyle\lim_{t\to+\infty}g_{abj}=g_{abj}^{c},\quad\lim_{t\to+\infty}\sigma_{abj}=\sigma_{abj}^{c}\
.$ (190)

We can then solve the flow equation and obtain the Lefschetz anti-thimbles
going through each saddle point and if they intersect with the original
integration contour the saddle point contributes to the integral.

For example in Figure 3 we illustrate examples of the anti-thimbles of the
saddle point

$\displaystyle g_{13}=1,\quad g_{24}=-1,\quad g_{14}=(-1)^{3/4},\quad
g_{23}=(-1)^{1/4},$ (191)
$\displaystyle\sigma_{13}=\text{i},\quad\sigma_{24}=-\text{i},\quad\sigma_{14}=(-1)^{3/4},\quad\sigma_{23}=-(-1)^{1/4}\,,$
(192)

that do not intersect with the original integration contour, namely the real
axis. This means the contribution of this saddle should not be included to the
integral.

Examples of anti-thimbles of another saddle point

$\displaystyle g_{13}=-(-1)^{1/4},\quad g_{24}=(-1)^{3/4},\quad
g_{14}=-1,\quad g_{23}=-1,$ (193)
$\displaystyle\sigma_{13}=(-1)^{1/4},\quad\sigma_{23}=(-1)^{3/4},\quad\sigma_{14}=-\text{i},\quad\sigma_{23}=-\text{i}\,,$
(194)

is shown in Figure 4. Again they do not intersect with the real axis so the
contribution from this saddle should not be included either.

Figure 4: Anti-thimble on the $g_{13}$ plane (left) and the $g_{24}$ plane
(right).

We can run this analysis over all the nontrivial saddles and find none of them
contribute to the integral. As a result, the path integral can be approximated
entirely by the trivial saddle.

Figure 5: The shaded region is where a non-trivial saddle in (195) dominates
over the trivial saddle. The plot for the other two non-trivial saddles can be
obtained from this plot by simple rotations.

$\bullet$ When $x=-y$, there are also nontrivial saddle points and a similar
analysis of Lefschetz thimbles demonstrate that they do not contribute to the
integral.

Actually, there is a quicker way to arrive at the same conclusion. We find
that the on-shell actions corresponding to these saddle points are

$\displaystyle\left(\frac{\sigma^{2}}{2}\right)^{\frac{N}{3}}e^{-N\pm\frac{3}{2}2^{\frac{1}{3}}Ne^{\frac{2\text{i}m\pi}{3}}\sigma^{\frac{4}{3}}},\quad
m=0,\pm 1,\quad\sigma\rightarrow\infty\ .$ (195)

However these saddle points should be saddle points of the entire multi-
dimensional integral including the integral over $\sigma$. As a result this
saddle should also satisfy the fall-off condition of the $\sigma$ integral,
otherwise they will not contribute to the $\sigma$ integral. Therefore we
should only consider the decaying saddle points namely

$\displaystyle\left(\frac{\sigma^{2}}{2}\right)^{\frac{N}{3}}e^{-N+\frac{3}{2}2^{\frac{1}{3}}Ne^{\pm\frac{2\text{i}\pi}{3}}\sigma^{\frac{4}{3}}},\quad\left(\frac{\sigma^{2}}{2}\right)^{\frac{N}{3}}e^{-N-\frac{3}{2}2^{\frac{1}{3}}N\sigma^{\frac{4}{3}}}\
.$ (196)

We plot the region where these non-trivial saddle dominates over the trivial
saddle in Figure 5, and it is easy to observe from the figure that the
wormhole saddle (317) of $\langle z^{2}\rangle$, located at $|\sigma|=1$, is
in the region where the trivial saddle dominates.

Another family of solutions to the equation of motion (180) has $x=0$ or
$y=0$. On shell actions on these saddles behave as

$\displaystyle\sigma^{\frac{2N}{3}}e^{-N+\frac{3}{2}Ne^{\pm\frac{2\text{i}\pi}{3}}\sigma^{\frac{4}{3}}},\quad\sigma^{\frac{2N}{3}}e^{-N-\frac{3}{2}N\sigma^{\frac{4}{3}}}\,,$
(197)

whose dominant regions are similar to Figure 5 and they are sub-leading
comparing with the trivial saddle.

Putting all the result together we confirm that the trivial saddle point
dominate in the $g_{ab}$ and $\sigma_{ab}$ integral and the wormhole saddle
(317) is self-averaging.

####  3.2.3 The unaveraged $z^{2}$: the linked half-wormhole saddles

The trivial saddle point discussed in the previous section gives vanishing
contribution at $\sigma\sim 0$, so we expect other saddle points dominate the
path integral here. In Saad:2021rcu  they are referred to as the (linked)
half-wormhole saddles. Here we provide some further details of the saddle
contribute at $\sigma\sim 0$ and show that it agrees with the exact result in
(170), ie

$\displaystyle\langle\Phi(0)^{2}\rangle_{\text{ext}}\sim 2qe^{-\frac{3}{2}N}\
.$ (198)

We can apply the same analysis, except that now we evaluate at $\sigma\sim 0$,
as in the previous section. As expected, the trivial saddle gives

$\displaystyle e^{N\log(\sigma)}\sim 0\ .$ (199)

The subleading non-trivial saddles (196) and (197) discussed in the previous
section has on-shell values

$\displaystyle\frac{e^{-\frac{3}{2}N}}{2^{N/2}},\quad e^{-\frac{3}{2}N}\ ,$
(200)

respectively when $\sigma=0$. So (197) dominates. Adding them up precisely
gives the exact solution (198)

$\displaystyle 2qe^{-\frac{3}{2}N}\,,$ (201)

The general lesson is that the linked half wormhole saddle points are always
in the integral, and furthermore they are also always saddles. It’s only that
they are, for most of the time, hidden behind the leading saddles. They can
only be exposed in regions where the leading saddle decreases faster, namely
the $\sigma\sim 0$ region in this case.

##  4 SYK at one time point: $\langle J_{a}\rangle\neq 0$

In the following, we will generalize the study of half-wormhole along several
directions. The main question we want to address is how the distribution of
the random coupling affects the wormhole and half-wormhole saddles.

First let us consider the case where the random coupling is drawn from a
general Gaussian distribution ${\cal N}(u,t^{2})$888When we write $J_{A}$, we
have in mind that the index set $A$ is automatically sorted, and all $J$’s
with other permutations of $A$ picks up signs accordingly.

$\displaystyle\langle J_{A}\rangle=J_{A}^{0}=u,\quad\langle
J_{A}^{2}\rangle-\langle J_{A}\rangle^{2}=\tau^{2}\frac{(q-1)!}{N^{q-1}}\equiv
t^{2}\,,$ (202)

in particular, the mean value of the random coupling could be non-vanishing.

The ensemble averaged quantities can be computed directly by first averaging
over the couplings and then integrating out the fermions

$\displaystyle\langle z\rangle$ $\displaystyle=$
$\displaystyle\text{PF}(J^{0})\,,$ (203) $\displaystyle\langle z^{2}\rangle$
$\displaystyle=$ $\displaystyle\int
d^{2N}\psi\exp\left(\text{i}^{q}t^{2}\sum_{A}\psi_{A}^{L}\psi_{A}^{R}+\text{i}^{q/2}J_{A}^{0}(\psi_{A}^{L}+\psi_{A}^{R})\right)$
(204) $\displaystyle=$
$\displaystyle\sum^{\prime}_{A,B}\text{sgn}(A)\text{sgn}(B)\left(J_{A_{1}}^{0}J_{B_{1}}^{0}+\delta_{A_{1}B_{1}}t^{2})\right)\dots\left(J_{A_{p}}^{0}J_{B_{p}}^{0}+\delta_{A_{p}B_{p}}t^{2})\right)\
.$ (205)

###  4.1 Half-wormhole saddle in $z$

Since $\langle z\rangle\neq 0$, we expect a disk saddle point in the path
integral presentation of $z$ that gives the contribution of $\langle
z\rangle$. Moreover, like linked half-wormhole contribution to $z^{2}$ in the
model with $u=0$, it is possible that there are also single half-wormhole
saddles contributing to $z$, 999This single half-wormhole saddle is related to
the half-wormhole saddle of JT gravity introduced in Blommaert:2021fob .  as
shown in Figure. 6. We will show in the following that such saddles indeed
exist and together with their contribute $\Theta_{1}$ the following
approximation is good

$\displaystyle z\approx\langle z\rangle+\Theta_{1}\ .$ (206)

Let us clarify the notation we use in this paper, we call the non-self-
averaged component in $z$ as “single half-wormhole” or simply “half-wormhole”,
and we refer to the non-self-averaged saddle in $z^{2}$ as “linked half-
wormhole”.

Figure 6: The single half-wormhole saddle of $z$.

