a ainstitutetext:  Department of Particle Physics and Astrophysics,
Weizmann Institute of Science, Rehovot 7610001, Israel b binstitutetext:
Department of Physics, Indian Institute of Technology, Palakkad 678557, India
c cinstitutetext:  School of Mathematics and Maxwell Institute for
Mathematical Sciences,
University of Edinburgh, Edinburgh EH9 3FD, United Kingdom

# Chord diagrams, exact correlators in spin glasses and black hole bulk
reconstruction

Micha Berkooz b , Prithvi Narayan c , Joan Simón
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The exact 2-point function of certain physically motivated operators in SYK-
like spin glass models is computed, bypassing the Schwinger-Dyson equations.
The models possess an IR low energy conformal window, but our results are
exact at all time scales. The main tool developed is a new approach to the
combinatorics of chord diagrams, allowing to rewrite the spin glass system
using an auxiliary Hilbert space, and Hamiltonian, built on the space of open
chord diagrams. We argue the latter can be interpreted as the bulk description
and that it reduces to the Schwarzian action in the low energy limit.

##  1 Introduction

Over the last few years the Sachdev-Ye-Kitaev (SYK) model Sachdev:1992fk ;
KitaevTalks ; Maldacena:2016hyu ; Polchinski:2016xgd ; Sachdev:2010um  has
attracted considerable attention as a model of quantum qravity which exhibits
the correct chaotic behaviour. The SYK model at large $N$ turns out to be
solvable, in the IR and other limits, and exhibits several interesting
features like conformal regime Sachdev:1992fk  and maximal chaos exponent
KitaevTalks ; Polchinski:2016xgd ; Maldacena:2016hyu ; Maldacena:2015waa . Its
pattern of symmetry breaking is encoded by the Schwarzian low energy effective
action Maldacena:2016hyu ; Jensen:2016pah ; Maldacena:2016upp , which is the
effective dynamics of gravity on $AdS_{2}$, coupled to a scalar field
Almheiri:2014cka .

There are many interesting generalizations of this model, starting from the
original work of Sachdev:1992fk , like those in higher dimensions
1999PhRvB..59.5341P ; Berkooz:2016cvq ; Gu:2016oyy ; Murugan:2017eto ;
Berkooz:2017efq . Models without disorder with SYK like physics have also been
proposed in Witten:2016iux ; Klebanov:2016xxf ; Gurau:2010ba . The long time
scales of the SYK model have been discussed and connections to random matrix
theory pointed out in Cotler:2016fpe ; Gharibyan:2018jrp . Higher point
functions in the model have been computed in Gross:2017hcz ; Gross:2017aos .
For work beyond large $N$, see Garcia-Garcia:2016mno ; a:2018kvh ; Garcia-
Garcia:2017pzl . The near $AdS_{2}$ spacetime interpretation was elaborated in
Sachdev:2015efa ; Mandal:2017thl ; Jensen:2016pah ; Cvetic:2016eiv ;
Engelsoy:2016xyb ; kitaev2017soft ; Das:2017hrt ; Das:2017wae ; Das:2017pif
(see Jevicki:2016bwu ; Jevicki:2016ito  for related work on bilocal fields).

In this work, we will be interested in a particular class of spin glass models
introduced in Erdos , which are close relatives of the SYK model, and derive a
formula for the exact 2-pt function of certain operators. The model is the
following. Consider $n$ sites with a spin $\frac{1}{2}$ degree of freedom on
each. Denote the Pauli matrices acting on site $i=1,2,\dots,n$ by
$\sigma^{(a)}_{i}$, with $a=1,2,3$. Given an integer p, we define a random
Hamiltonian $H^{(p)}$ as follows. Let $e=(i_{1},....i_{p})$ be a vector of
length $p$ of distinct integers defining a subset of the $n$ sites, and let
$a=(a_{1},...a_{p})$ be a second vector of length $p$, with entries being
either 1,2 or 3. Denoting the pair $(a,e)$ by J, we define

$\sigma_{J}=\sigma_{(a,e)}=\sigma_{i_{1}}^{(a_{1})}\sigma_{i_{2}}^{(a_{2})}....\sigma_{i_{p}}^{(a_{p})}$
(1)

and the spin glass Hamiltonian is

$H^{(p)}=3^{-p/2}{n\choose p}^{-1/2}\sum_{J}\alpha_{J}\sigma_{J}$ (2)

where the sum runs over all possible $J$’s, and $\alpha_{J}$ are independent
Gaussian variables with zero mean and unit standard deviation (we will drop
the superscript p from now on). The relevant parameter controlling the
asymptotic density of states is Erdos

$q=e^{-\lambda}\quad\text{with}\quad\lambda=\frac{4}{3}\ \frac{p^{2}}{n}\ ,$
(3)

and the exact asymptotic density of states of the model (2) was computed in
Erdos  in the limit

$\lambda\ \text{fixed},\hskip 28.45274pt\ n\to\infty$ (4)

We will refer to this as the $\lambda$-scaling limit. We will be interested in
the limit of $\lambda\to 0$, where the distribution of eigenvalues approaches
a Gaussian distribution (point-wise) and hence we will refer to these models
as “Almost Gaussian” spin glass models.