To demonstrate (206) explicitly, recall that the partition function is given
by

$\displaystyle z=\int\text{d}^{N}\psi\exp\left(\text{i}^{q/2}\sum
J_{i_{1}\dots i_{q}}\psi_{i_{1}\dots i_{q}}\right)\ .$ (207)

The ensemble averaged quantity $\langle z\rangle$ does not vanish

$\displaystyle\langle z\rangle=\int\text{d}^{N}\psi\exp(\text{i}^{q/2}\sum
J^{(0)}_{i_{1}\dots i_{q}}\psi_{i_{1}\dots
i_{q}})=u^{p}\frac{(pq/2)!}{p!((q/2)!)^{p}}\equiv m_{p}u^{p}\,,\quad pq=N\ .$
(208)

In the following we present a heuristic but simple proof of this result. A
more rigorous but technical proof is presented in Appendix G. For simplicity
let us first consider the $q=4$ case

$\displaystyle\langle z\rangle=\int d^{N}\psi\,e^{-u\sum_{A}\psi_{A}},\quad
A=\\{a_{1}<\dots<a_{4}\\}\ .$ (209)

We introduce the collective variable $G$

$\displaystyle G=\frac{1}{N}\sum_{1\leq i<j\leq N}\psi_{i}\psi_{j},\quad
G^{2}=\frac{2!}{N^{2}}\sum_{A}\psi_{A}\,,$ (210)

then $\langle z\rangle$ can be rewritten as

$\displaystyle\langle
z\rangle=\int_{\mathbb{R}}\text{d}G\int_{\text{i}\mathbb{R}}\frac{d\Sigma}{2\pi\text{i}/N}\text{d}^{N}\psi\,e^{-\frac{u}{2}N^{2}G^{2}}e^{-\Sigma(NG-\sum_{i<j}\psi_{i}\psi_{j})}\
.$ (211)

Now we can integrate the out the fermions to get

$\displaystyle\int
d^{N}\psi\,e^{\Sigma\sum_{i<j}\psi_{i}\psi_{j}}=(\Sigma)^{N/2}m_{p}\,|_{(q=2)}=\Sigma^{N/2}\
.$ (212)

Then (211) becomes

$\displaystyle\langle z\rangle_{q=4}$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}}\text{d}G\int_{\text{i}\mathbb{R}}\frac{d\Sigma}{2\pi\text{i}/N}\Sigma^{N/2}e^{-\frac{uN^{2}G^{2}}{2}}e^{-N\Sigma
G}\,$ (213) $\displaystyle=$ $\displaystyle
N^{-N/2}(\partial_{G})^{N/2}e^{-\frac{uN^{2}G^{2}}{2}}\,|_{G=0}\,=\left(\frac{u}{2}\right)^{N/4}\frac{(N/2)!}{(N/4)!}=m_{p}u^{p}|_{q=4}\,.$

For general $q$, the proof is similar with the modification

$\displaystyle\sum_{A}\psi_{A}=\frac{N^{q/2}}{(q/2)!}G^{q/2}\,.$ (214)

In summary, we have generalized the $G,\Sigma$ trick and derived an effective
action to compute $\langle z\rangle$:

$\displaystyle\langle
z\rangle=\int_{\mathbb{R}}\text{d}G\int_{\text{i}\mathbb{R}}\frac{d\Sigma}{2\pi\text{i}/N}\Sigma^{N/2}e^{u\text{i}^{q/2}\frac{N^{q/2}}{(q/2)!}G^{q/2}}e^{-N\Sigma
G}\,.$ (215)

It would be convenient to rotate the integral contour as

$\displaystyle\Sigma\rightarrow\text{i}e^{-\text{i}\frac{2\pi}{q}}\sigma,\quad
G\rightarrow e^{\text{i}\frac{2\pi}{q}}g$ (216)

such that we obtain a “standard” action:

$\displaystyle\langle
z\rangle=\int_{\mathbb{R}}\frac{dgd\sigma}{2\pi/N}\exp\left\\{\frac{N}{2}\left(\log(\text{i}e^{-\frac{2\pi\text{i}}{q}}\sigma)-2\text{i}\sigma
g-\frac{2\mu}{q}g^{q/2}\right)\right\\},$ (217)

where we define

$\displaystyle\mu\equiv\text{i}^{q/2}u\frac{2{N}^{q/2-1}}{(q/2-1)!},\quad\leftrightarrow\quad
u=(-\text{i})^{q/2}\mu\frac{(q/2-1)!}{2N^{q/2-1}}.$ (218)

Rescaling $\mu$ to 1, the saddle point equations are then

$\displaystyle\frac{1}{\sigma}-2\text{i}g=0,\quad-2\text{i}\sigma-\mu
g^{q/2-1}=0,\quad\rightarrow\quad\mu g^{q/2}=-1\ .$ (219)

Comparing (217) with (172) it is easy to find that to reproduce the exact
result (208) we have to added the contributions from all the $q/2$ saddles.

Having found the suitable saddle contributions to the averaged partition
function $\langle z\rangle$, we proceed to analyze the difference between the
non-averaged quantity and the mean value $z-\langle z\rangle$. We start with
inserting the identity

$\displaystyle
1=\int_{-\infty}^{\infty}dG_{h}\int_{-\text{i}\infty}^{\text{i}\infty}\frac{Nd\Sigma_{h}}{2\pi\text{i}}e^{-\Sigma_{h}(NG_{h}-\sum_{i<j}\psi_{i}\psi_{j})+\frac{N\mu}{q}\left(G_{h}^{q/2}-\left(\frac{1}{N}\sum_{i<j}\psi_{i}\psi_{j}\right)^{q/2}\right)}\,,$

into the non-averaged partition function $z$. To make the integral well
defined, we again rotate the contour by
$\Sigma_{h}=\text{i}e^{-2\text{i}\pi/q}\sigma_{h},G_{h}=e^{2\text{i}\pi/q}g_{h}$,
then $z$ can be cast into the form

$\displaystyle
z=\int_{-\infty}^{\infty}\frac{N\text{d}\sigma_{h}}{2\pi}\Psi(\sigma_{h})\hat{\Theta}(\sigma_{h})\,,$
(221)

where the first factor is similar to (164)

$\displaystyle\Psi(\sigma_{h})=\int_{\mathbb{R}}\frac{\text{d}g_{h}}{2\pi/N}\exp[N(-\text{i}\sigma_{h}g_{h}-\frac{\mu}{q}g_{h}^{q/2})]\,,$
(222)

and the second factor is

$\displaystyle\hat{\Theta}(\sigma_{h})=\int\text{d}^{N}\psi\exp[\text{i}e^{-\frac{2\text{i}\pi}{q}}\sigma_{h}\sum_{i<j}\psi_{i}\psi_{j}+\text{i}^{q/2}J_{A}\psi_{A}-\text{i}^{q/2}u\sum_{A}\psi_{A}]\
.$ (223)

Averaging over the coupling, we get back to the computation in (217) where
$\sigma_{h}=\frac{1}{2i}\left(\mu^{-2/q}e^{4\pi i(n+\frac{1}{2})/q}\right)$.
We expect a separate saddle point to appear in this integral which leads to
the difference $z-\langle z\rangle$. The $\Psi(\sigma_{h})$ is peaked at
$\sigma_{h}=0$, so we look for dominant contributions around
$\sigma_{h}\approx 0$, which is

$\displaystyle\Theta_{1}=\hat{\Theta}(0)=\text{Pf}(J-J^{0})=\sum^{\prime}_{A}\text{sgn}(A)(J_{A_{1}}-J_{A_{1}}^{0})\dots(J_{A_{p}}-J_{A_{p}}^{0})\
.$ (224)

It is clear that its average vanishes $\langle\Theta_{1}\rangle=0$. Then we
propose the approximation

$\displaystyle z\approx\langle z\rangle+\Theta_{1}\ .$ (225)

which is (206). According to the power of $J^{0}_{A}=u$, we can further expand

$\displaystyle\Theta_{1}$ $\displaystyle=\sum_{k=0}^{p}\Theta_{1}^{(k)}u^{k}\
.$ (226)

To verify this approximation, we define the error function

$\displaystyle\text{Error}=z-\langle z\rangle-\Theta_{1}\ .$ (227)

A direct calculation gives

$\displaystyle\langle\text{Error}^{2}\rangle=\langle z^{2}\rangle-\langle
z\rangle^{2}+\langle\Theta^{2}\rangle-2\langle z\Theta\rangle$ (228)

The quantities $\langle z^{2}\rangle,\langle\Theta^{2}\rangle,\langle
z\Theta\rangle$ can be computed with the Feynman diagrams as shown in Fig. 7.

Figure 7: Feynman diagrams for $\langle
z^{2}\rangle,\langle\Theta_{1}^{2}\rangle,\langle z\Theta_{1}\rangle$. Each
black dot represents a $z$ or $\Theta_{1}$, each red dot and the attached line
represents a contraction with the $J_{A}^{0}$ source, and each blue line is a
contraction of a pair of $J_{A}$.