The spin glass model (2) is quite similar to the SYK model. Apart from
replacing Majorana fermions with Pauli matrices, the more critical difference
is that in the $\lambda$-scaling limit (4) $p$ is scaled with $\sqrt{n}$,
leaving $\lambda$ as a parameter, whereas in SYK, $p$ is held fixed as
$n\to\infty$, while scaling the energies properly to obtain a solution of the
model. However, the $\lambda$-scaling model with Majorana fermions was
discussed in Cotler:2016fpe  where it was shown to have a low energy limit
$\lambda\to 0,E\to 0$, (dubbed “double scaled SYK” model) where the density of
states is that of the Schwarzian theory. Hence, models (2) can just as well be
used to study the physics of $AdS_{2}$. In this work, we discuss the full
Almost Gaussian model and use the ”double scaled limit” to check our results.

The main results in the paper are

  * •

We motivate why random operator observables are relevant for black hole
physics, i.e., not just random Hamiltonians. This is done in section 2, where
we also survey existing results and state the new result on the 2-pt function.

  * •

A new method of computing the distribution of eigenvalues of the Hamiltonian
(2) in the $\lambda$-scaling limit. The new method relies on the reduction in
Erdos  of the spin glass Hamiltonian to chord diagrams but then takes a
different route in evaluating the latter. This is done in section 3. Section 4
is an analysis of the $\lambda\to 0$ limit which parallels Appendix B of
Cotler:2016fpe  in our notation.

  * •

The derivation in section 3 relies on an auxiliary Hilbert space and a
Hamiltonian acting on it, which we denote by $T$. This new Hamiltonian is
equivalent111With one important exception : the trace is replaced by some
choice of initial and final states. to the full Hamiltonian of the spin glass
in that whenever the unitary operator $e^{iHt}$ appears, acting on the
original Hilbert space, it can be replaced by $e^{iTt}$ acting on the
auxiliary Hilbert space. In section 5 we suggest that this is the analogue of
the bulk Hamiltonian and show in what limit it reduces to the Schwarzian
effective action in its Liouville form.

  * •

In section 6 we compute the exact time dependent 2-pt function of an
additional random operator of length $p^{\prime}\sim\sqrt{n}$. This can be
reduced to another chord partition function in which one chord is marked. We
use the technique developed in section 3 to evaluate it.

##  2 Motivation, Setup and Summary of results

We will analyse the spin glass Hamiltonian model (1)-(2). However, we will
probe it using a random operator. The latter will be of a similar statistical
type as the Hamiltonian, i.e. it will be defined by the same equation (1)-(2)
but with

  * •

a different length parameter $p^{\prime}\not=p$, and

  * •

a new set of independently drawn coefficients.

In subsection 2.1 we motivate this specific choice of operator. The rest of
the section is an “executive summary” of the setup of the model and known
results in 2.2, and a summary of the new results in 2.3.

###  2.1 Motivation - random observables and factorization

####  2.1.1 Why random operator probes ?

Since black holes share some properties with chaotic systems Sekino:2008he ;
Shenker:2013pqa ; Maldacena:2013xja ; Roberts:2014isa , they can be thought of
as described by a suitable random Hamiltonian. In particular, for AdS black
holes, we might want to think about some core of states in the spectrum
governed by a random Hamiltonian, describing the near horizon black hole
physics, dressed by a “structured” non-random Hamiltonian describing
excitations well separated from the horizon. In this picture, one needs to
specify the statistical class of the random Hamiltonian. This is precisely
what the SYK model achieves, as the relevant class for nearly-AdS2 spacetimes.

The next step is to probe the black hole (BH) using the available bulk probes,
such as single trace operators or their analogues. The Hamiltonian in quantum
mechanics, or the local energy-momentum tensor in higher dimensions, is one
such operator. Probing with the full Hamiltonian does not provide any more
information beyond the partition function, but the local energy momentum
operator does. In practice, it is another massless field for which we can put
sources on the boundary. Just as the full Hamiltonian is indistinguishable
from a random operator when acting on the BH states, we can expect that the
local energy momentum operator will also be effectively described by some
random (local) operator acting on the Hilbert space of the states of the black
hole.