Recall that value of $\langle z\rangle$ is given by the star diagram that is
one connected component of the last term in Fig. 7

$\displaystyle\langle z\rangle=\frac{(pq/2)!}{p!((q/2)!)^{p}}\mu^{p}\equiv
m_{p}\mu^{p}\,,$ (229)

The value of $\langle z^{2}\rangle$ can be computed either from summing over
the diagrams,

$\displaystyle\langle
z^{2}\rangle=\sum_{k=0}^{p}c_{k}m_{p-k}^{2}t^{2k}u^{2p-2k}\equiv\sum_{k}z_{2}^{(k)}\,,$
(230)

where

$\displaystyle c_{k}=\frac{1}{k!}{N\choose q}{N-q\choose q}\dots
N-(k-1)q\choose q=\frac{N!}{k!(q!)^{k}(N-kq)!}\,,$ (231)

or by introducing the collective variables

$\displaystyle G_{LR}=\frac{1}{N}\sum_{i}\psi_{i}^{L}\psi_{i}^{R},\quad
G_{L}=\frac{1}{N}\sum_{i<j}\psi_{i}^{L}\psi_{j}^{L},\quad
G_{R}=\frac{1}{N}\sum_{i<j}\psi_{i}^{R}\psi_{j}^{R}\,,$ (232)

and doing the path integral

$\displaystyle\langle z^{2}\rangle$ $\displaystyle=$
$\displaystyle\int_{R}\text{d}^{3}G_{i}\int_{\text{i}\mathbb{R}}d^{3}\Sigma_{i}\,e^{\frac{N}{q}(\tau^{2}G_{LR}^{q}+\mu
G_{L}^{q/2}+\mu G_{R}^{q/2})-N(\Sigma_{i}G_{i})}\int\text{d}^{2N}\psi
e^{\frac{1}{2}{\Psi}M{\Psi}},$ $\displaystyle=$
$\displaystyle\int_{R}\text{d}^{3}G_{i}\int_{\text{i}\mathbb{R}}d^{3}\Sigma_{i}\,e^{\frac{N}{q}(\tau^{2}G_{LR}^{q}+\mu
G_{L}^{q/2}+\mu
G_{R}^{q/2})-N(\Sigma_{i}G_{i})}\sqrt{\text{det}[\Sigma_{L}\Sigma_{R}A^{2}+\Sigma_{LR}^{2}]}$
$\displaystyle=$
$\displaystyle\int_{R}\text{d}^{3}G_{i}\int_{\text{i}\mathbb{R}}d^{3}\Sigma_{i}\,e^{\frac{N}{q}(\tau^{2}G_{LR}^{q}+\mu
G_{L}^{q/2}+\mu
G_{R}^{q/2})-N(\Sigma_{i}G_{i})}\text{det}[\text{i}\sqrt{\Sigma_{L}\Sigma_{R}}A+\Sigma_{LR}]$
$\displaystyle=$
$\displaystyle\int_{R}\text{d}^{3}G_{i}\int_{\text{i}\mathbb{R}}d^{3}\Sigma_{i}\,e^{\frac{N}{q}(\tau^{2}G_{LR}^{q}+\mu
G_{L}^{q/2}+\mu
G_{R}^{q/2})-N(\Sigma_{i}G_{i})}\frac{1}{2}\left((\Sigma_{LR}+\text{i}\sqrt{\Sigma_{L}\Sigma_{R}})^{N}+(\Sigma_{LR}-\text{i}\sqrt{\Sigma_{L}\Sigma_{R}})^{N}\right)$
$\displaystyle=$
$\displaystyle\int_{R}\text{d}^{3}G_{i}\int_{\text{i}\mathbb{R}}d^{3}\Sigma_{i}\,\sum_{m=0}^{N/2}{N\choose
2m}(\Sigma_{LR})^{2m}(\text{i}^{2}\Sigma_{L}\Sigma_{R})^{\frac{N}{2}-m}e^{\frac{N}{q}(\tau^{2}G_{LR}^{q}+\mu
G_{L}^{q/2}+\mu G_{R}^{q/2})}e^{-N(\Sigma_{i}G_{i})}\,,$

where we have defined

$\displaystyle\Psi=\left(\psi_{1}^{L},\dots,\psi_{N}^{L},\psi_{1}^{R},\dots,\psi_{N}^{R}\right),\quad
M=\begin{pmatrix}\Sigma_{L}A&\Sigma_{LR}I_{N}\\\
-\Sigma_{LR}I_{N}&\Sigma_{R}A\\\ \end{pmatrix},$ (234) $\displaystyle
A=-A^{T},\quad A_{ij}=1,\quad\forall i<j.$ (235)

Using the same tricks as (213), (4.1) can be evaluated exactly as

$\displaystyle\langle z^{2}\rangle$ $\displaystyle=$ $\displaystyle
N^{-N}\sum_{k=0}^{p}{N\choose
kq}(\partial_{G_{LR}})^{kq}(\text{i}^{2}\partial_{G_{L}}\partial_{G_{R}})^{\frac{N-kq}{2}}e^{\frac{N}{q}(\tau^{2}G_{LR}^{q}+\mu
G_{L}^{q/2}+\mu G_{R}^{q/2})}|_{G_{i}=0}$ (236) $\displaystyle=$
$\displaystyle N^{-N}\sum_{k=0}^{p}\text{i}^{N-kq}{N\choose
kq}\frac{(kq)!}{k!}\left(\frac{N\tau^{2}}{q}\right)^{k}\left[\frac{(\frac{q(p-k)}{2})!}{(p-k)!}\right]^{2}\left(\frac{N\mu}{q}\right)^{2p-2k}$
(237) $\displaystyle=$
$\displaystyle\sum_{k=0}^{p}c_{k}m_{p-k}^{2}t^{2k}u^{2p-2k},$ (238)

which agrees with (230) as it should be.

Furthermore, from this result we find $z_{2}^{(0)}=\langle z\rangle^{2}$ which
is given by the last diagram in Fig. 7 and $z_{2}^{(p)}=\langle
z^{2}\rangle_{\mu=0}$ which is given by the first diagram in Fig. 7. The
expression of $\Theta_{1}$ (224) implies that
$\langle\Theta_{1}^{2}\rangle=\langle\Theta_{1}z\rangle=z_{2}^{(p)}$,
therefore we find

$\displaystyle\langle\text{Error}^{2}\rangle=\sum_{k=1}^{p-1}c_{k}m_{p-k}^{2}t^{2k}u^{2p-2k}\equiv\sum_{k=1}^{p-1}z_{2}^{(k)}\,,$
(239)

where $m_{p}$ is defined in (208). In the large-$N$ limit, some of the terms
in the summation (230) dominate. If $z_{2}^{(p)}$ or $z_{2}^{(0)}$ dominates
then the error is small.

However the dominant term is not always given by a fixed $z_{2}^{(k)}$. A
simple argument is the following. To find the dominant term we can compute the
ratio101010Recall that $p=N/q$.

$\displaystyle
r_{k}=\frac{z_{2}^{(k)}}{z_{2}^{(k-1)}}=\frac{t^{2}(-k+p+1)(-4k+4p+1)(-4k+4p+3)}{3u^{2}(2k(p-k)+k)}\,,$
(240) $\displaystyle r_{p}=\frac{t^{2}}{pu^{2}},\quad
r_{1}\sim\frac{p^{2}t^{2}}{u^{2}}\,,$ (241)

here for simplicity we have chosen $q=4$. First we notice that $r_{k}$
decreases with respect to $k$. Therefore if $r_{1}\leq 1$ i.e.

$\displaystyle\frac{u}{t}\geq{p}\,,$ (242)

then the dominant term will be $z_{2}^{(0)}$. It means that all the wormhole
saddles are suppressed. However if $r_{p}\geq 1$ i.e.

$\displaystyle\frac{u}{t}\leq\frac{1}{\sqrt{p}}$ (243)

then the dominant term will be $z_{2}^{(p)}$, in other words the effect of
$\mu$ can be neglected. For other cases with

$\displaystyle\frac{1}{\sqrt{p}}<\frac{u}{t}<p,$ (244)

by fine tuning the value of $u/t$, every diagram in Fig. (7) is possible to be
dominant. For the choices (202) and (218) which lead to reasonable large $N$
behavior we have

$\displaystyle\frac{u}{t}\sim\frac{\mu}{\tau}\frac{(q/2-1)!}{\sqrt{(q-1)!}}N^{\frac{1}{2}}\sim\sqrt{p},$
(245)

which exactly lies in the (244). It also implies there should be other saddles
contributing to (223).

On the other hand, the can derive the saddle point equations

$\displaystyle G_{L(R)}^{-1+\frac{q}{2}}=\frac{2}{\mu}\Sigma_{L(R)},\quad
G_{LR}^{-1+q}=\frac{1}{\tau^{2}}\Sigma_{LR},$ (246) $\displaystyle
G_{L(R)}=\frac{\text{i}\Sigma_{R(L)}}{2\sqrt{\Sigma_{L}\Sigma_{R}}}\frac{f_{+}^{n-1}-f_{-}^{n-1}}{f_{+}^{n}+f_{-}^{n}},\quad
G_{LR}=\frac{f_{+}^{n-1}+f_{-}^{n-1}}{f_{+}^{n}+f_{-}^{n}}\,,$ (247)

where $f_{\pm}=\Sigma_{LR}\pm\text{i}\sqrt{\Sigma_{L}\Sigma_{R}}$. Again for
simplicity we will choose $\tau^{2}=\mu=1$. There are always two types of
trivial solutions

$\displaystyle\text{wormhole solution}:\quad G_{L}=G_{R}=0,\quad
G_{LR}=e^{\frac{2\text{i}m\pi}{q}},$ (248) $\displaystyle\text{disconnect
solution}:\quad G_{LR}=0,\quad G_{L}=e^{\frac{4\text{i}m_{L}\pi}{q}},\quad
G_{R}=e^{\frac{4\text{i}m_{R}\pi}{q}}$ (249)

with on-shell action

$\displaystyle\text{wormhole solution}:\quad\langle
z^{2}\rangle_{\text{wh}}=e^{-N(1-\frac{1}{q})}e^{\frac{2\text{i}m\pi N}{q}}$
(250) $\displaystyle\text{disconnect solution}:\quad\langle
z^{2}\rangle_{\text{dis}}={2^{-N}}e^{-N(1-\frac{2}{q})}{e^{\frac{4\text{i}m\pi
N}{q}}}.$ (251)

Note that the ratio of these two contribution is

$\displaystyle\frac{\langle z^{2}\rangle_{\text{wh}}}{\langle
z^{2}\rangle_{\text{dis}}}=\left(2e^{-1/q}\right)^{N},$ (252)

so when $q\geq 2$ it is the wormhole saddle dominates. The general analytic
solution is hard to obtain. However in the large $N$ limit we expect that only
$f_{+}$ or $f_{-}$ will survive. Assuming $f^{N}_{-}\rightarrow
0,N\rightarrow\infty$, (247) get dramatically simplified

$\displaystyle
G_{L(R)}=\frac{\Sigma_{R(L)}}{-2\text{i}\sqrt{\Sigma_{R}\Sigma_{L}}}\frac{1}{\Sigma_{LR}+\text{i}\sqrt{\Sigma_{L}\Sigma_{R}}},\quad
G_{LR}=\frac{1}{\Sigma_{LR}+\text{i}\sqrt{\Sigma_{L}\Sigma_{R}}},$ (253)

from which we obtain

$\displaystyle G_{LR}^{q}+G_{R}^{q/2}+G_{L}^{q/2}=1,\quad
G_{R}^{q/2}=G_{L}^{q/2}.$ (254)

For the case of $q=4$, (246) and (253) can be solved explicitly and it
contributes the on-shell action

$\displaystyle\langle z^{2}\rangle_{\text{non-trivial}+}\approx
e^{-0.63N}e^{\frac{2m\text{i}\pi N}{4}}>\langle
z^{2}\rangle_{\text{wh}}=e^{-0.75N}e^{\frac{2m\text{i}\pi N}{4}}.$ (255)

We also checked that for these solutions
$\lim_{N\rightarrow\infty}f_{-}^{N}=0$. Similar saddles can also be found for
the case of $f_{+}^{N}=0$. Therefore we conclude that in the large $N$ limit
the dominate saddles are the non-trivial ones.

In the regime of (244), the ansatz (224) of half-wormhole saddle is not
adequate. We have to consider the contribution from the $\sigma_{h}$
fluctuation to $\Theta$. This can be done by expanding
$\hat{\Theta}(\sigma_{h})$ with respect $\sigma_{h}$, substituting into $z$
and integrating over $\sigma_{h}$. Equivalently this can be done by expanding
the exact value of $z$

$\displaystyle z$ $\displaystyle=$
$\displaystyle\text{PF}(J_{A})=\text{PF}(u+J_{A}-J_{A}^{0})$ (256)
$\displaystyle=$
$\displaystyle\sum^{\prime}_{A}\text{sgn}(A)(u+J_{A_{1}}-J_{A_{1}}^{0})\dots(u+J_{A_{p}}-J_{A_{p}}^{0})\equiv\sum_{n=0}^{p}\Theta^{(n)}\,,$

with respect to $u$. For examples

$\displaystyle\Theta^{(p-1)}=\sum_{A}^{\prime}\text{sgn}(A)(J_{A_{1}}-J_{A_{1}}^{0})\dots
J_{A_{i}}^{0}\dots(J_{A_{p}}-J_{A_{p}}^{0})\,,$ (257)
$\displaystyle\Theta^{(0)}=\langle z\rangle\,,\quad\Theta^{(p)}=\Theta.$ (258)

Then from the Feynman diagrams it is not hard to find in Fig. 7 that

$\displaystyle\langle{\Theta^{(k)}}{\Theta^{(k)}}\rangle=\langle\Theta^{(k)}z\rangle=z_{2}^{(k)}.$
(259)

So if $z_{2}^{(k)}$ is the dominant term, we can choose the half-wormhole
saddle to be $\Theta^{(k)}$. Or we can think of that for each wormhole saddle
$z_{2}^{(k)}$ there is a corresponding half-wormhole saddle $\Theta^{(k)}$
such that

$\displaystyle z\approx\langle z\rangle+\Theta^{(k)}.$ (260)

We will present a further analysis on this model somewhere else.

###  4.2 Linked half-wormhole saddles in $z^{2}$

In this section we study the linked half-wormhole contribution to $z^{2}$,
and, in particular, we would like to understand the relation with the single
half-wormhole saddles in $z$,

To get a general picture, we first compute $\langle z^{4}\rangle$ from the
Feynman diagrams shown in Fig.8. In general it is a cumbersome combinatorial
problem but in the large $N$ limit we know that it should be factorized into
disconnected diagrams as

$\displaystyle\langle z^{4}\rangle\approx 3{z_{2}^{(k)}}^{2}\,,\qquad\langle
z^{2}\rangle\approx z_{2}^{(k)}\,,$ (261)

which is shown in Fig.9 and here we have assumed that $z_{2}^{(k)}$ is the
dominant wormhole saddles.

This means there are more refined structures of the nontrivial saddles in
$z^{2}$, comparing with the general discussion in Saad:2021rcu . Inspired by
our analysis of the single half-wormhole for $z$, we insert another two copies
of identities (4.1) in $z^{2}$

$\displaystyle
z^{2}=\int\text{d}\sigma_{w}\text{d}\sigma_{h_{L}}\text{d}\sigma_{h_{R}}\Psi(\sigma_{w},\sigma_{h_{L}},\sigma_{h_{L}})\hat{\Lambda}(\sigma_{w},\sigma_{h_{L}},\sigma_{h_{L}})\,,$
(262)
$\displaystyle\Psi(\sigma_{w},\sigma_{h_{L}},\sigma_{h_{L}})=\Psi(\sigma_{w})\Psi(\sigma_{h_{L}})\Psi(\sigma_{h_{R}})\,,$
(263)
$\displaystyle\hat{\Lambda}(\sigma_{w},\sigma_{h_{L}},\sigma_{h_{L}})=\int\text{d}^{2N}\psi\exp[\text{i}e^{-\frac{2\text{i}\pi}{q}}\sigma_{h_{L}}\sum_{i<j}\psi^{L}_{ij}+\text{i}e^{-\frac{2\text{i}\pi}{q}}\sigma_{h_{R}}\sum_{i<j}\psi^{R}_{ij}+\text{i}e^{\frac{\text{i}\pi}{q}}\sigma_{w}\psi_{i}^{L}\psi_{i}^{R}$
$\displaystyle\qquad\qquad\qquad\qquad+\text{i}^{q/2}J_{A}(\psi_{A}^{L}+\psi_{A}^{R})-\text{i}^{q/2}u\sum_{A}(\psi_{A}^{L}+\psi_{A}^{R})-\text{i}^{q}t^{2}\psi_{A}^{L}\psi_{A}^{R}],$
(264)

where we have introduced three pairs of $G,\Sigma$ variables

$\displaystyle G_{w}=\frac{1}{N}\psi_{i}^{L}\psi_{i}^{R},\quad
G_{h_{L}}=\frac{1}{N}\sum_{i<j}\psi^{L}_{ij},\quad
G_{h_{R}}=\frac{1}{N}\sum_{i<j}\psi^{R}_{ij},$ (265)

and rotated the contour as before. As before, the function $\Psi$ is highly
peaked around $\Psi(0,0,0)$ so we expect that there is a half-wormhole saddle
point

$\displaystyle\Lambda=\hat{\Lambda}(0,0,0)$
$\displaystyle=\sum^{\prime}_{A,B}\text{sgn}(A)\text{sgn}(B)\prod_{k=1}^{p}\left((J_{A_{k}}-J_{A_{k}}^{0})(J_{B_{k}}-J_{B_{k}}^{0})-\delta_{A_{k}B_{k}}t^{2}\right)\,,$
(266)

whose average manifestly vanishes $\langle\Lambda\rangle=0$ and it further
satisfies $\langle\Lambda^{2}\rangle=2{z_{2}^{(p)}}^{2}$.