But the local energy-momentum tensor is just one of a tower of single trace
operators with which we can probe the system. In ${\cal N}=4$ SYM we can use
its primary $\text{tr}(X^{2})$ to probe the black hole, or we can just as well
use any other of the $\text{tr}(X^{n})$ operators. If the former is a random
operator on the states of the black hole, why should we not expect that all
single trace operators be of a similar nature ? We would like to suggest that
the relevant probes appearing in General Relativity are random operators on
the BH states222A similar suggestion was made in Balasubramanian:2014gla  for
a different ensemble, and a related discussion for long time scales appears in
Barbon:2014rma .. The main issue would then be from what ensemble these
operators are drawn. If we have some idea about the statistical ensemble of
the Hamiltonian, we can try and guess what is the ensemble for the other
single trace operators.

Another way to phrase the argument is that the SYK model is dual to AdS2 in an
appropriate large N and energy regimes. But there are other models which
realize the same universality class (for example, the one discussed in this
paper is based on different spin matrices). So there may be many ways of
defining the statistics of the random Hamiltonian which give rise to the same
physics - some may be similar to SYK and others may be different. Focusing on
the computation of specific operators used to define a specific realization,
such as $\chi_{i}$ in the SYK model, certainly yields the maximal amount of
information about the model but it may not be universal enough throughout the
different models. Rather, motivated by the fact that the local energy-momentum
tensor is a one ”single trace operator” out of many, we would like to suggest
that useful probes are random operators appropriately made out of the basic
constituents of the theory, just like the Hamiltonian is. The statistical
class of these random operators may be more universal throughout the different
ways of building models (as we will see in our case).

Yet another argument is the following. In the SYK model, the Hamiltonian is a
sum of finite rank polynomials of the $\chi$ fields with random couplings.
Viewing the $\chi$’s as the analogues of the single trace operators in higher
dimensions (which is anyhow problematic since they live in $\mathrm{SO(N)}$
representations) implies that the Hamiltonian in the black hole regime can be
written as a sum of polynomials of single trace operators. This seems to be a
very strong assumption for the higher dimensional AdS/CFT dualities. A weaker
assumption is that both the local energy momentum tensor and all other single
trace operators can be written using some other operators which act on the BH
states, which are just used to define the statistical class of the operator
and probes. These operators need not be asymptotic observables outside the
black hole but rather they just need to be a rich enough set to allow for the
correct definition of the statistical class of the observables.

This is somewhat against the usual application of the AdS/CFT correspondence
where, in this context, the SYK model is taken to be the microscopic theory
which defines all of spacetime. In this approach, one is committed to all the
operators defined in the model. However, in practice if one is interested in
the AdS2 part, one glues it to an external region in order to break conformal
invariance (and the gluing might eventually vary if, for example, one thinks
about an AdS2 near horizon of an object in higher dimensions). It is not clear
to what extent the full SYK provides an extension which has an adequate
gravity dual outside the AdS2 region, and even if it does, it is not clear
whether it is universal. The right probes on AdS2 are determined just as much
by this outside-of-AdS2 region since the probe must be defined on the
boundary. This means the choice of right probes in the AdS2 region, within a
given model, might be ambiguous in general.

####  2.1.2 What random operator probes ?

Having argued that random operators are suitable probes, with ensembles
related to the one from which the Hamiltonian is taken, in this subsection we
would like to discuss another constraint on the ensemble from which probes are
drawn, originating from requiring factorization of correlation functions. We
will see that it again points us in the direction of almost Gaussian random
operators, similar to $H^{(p)}$.

Within the AdS/CFT correspondence correlation functions of single trace
operators factorize at leading order when evaluated in the ground state or in
any other state well described by a semiclassical background. This is usually
taken to include black holes, although this assertion is on less solid footing
there, as the detailed quantum state of the black hole may matter (and surely
does over long enough time scales). So the extent to which correlation
functions do not factorize will teach us about the role of the quantum state
of the black hole, and may also teach us about deviations from the standard
Einstein-Hilbert low energy effective action.

In the field theory side of the AdS/CFT correspondence, factorization is a
consequence of the large N limit when evaluated around the ground state.
Around the black hole background it implies a non-trivial constraint on the
statistics of probe operators Balasubramanian:2014gla . Consider a
microcanonical ensemble with a small enough energy spread, and consider the
4-pt function333We will insert the operators at distinct but close enough
times. The argument does not hinge on these details.