Figure 8: Feynman diagrams for $\langle z^{4}\rangle$ Figure 9: $\langle
z^{4}\rangle\approx 3{z_{2}^{(k)}}^{2}$

However because of the large $N$ behavior (261), again we have to consider the
fluctuations of $\sigma_{h}$. It is achieved by expand
$\hat{\Lambda}(0,\sigma_{h_{L}},\sigma_{h_{R}})$ with respect to
$\sigma_{h_{L(R)}}$ or equivalently by expanding

$\displaystyle\sum^{\prime}_{A,B}\text{sgn}(A)\text{sgn}(B)\prod_{k=1}^{p}\left((u+J_{A_{k}}-J_{A_{k}}^{0})(u+J_{B_{k}}-J_{B_{k}}^{0})-\delta_{A_{k}B_{k}}t^{2}\right)\equiv\sum_{n=0}^{p}\Lambda^{(k)}.$
(267)

Some examples are

$\displaystyle\Lambda^{(p-1)}=\sum_{i}\sum^{\prime}_{A,B}\text{sgn}(A)\text{sgn}(B)\left((J_{A_{1}}-J_{A_{1}}^{0})(J_{B_{1}}-J_{B_{1}}^{0})-\delta_{A_{1}B_{1}}t^{2}\right)\dots$
$\displaystyle
J_{A_{i}}^{0}J_{B_{i}}^{0}\dots\left((J_{A_{p}}-J_{A_{p}}^{0})(J_{B_{p}}-J_{B_{p}}^{0})-\delta_{A_{p}B_{p}}t^{2}\right),\quad\Lambda^{(0)}=\langle
z\rangle^{2},\quad\Lambda^{(p)}=\Lambda\ .$

Then similarly one can find that

$\displaystyle\langle\Lambda^{(k)}\Lambda^{(k)}\rangle=\langle
z\Lambda^{(k)}\rangle=2{z_{2}^{(k)}}^{2}$ (268)

so that when $z_{2}^{(k)}$ is the dominant wormhole saddle in the large $N$
limit the

$\displaystyle z^{2}\approx\langle z^{2}\rangle+\Lambda^{(k)}\approx
z_{2}^{(k)}+\Lambda^{(k)}\,,$ (269)

is a good approximation.

##  5 SYK at one time point: $\langle J_{a}\rangle=0,\quad\langle
J_{a}^{4}\rangle_{c}\neq 0$

Another class of interesting distributions of the random coupling is non-
Gaussian. In this section we consider a special subset of them that have
vanishing mean values, namely

$\displaystyle\langle J_{A}\rangle=0\,,\qquad\langle
J_{A}^{2}\rangle=t^{2}\,,\qquad\langle J_{A}^{4}\rangle=v^{4}+3\langle
J_{A}^{2}\rangle^{2}\ .$ (270)

It is easy to compute that the partition function of the 0d SYK model with
such random couplings are

$\displaystyle\langle z\rangle=0,\quad\langle
z^{2}\rangle=\frac{N!}{p!(q!)^{p}}t^{2},\ .$ (271)

The higher moments of $J_{A}$ in (6) contributes nontrivially to $\langle
z^{4}\rangle$

$\displaystyle\langle z^{4}\rangle$ $\displaystyle=$
$\displaystyle\sum_{A,B,C,D}^{\prime}\text{sgn}(A)\text{sgn}(B)\text{sgn}(C)\text{sgn}(D)\langle
J_{A_{1}}J_{B_{1}}J_{C_{1}}J_{D_{1}}\dots
J_{A_{p}}J_{B_{p}}J_{C_{p}}J_{D_{p}}\rangle\,,$ (272)

which can be expanded

$\displaystyle\langle
z^{4}\rangle=\sum_{k=0}^{p}c_{k}n_{N-qk}v^{4k}t^{4(p-k)}\equiv\sum_{k}z_{4}^{(k)},$
$\displaystyle
n_{N}=\frac{N!}{(q!)^{2N/q}}\sum_{\begin{subarray}{c}n_{1}+n_{2}+n_{3}=N/q\\\
n_{i}\geq
0\end{subarray}}\frac{(qn_{1})!(qn_{2})!(qn_{3})!}{(n_{1}!n_{2}!n_{3}!)^{2}},$
$\displaystyle
c_{k}n_{N-qk}=\frac{N!}{k!(q!)^{2p-k}}\sum_{\begin{subarray}{c}n_{1}+n_{2}+n_{3}=N/q-k\\\
n_{i}\geq
0\end{subarray}}\frac{(qn_{1})!(qn_{2})!(qn_{3})!}{(n_{1}!n_{2}!n_{3}!)^{2}}$
(273)

where $c_{k}$ is the number of ways to choose $k$ $q$-subsets out of $N$ and
$n_{N}$ is the multiplicities coming from the different Wick contractions,
i.e.

$\displaystyle\langle z^{4}\rangle_{v=0}=n_{N}t^{4p}.$ (274)

To find the dominant term in the large $N$ limit let us define the ratio

$\displaystyle\tilde{r}_{k}=\frac{z_{4}^{(k)}}{z_{4}^{(k-1)}}\sim\frac{v^{4}}{t^{4}}\frac{1-k+p}{k}\frac{4!(4p-kp)!}{(4p-4k+4)!},$
(275)
$\displaystyle\tilde{r}_{1}\sim\frac{v^{4}}{t^{4}}\frac{1}{p^{2}},\quad\tilde{r}_{p}\sim\frac{\upsilon^{4}}{t^{4}}\frac{1}{p}\,,$
(276)

where we have taken $q=4$ for simplicity. By taking the derivative with
respect to $k$ we find that $\tilde{r}_{k}$ will initially decrease and then
increase with increasing $k$ so $\tilde{r}_{p}$ is the maximal value. If
$\tilde{r}_{p}\leq 1$ i.e.

$\displaystyle\frac{v^{4}}{t^{4}}\leq p\,,$ (277)

then the dominant term will be $z_{4}^{(0)}$ therefore the contributions of
higher moments can be ignored in this limit. Recall that the half-wormhole
saddle of $z^{2}$ when $\langle J_{A}\rangle=0$ can be written as

$\displaystyle\Phi=\sum^{\prime}_{A,B}\text{sgn}(A)\text{sgn}(B)\left(J_{A_{1}}J_{B_{1}}-\delta_{A_{1}B_{1}}t^{2}\right)\dots\left(J_{A_{p}}J_{B_{p}}-\delta_{A_{p}B_{p}}t^{2}\right)\,,$
(278)

such that

$\displaystyle\langle\Phi^{2}\rangle\approx\langle\Phi z^{2}\rangle\approx
2\langle z^{2}\rangle^{2},$ (279)

and

$\displaystyle\langle\text{Error}^{2}\rangle$ $\displaystyle=$
$\displaystyle\langle z^{4}\rangle-\langle
z^{2}\rangle^{2}+\langle\Phi^{2}\rangle-2\langle z^{2}\Phi^{2}\rangle$ (280)
$\displaystyle\approx$ $\displaystyle 3\langle z^{2}\rangle^{2}-\langle
z^{2}\rangle^{2}+2\langle z^{2}\rangle^{2}-4\langle z^{2}\rangle^{2}=0,$

in the leading order of $N$ as before. However if $\tilde{r}_{p}>1$, then it
will be possible that $z_{4}^{(p)}$ is the leading term whose corresponding
Feynman diagram is shown in Fig.10.

Figure 10: $z_{4}^{(p)}$

Therefore there will be no half-wormhole saddle anymore since the (two-mouth)
wormhole saddles are not dominant.

One can consider more general distribution with all the cumulants to be non-
vanishing. The analysis and the results will be similar. If $v$ is very large
then it is the four-way wormhole saddle that dominate. It is therefore
possible to introduce a new ”four-linked-wormhole” saddle as we show in next
section. However, if $v$ is relatively small it is still the two-mouth
wormhole (with some legs as shown in Fig.7) that dominates. We will present a
more thorough analysis of these points separately.

##  6 SYK at one time point: $\langle J_{a}\rangle=\langle
J_{a}^{2}\rangle=\langle J_{a}^{3}\rangle=0$

In this section, we consider a special model where we could focus on the
“multi-linked” wormhole saddle points. In this model the random coupling only
have non-vanishing $4^{\text{th}}$ cumulant

$\displaystyle\langle J_{a}\rangle=\langle J_{a}^{2}\rangle=\langle
J_{a}^{3}\rangle=0,\quad\langle J_{a}^{4}\rangle=v^{4}\ .$ (281)

Such a distribution could also be considered as an extremal limit of other
distributions.

###  6.1 Averaged quantities: $\langle z^{4}\rangle$ and $\langle
z^{8}\rangle$

Due to our special choice (281) the first non-vanishing averaged quantity is

$\displaystyle\langle z^{4}\rangle$ $\displaystyle=$
$\displaystyle\int\text{d}^{4N}\psi\exp\left(v^{4}\sum_{A_{1}<\dots<A_{q}}\psi_{A_{1}}^{1}\psi_{A_{1}}^{2}\psi_{A_{1}}^{3}\psi_{A_{1}}^{4}\dots\psi_{A_{q}}^{1}\psi_{A_{q}}^{2}\psi_{A_{q}}^{3}\psi_{A_{q}}^{4}\right)$
(282) $\displaystyle=$
$\displaystyle\int\text{d}^{4N}\psi\exp\left(\frac{v^{4}}{q!}(\sum_{i}^{N}\psi_{i}^{1}\psi_{i}^{2}\psi_{i}^{3}\psi_{i}^{4})^{q}\right)\,.$

Then we can introduce the $G,\Sigma$ trick

$\displaystyle\langle z^{4}\rangle$ $\displaystyle=$
$\displaystyle\int\text{d}^{4N}\psi\int\text{d}G\,\delta(G_{4}-\sum_{i}^{N}\psi_{i}^{1}\psi_{i}^{2}\psi_{i}^{3}\psi_{i}^{4})\exp\left(\frac{v^{4}}{q!}G_{4}^{q}\right)$
(283) $\displaystyle=$
$\displaystyle\int\text{d}^{4N}\psi\int\text{d}G\frac{\text{d}\Sigma}{2\pi\text{i}}\exp\left(-\Sigma(G_{4}-\sum_{i}^{N}\psi_{i}^{1}\psi_{i}^{2}\psi_{i}^{3}\psi_{i}^{4})\right)\exp\left(\frac{v^{4}}{q!}G_{4}^{q}\right)$
$\displaystyle=$
$\displaystyle\int\text{d}G\int\frac{\text{d}\Sigma}{2\pi\text{i}}\exp\left(N\log\Sigma-\Sigma
G_{4}+\frac{v^{4}}{q!}G^{q}\right)$ $\displaystyle=$
$\displaystyle(\partial_{G_{4}})^{N}\exp\left(\frac{\upsilon^{4}}{q!}G_{4}^{q}\right)\,|_{G_{4}=0}=\left(\frac{v^{4}}{q!}\right)^{N/q}\frac{N!}{(N/q)!}=v^{4p}\frac{N!}{p!(q!)^{p}}\,.$

Alternatively, we can obtain this result by integrating out the fermions first
to get the hyperpfaffin, taking the $4^{\text{th}}$ power, and then do the
average

$\displaystyle\langle z^{4}\rangle=\sum_{ABCD}\text{sgn}(A,B,C,D)\langle
J_{A_{1}}J_{B_{1}}J_{C_{1}}J_{D_{1}}\dots
J_{A_{p}}J_{B_{p}}J_{C_{p}}J_{D_{p}}\rangle=v^{4p}\sum_{A}\,1=v^{4p}\frac{N!}{p!(q!)^{p}}\,.$
(284)

The computation of $\langle z^{8}\rangle$ is more involved

$\displaystyle\langle
z^{8}\rangle=\int\text{d}^{8N}\psi\exp\left(\frac{v^{4}}{q!}(\sum_{i}^{N}\psi_{i}^{a}\psi_{i}^{b}\psi_{i}^{c}\psi_{i}^{d})^{q}\right)\,,$
(285)

where

$\displaystyle(a,b,c,d)\in\\{1\leq a<b<c<d\leq 8\\}\ .$ (286)

In the following we use the collective index $A^{\prime}$ to label the
$4$-element subset. Then we introduce antisymmetric tensors
$G_{abcd}=G_{A^{\prime}}$ and $\Sigma_{abcd}=\Sigma_{A^{\prime}}$ as the
collective field variables such that (284) can be expressed as

$\displaystyle\langle z^{8}\rangle$ $\displaystyle=$
$\displaystyle\int\frac{\text{d}G_{A^{\prime}}\text{d}\Sigma_{A^{\prime}}}{(2\pi\text{i})^{70}}(\text{PF}(\Sigma_{A^{\prime}}))^{N}\exp\left(-\sum_{A^{\prime}}[\Sigma_{A^{\prime}}G_{A^{\prime}}+\frac{v^{4}}{q!}G_{A^{\prime}}^{q}]\right)$
(287) $\displaystyle=$
$\displaystyle\left(\sum^{\prime}_{A^{\prime}_{1}<A^{\prime}_{2}}\text{sgn}(A^{\prime})\partial_{G_{A^{\prime}_{1}}}\partial_{G_{A^{\prime}_{2}}}\right)^{N}\exp\left(\frac{v^{4}}{q!}G_{A^{\prime}}^{q}\right)|_{G_{A^{\prime}}=0}$
$\displaystyle\approx$
$\displaystyle\left(\frac{v^{4}}{q!}\right)^{\frac{2N}{q}}\frac{N!^{2}}{p!^{2}}\frac{1}{2}{8\choose
4}=35\left(\frac{v^{4}}{q!}\right)^{\frac{2N}{q}}\frac{N!^{2}}{p!^{2}}\,,$

where in the last line we have taken the large $N$ limit. In this limit we
have

$\displaystyle\langle z^{8}\rangle\approx 35\langle z^{4}\rangle^{2}\ .$ (288)

###  6.2 The un-averaged $z^{4}$

Following similar ideas as in the previous sections, we insert a suitable
identity to the expression of $z^{4}$

$\displaystyle z^{4}$ $\displaystyle=$
$\displaystyle\int\text{d}^{4N}\psi\exp\left(\text{i}^{q/2}\sum_{A,i}J_{A}\psi_{A}^{i}\right)\int\text{d}G_{4}\delta(G_{4}-\sum_{i}^{N}\prod_{a=1}^{4}\psi_{i}^{a})\exp\left(\frac{v^{4}}{q!}[G_{4}^{q}-(\sum_{i}^{N}\prod_{a=1}^{4}\psi_{i}^{a})^{q}]\right)\,$

Rotating the contour as before we can rewrite $z^{4}$ as

$\displaystyle z^{4}=\int\text{d}\sigma\Psi(\sigma)\hat{\Gamma}(\sigma)\,,$
(290)

where $\Psi(\sigma)$ is same as (164) and the second factor is

$\displaystyle\hat{\Gamma}(\sigma)=\int\text{d}^{4N}\psi\exp\left(\text{i}e^{-\frac{\text{i}\pi}{q}}\sigma\prod_{a}\psi^{a}_{i}+\text{i}^{q/2}\sum_{A,a}J_{A}\psi^{a}_{A}-v^{4}\sum_{A}\prod_{a}\psi_{A}^{a}\right).$
(291)

Therefore we expect the half-wormhole saddle is given by

$\displaystyle\Gamma=\hat{\Gamma}(0)$ $\displaystyle=$
$\displaystyle\sum_{ABCD}\text{sgn}(A,B,C,D)\prod_{k=1}^{p}(J_{A_{k}}J_{B_{k}}J_{C_{k}}J_{D_{k}}-\delta_{A_{k}}^{B_{k}}\delta_{C_{k}}^{B_{k}}\delta_{C_{k}}^{D_{k}}v^{4})\,,$
(292)

which satisfies

$\displaystyle\langle\Gamma\rangle=0\,,\qquad\langle\Gamma^{2}\rangle=\langle\Gamma
z^{4}\rangle\approx 34\langle z^{4}\rangle^{2}\,,$ (293)
$\displaystyle\langle(z^{4}-\langle z^{4}\rangle-\Gamma)^{2}\rangle=\langle
z^{8}\rangle-\langle z^{4}\rangle^{2}+\langle\Gamma^{2}\rangle-2\langle\Gamma
z^{4}\rangle\approx 0\,.$ (294)

We find clearly that the contribution from this four-linked-wormhole saddle is
not equal to the square of (two-linked) half-wormhole saddle. Even though we
derive it in the 0-SYK toy model, it should exist in other SYK-like theory as
long as the $G,\Sigma$ trick can be applied. We will present some more details
about these more general discussions somewhere else.

##  7 SYK at one time point: Poisson distribution

Up to now we have only considered random couplings with continuous probability
distributions. It is also interesting to consider random couplings that take
discrete values such as the Poisson distribution.

In fact the Poisson distribution, whose PDF and moments are given by (596) and
(597), can be regarded as an opposite extremum to what we have considered
above in the sense that all the cumulants are equal $\langle
J^{n}\rangle_{c}=N\lambda$, $\forall n$. From the gravity point of view, it
means that all the wormholes with different number of boundaries have the same
amplitude. Ensemble theory or theories with random coupling with Poisson
distribution have been studied in Marolf:2020xie ; Peng:2020rno ; Peng:2021vhs
. If we view the index $i$ of $\psi^{i}$ as the label of different time
points, then the effect of ensemble average is to introduce (“non-local”)
interaction between different time points. In particular, starting with action
(154) we can compute the first few moments111111Here we have rescaled
$q\rightarrow 2q$, $N\rightarrow 2N$.

$\displaystyle\langle z\rangle$
$\displaystyle=\int\text{d}^{2N}\psi\,e^{N\text{i}^{q}\lambda\sum_{A}\psi^{1}_{A}},$
(295) $\displaystyle\langle z^{2}\rangle$
$\displaystyle=\int\text{d}^{4N}\psi\,e^{N\text{i}^{q}\lambda\sum_{A}(\psi_{A}^{1}+\psi_{A}^{2})}e^{N\text{i}^{2q}\lambda\sum_{A}\psi_{A}^{1}\psi_{A}^{2}},$
(296) $\displaystyle\langle z^{3}\rangle$
$\displaystyle=\int\text{d}^{6N}\psi\,e^{N\text{i}^{q}\lambda\sum_{A}(\psi_{A}^{1}+\psi_{A}^{2}+\psi_{A}^{3})}e^{N\text{i}^{2q}\lambda\sum_{A}(\psi_{A}^{1}\psi_{A}^{2}+\psi_{A}^{1}\psi_{A}^{3}+\psi_{A}^{2}\psi_{A}^{3})}e^{N\text{i}^{3q}\lambda\sum_{A}\psi_{A}^{1}\psi_{A}^{2}\psi_{A}^{3}}\
.$ (297)

For a generic $k$, we find

$\displaystyle\langle z^{k}\rangle=\int\text{d}^{2kN}\psi
e^{\lambda\sum_{A}\sum_{n=1}^{k}\frac{1}{n!}(\text{i}^{q}\sum^{k}_{i=1}(\psi_{A}^{i}))^{n}}\
.$ (298)

Formally we can define

$\displaystyle{\cal Z}(\lambda)\equiv\langle
z^{\infty}\rangle=\int\text{d}\psi\exp\left\\{N\lambda\sum_{A}(e^{\text{i}^{q}\sum_{i=1}\psi_{A}^{i}}-1)\right\\}\,.$
(299)

We can compute these moments by integrating out the fermions directly

$\displaystyle\langle z^{n}\rangle=\langle\text{Pf}(J_{A})^{n}\rangle\ .$
(300)

However the ensemble average of $\text{PF}(J_{A})^{n}$ is very complicated.
Alternatively, if we only care about the large $N$ behavior we can use the
$G,\Sigma$ trick and do a saddle point approximation. For example, the
$G,\Sigma$ expression of $\langle z\rangle$ is similar to (215)

$\displaystyle\langle z\rangle=\int d\Sigma
dG(-\text{i})^{N}\Sigma^{N}e^{N\text{i}^{{q}}\lambda\frac{G^{{q}}}{{q}!}}e^{\text{i}N\Sigma
G}.$ (301)

The saddle point equations are

$\displaystyle\Sigma
G=\text{i},\quad\frac{\lambda}{(q-1)!}(\text{i}G)^{q}=1\,,$ (302)

whose solutions are

$\displaystyle\text{i}G=\left(\frac{(q-1)!}{\lambda}\right)^{1/q}e^{\frac{2m\pi\text{i}}{q}},\quad
m=1,\dots,q\,.$ (303)

It has been argued in Saad:2021rcu  these $q$ saddle points should be added
together to reproduce the correct large $N$ behavior in a very similar
calculation. We expect the same to apply in the current situation121212Here we
have dropped the normalization factor $\text{i}^{N}$.

$\displaystyle\langle
z\rangle_{\text{Disk}}=e^{-N(1-\frac{1}{q})}\left(\frac{N^{q}\lambda}{(q-1)!}\right)^{p}\sum_{m}e^{\frac{2m\pi\text{i}}{q}}=qe^{-N(1-\frac{1}{q})}\left(\frac{N^{q}\lambda}{(q-1)!}\right)^{p},$
(304)

where $p=N/q$ as before. Adding the 1-loop factor $1/\sqrt{q}$ we end up with
the correct large-$N$ behavior

$\displaystyle\langle z\rangle_{\text{Disk}+1\text{
loop}}=\frac{1}{\sqrt{q}}e^{-N(1-\frac{1}{q})}\left(\frac{N^{q}\lambda}{(q-1)!}\right)^{p}.$
(305)

Other moments can be computed similarly. For example, to compute $\langle
z^{2}\rangle$, we need to introduce three collective variables

$\displaystyle G_{1}=\sum_{i<j}\text{i}\psi_{i}^{1}\psi_{j}^{1},\quad
G_{2}=\sum_{i<j}\text{i}\psi_{i}^{2}\psi_{j}^{2},\quad
G_{12}=\sum_{i}\psi_{i}^{1}\psi_{i}^{2}$ (306)

such that

$\displaystyle\text{i}^{q}\sum_{A}\psi_{A}^{1}=\frac{G_{1}^{q}}{q!},\quad\text{i}^{q}\sum_{A}\psi_{A}^{2}=\frac{G_{2}^{q}}{q!},\quad\text{i}^{2q}\sum_{A}\psi_{A}^{1}\psi_{A}^{2}=\frac{G_{12}^{2q}}{(2q)!}.$
(307)

Imposing these relations with the help of a set of Lagrangian multiplier
fields $\Sigma_{1}$, $\Sigma_{2}$ and $\Sigma_{12}$, the $\langle
z^{2}\rangle$ can be expressed as

$\displaystyle\langle z^{2}\rangle$ $\displaystyle=$
$\displaystyle\int[\text{d}^{3}G_{i}d^{3}\Sigma_{i}]e^{N\frac{\lambda}{q!}(G_{1}^{q}+G_{2}^{q}+\frac{q!}{(2q)!}G_{12}^{2q})}e^{\text{i}N\sum_{i}(\Sigma_{i}G_{i})}\int\text{d}^{2N}\psi
e^{\frac{1}{2}{\Psi}M{\Psi}},$ (308) $\displaystyle=$
$\displaystyle\int[\text{d}^{3}G_{i}d^{3}\Sigma_{i}]\sqrt{\text{det}[\Sigma_{1}\Sigma_{2}A^{2}-\Sigma_{12}^{2}I_{2N}]}e^{\frac{N\lambda}{q!}(G_{1}^{q}+G_{2}^{q}+\frac{q!}{(2q)!}G_{12}^{2q})}e^{\text{i}N\sum_{i}(\Sigma_{i}G_{i})}$
(309) $\displaystyle=$
$\displaystyle\int[\text{d}^{3}G_{i}d^{3}\Sigma_{i}]\text{i}^{2N}\text{det}[\sqrt{\Sigma_{1}\Sigma_{2}}A+\Sigma_{12}I_{N}]e^{N\frac{\lambda}{q!}(G_{1}^{q}+G_{2}^{q}+\frac{q!}{(2q)!}G_{12}^{2q})}e^{\text{i}N\sum_{i}(\Sigma_{i}G_{i})}$
$\displaystyle=$
$\displaystyle\int[\text{d}^{3}G_{i}d^{3}\Sigma_{i}]\frac{\text{i}^{2N}}{2}\left((\Sigma_{12}+\sqrt{\Sigma_{1}\Sigma_{2}})^{2N}+(\Sigma_{12}-\sqrt{\Sigma_{1}\Sigma_{2}})^{2N}\right)e^{N\frac{N\lambda}{q!}(G_{1}^{q}+G_{2}^{q}+\frac{q!}{(2q)!}G_{12}^{2q})}e^{N\text{i}\sum_{i}(\Sigma_{i}G_{i})}$
$\displaystyle=$
$\displaystyle\int[\text{d}^{3}G_{i}d^{3}\Sigma_{i}]\text{i}^{2N}\sum_{k=1}^{N}{2N\choose
2k}\Sigma_{12}^{2N-2k}(\Sigma_{1}\Sigma_{2})^{k}e^{N\frac{\lambda}{q!}(G_{1}^{q}+G_{2}^{q}+\frac{q!}{(2q)!}G_{12}^{2q})}e^{N\text{i}\sum_{i}(\Sigma_{i}G_{i})}$
(311)

where we have defined

$\displaystyle\Psi=\left(\psi_{1}^{1},\dots,\psi_{2N}^{1},\psi_{1}^{2},\dots,\psi_{2N}^{2}\right),\quad
M=\begin{pmatrix}\Sigma_{1}A&-\text{i}\Sigma_{12}I_{2N}\\\
\text{i}\Sigma_{12}I_{2N}&\Sigma_{2}A\\\ \end{pmatrix},$ (312) $\displaystyle
A=-A^{T},\quad A_{ij}=1,\quad\forall i<j.$ (313)

The saddle point equations lead to

$\displaystyle\text{i}\Sigma_{i}+\frac{\lambda}{(q-1)!}G_{i}^{q-1}=0,\quad
i=1,2,$ (314)
$\displaystyle\text{i}\Sigma_{12}+\frac{\lambda}{(2q-1)!}G_{12}^{2q-1}=0\,,\quad\sum_{i}\Sigma_{i}G_{i}=2\text{i}\
.$ (315)

This set of equations have multiple solutions. For example, the wormhole
saddle is

$\displaystyle G_{1}=G_{2}=\Sigma_{1}=\Sigma_{2}=0,\quad
G_{12}=\left(\frac{2(2q-1)!}{\lambda}\right)^{1/2q}e^{\frac{2m\pi\text{i}}{2q}},$
(316) $\displaystyle\langle
z^{2}\rangle_{WH+1\text{loop}}=\frac{1}{\sqrt{2q}}e^{-2N(1-\frac{1}{2q})}\left(\frac{(2N)^{2q}\lambda}{2(2q-1)!}\right)^{p}$
(317)

and the disconnected saddle is

$\displaystyle G_{12}=\Sigma_{12}=0,\quad
G_{1}=G_{2}=\left(\frac{(q-1)!}{\lambda}\right)^{1/q},$ (318)
$\displaystyle\langle
z^{2}\rangle_{disc+1\text{loop}}=\frac{1}{q}e^{-2N(1-\frac{1}{q})}\left(\frac{N^{q}\lambda}{(q-1)!}\right)^{2p}=\langle
z\rangle_{\text{Disk}+1\text{loop}}^{2}.$ (319)

The ratio of these two saddles is

$\displaystyle\frac{\langle z^{2}\rangle_{WH+1\text{loop}}}{\langle
z^{2}\rangle_{disc+1\text{loop}}}=\sqrt{\frac{q}{2}}\left(\frac{q!^{2}2^{2q}}{e\lambda
q(2q)!}\right)^{p}\,.$ (320)

In the large $N$ or $p=N/q$ limit, the wormhole saddle can dominate only when
$\lambda<\frac{q!^{2}2^{2q}}{e\lambda
q(2q)!}\left(\frac{q}{2}\right)^{\frac{1}{2p}}$ which is consistent with our
previous results.

Then a natural question is that in this limit how about other n-boundary
wormhole saddles? In the following let us focus on a particular $n$-linked-
wormhole saddles. When $n=2k$ is even, the situation is similar to the one in
section 6:

$\displaystyle\langle z^{2k}\rangle_{\text{connected}}$ $\displaystyle=$
$\displaystyle\int
d^{4kN}\psi\text{d}G\frac{\text{d}\Sigma}{2\pi}\exp\left(\text{i}N\Sigma\left(G-\sum_{i}^{2N}\prod_{a=1}^{2k}\psi_{i}^{a}\right)\right)\exp\left(N\frac{\lambda}{(2q)!}G^{2q}\right)$
(321) $\displaystyle=$
$\displaystyle\int\text{d}G\frac{\text{d}\Sigma}{2\pi}(\text{i}\Sigma)^{2N}\exp\left(\frac{N\lambda}{{(2q)}!}G^{2q}+\text{i}N\Sigma
G\right)\,,$ (322)

where the collective variable $G$ is

$\displaystyle G=\sum_{i}^{2N}\prod_{a=1}^{2k}\psi_{i}^{a}\ .$ (323)

The expression (322) is of the same form as (301) so the saddle point
approximation is

$\displaystyle\langle z^{2k}\rangle_{2k-WH+1\text{loop}}=\langle
z^{2}\rangle_{2-WH+1\text{loop}}=\frac{1}{\sqrt{2q}}e^{-2N(1-\frac{1}{2q})}\left(\frac{(2N)^{2q}\lambda}{2(2q-1)!}\right)^{p}.$
(324)

When $n=2k+1$ is odd, the situation is similar to the one of $n=1$:

$\displaystyle\langle z^{2k+1}\rangle_{\text{connected}}$ $\displaystyle=$
$\displaystyle\int
d^{(4k+2)N}\psi\text{d}G\frac{\text{d}\Sigma}{2\pi}\exp\left(\text{i}N\Sigma(G-\sum_{i<j}^{2N}\prod_{a=1}^{2k+1}\psi_{i}^{a}\prod_{a=1}^{2k+1}\psi_{j}^{a}\right)\exp\left(\frac{N\lambda}{q!}G^{q}\right)$
(325) $\displaystyle=$
$\displaystyle\int\text{d}G\frac{\text{d}\Sigma}{2\pi}(\text{i}\Sigma)^{2N}\exp\left(\frac{N\lambda}{{q}!}G^{q}+\text{i}N\Sigma
G\right),$

where the collective variable $G$ is obviously defined as

$\displaystyle
G=\sum_{i<j}^{2N}\prod_{a=1}^{2k+1}\psi_{i}^{a}\prod_{a=1}^{2k+1}\psi_{j}^{a},$
(326)

therefore the saddle point approximation is

$\displaystyle\langle z^{2k+1}\rangle_{2k+1-HW+1\text{loop}}=\langle
z\rangle_{\text{Disk}+1\text{loop}}=\frac{1}{\sqrt{q}}e^{-N(1-\frac{1}{q})}\left(\frac{N^{q}\lambda}{(q-1)!}\right)^{p}\
.$ (327)

These higher $n$-linked-wormholes should be compared with the corresponding
powers of the disk solution, and furthermore since $\langle
z^{2}\rangle_{2-WH+1\text{loop}}\gg 1$, we conclude that all these multiple-
linked-wormholes are suppressed. In other words, the ensemble of $z$ can be
approximated by a Gaussian when the ratio (320) is of order 1.

##  8 The Brownian SYK model

In this section, we study the wormhole and half-wormholes saddles in the
Brownian SYK model Saad:2018bqo . In the Brownian SYK model, the couplings are
only correlated at the same instant of time so that after integrating over the
coupling we end up with a local effective action131313See Appendix (F) for
general discussion on averaged model.. The quantity that is analogous to the
partition function but with some information of real time evolution is

$\displaystyle U(T)=\mathbf{T}e^{-\text{i}\int_{0}^{T}dtH(t)}\ .$ (328)

To check the nature of its fluctuations that is not caused by the phase
factor, we consider the norm square of its trace

$\displaystyle\left|{\rm Tr}\,U(T)\right|^{2}\ .$ (329)

This quantity is manifest real in the sense the complex conjugate maps ${\rm
Tr}\,U(T)$ to ${\rm Tr}\,U(T)^{*}$. The trace is over the Hilbert space, which
has a path integral interpretation

$\displaystyle{\rm Tr}\,U(T)$
$\displaystyle=\int\mathcal{D}\psi_{a}\exp\left\\{-\text{i}\int_{0}^{T}dt\left[-\frac{\text{i}}{2}\psi_{a}\partial_{t}\psi_{a}+J_{a_{1}\ldots
a_{q}}(t)\text{i}^{\frac{q}{2}}\psi_{a_{1}\ldots a_{q}}\right]\right\\}\,,$
(330)

where the Lagrangian density is manifestly real.

To compute (329), we introduce two replicas of fermions; $\psi^{(L)}$
constitute the fermions in $H$ of $U$ and $\psi^{(R)}$ in $U^{*}$. Therefore
the complex conjugate should map between $\psi^{(L)}$ and $\psi^{(R)}$. One
conventional way to define $\psi^{(R)}$ from $\psi^{(L)}$ is

$\displaystyle\psi^{(R)}_{a}=\left(\psi^{(L)}_{a}\right)^{*}\ .$ (331)

Then the complex conjugation of (330) is

$\displaystyle{\rm Tr}\,U(T)^{*}$
$\displaystyle=\int\mathcal{D}\psi_{a}^{(R)}\exp\left\\{-\text{i}\int_{0}^{T}dt\left[\frac{\text{i}}{2}\psi^{(R)}_{a}\partial_{t}\psi^{(R)}_{a}-J_{a_{1}\ldots
a_{q}}(t)\text{i}^{\frac{q}{2}}\psi^{(R)}_{a_{1}\ldots
a_{q}}\right]\right\\}\,,$ (332)

We can further do a field redefinition $\psi^{(R)}\to\text{i}\psi^{(R)}$ so
that the kinetic term has the “right” sign141414Here we choose to absorb an
extra $i^{N}$ phase factor into the definition of the path integral measure.
There might be $N\bmod 4$ effects that we will discuss separately.

$\displaystyle{\rm Tr}\,U(T)^{*}$
$\displaystyle=\int\mathcal{D}\psi_{a}^{(R)}e^{-\text{i}\int_{0}^{T}dt\left[-\frac{\text{i}}{2}\psi^{(R)}_{a}\partial_{t}\psi^{(R)}_{a}-J_{a_{1}\ldots
a_{q}}(t)(-\text{i})^{\frac{q}{2}}\psi^{(R)}_{a_{1}\ldots a_{q}}\right]}\,,$
(333)

Combining (330), with $\psi_{a}$ replaced by $\psi_{a}^{(L)}$, and (333), the
quantity we would like to compute is

$\displaystyle|\operatorname{Tr}U(T)|^{2}=\int\mathcal{D}\psi_{a}^{(L)}\mathcal{D}\psi_{a}^{(R)}e^{\text{i}\int_{0}^{T}dt\left[\frac{\text{i}}{2}\psi_{a}^{(j)}\partial_{t}\psi_{a}^{(j)}-J_{a_{1}\ldots
a_{q}}(t)\left(\text{i}^{\frac{q}{2}}\psi_{a_{1}\ldots
a_{q}}^{(L)}-(-\text{i})^{\frac{q}{2}}\psi_{a_{1}\ldots
a_{q}}^{(R)}\right)\right]}\ .$ (334)

A side remark is that the complex conjugation is closely related to time