$\text{Tr}\bigl{(}MM^{\dagger}M^{\dagger}M\bigr{)},$ (5)

where the trace is over the states in this energy band. Since black holes are
strongly mixing systems, one might have expected that M - when acting in this
energy band - will be described by one of the ordinary random matrix
ensembles. An example of this is the often assumed strong form of ETH

$M_{i}^{j}{M^{\dagger}}_{k}^{l}\propto\delta_{i}^{l}\delta_{k}^{j}.$ (6)

where the most straightforward interpretation of this formula is as a
statement about statistics of the matrix elements444We will assume that the
operator has no 1-pt function, and in any case we can shift it away.. This
relation actually comes about by a minimal set of assumptions - that A) only
pairwise contraction of the operators matter - after all, we would like to
obtain factorization, and B) that all the states in the microcanonical energy
band are equivalent and hence the statistics should have full unitary
invariance in this energy band - this is also an assumption often made in
statistical physics.

However, under these circumstances correlation functions do not factorize
properly. The ansatz in (6) is the same as drawing the operator M from a
distribution with measure

$e^{-{\cal N}Tr(MM^{\dagger})},$ (7)

where ${\cal N}$ is the number of states in the energy band. So we only need
to compute a Gaussian integral. With this measure, in the large ${\cal N}$
limit, the 4-pt function (5) receives only one (planar) contribution. However,
factorization implies that there are two contractions. It seems difficult to
remedy this within the ordinary ensembles (for example, by changing the
measure to $e^{-{\cal N}V(M,M^{\dagger})}$ for a more general $V$).

Since there are restrictions to implementing factorization in the simplest
ensembles, it is interesting to find additional examples in which correlators
factorize. More precisely we would like them to almost factorize - the
deviation from exact factorization is then interpreted as bulk interactions.
At the level of a single operator, the most naive indicator of factorization -
neglecting for the moment the issue of time dependence - is that

$E\bigl{(}\langle M^{2k}\rangle\bigr{)}\sim A^{2k}(2k-1)!!$ (8)

for an hermitian operator M, where $E()$ is the statistical average over the
ensemble from which the operator is drawn (and $A$ is a factor set by the
normalization of the operator).

The ensembles in Erdos  are precisely of this type. For any operator of the
form (1)-(2), the distribution of eigenvalues approaches the Gaussian one in
the limit $\lambda\rightarrow 0$, so all operators with ${p\over\sqrt{n}}\ll
1$ will be approximately Gaussian. If the Hamiltonian has a specific (small)
$\lambda$, then operators for all other values of (small) $\lambda$ are in
qualitatively a similar statistical class and approximately factorize. We will
use them as our probes.

###  2.2 Set up of the model and summary of known results

The model discussed in Erdos  is defined in equations (1) to (4). One of the
main results in that paper is that the asymptotic distribution of eigenvalues,
in the limit

$n\to\infty\hskip 56.9055pt\lambda\text{ fixed},$ (9)

is given by (recall $q=e^{-\lambda}$)

$v(E|q)=\frac{\sqrt{1-q}}{\pi\sqrt{1-\frac{\sqrt{1-q}}{4}\,E^{2}}}\prod_{k=0}^{\infty}\left[{1-q^{2k+2}\over
1-q^{2k+1}}\left(1-{(1-q)q^{k}\over(1+q^{k})^{2}}\,E^{2}\right)\right]$ (10)

in the range
$E\in\left[-\frac{2}{\sqrt{1-e^{-\lambda}}},\,\frac{2}{\sqrt{1-e^{-\lambda}}}\right]$,
and vanishes outside this region.

The proof proceeds by computing the moments

$m_{L}={1\over 2^{n}}E\bigl{(}\text{Tr}(H^{L})\bigr{)}$ (11)

in the following steps:

1) For the first step one needs to define what are chord diagrams. Consider
$L=2n$ dots on a circle - a chord diagram is a pairing of these dots into $n$
pairs. We draw a line connecting each paired dots. i.e., a total of $n$ lines.
Denote a specific chord diagram by $\pi$. We then denote by $k(\pi)$ the
number of crossings of lines (when we draw the diagram such that each pair of
lines intersects at most once). An example of a chord diagram is shown in
Figure 1 with $n=8$ and with number of crossings $k=2$.

Figure 1: A sample Chord diagram.

In the first step one shows that

$m_{L}=\sum_{\pi}e^{-\lambda k(\pi)}$ (12)

where the sum is over all the chord diagrams. The expression on the RHS is
called the chord partition function, and $q=e^{-\lambda}$ was defined before
in (3) in terms of the parameters of the spin glass. For example, the
contribution to the sum by the chord diagram shown in Figure 1 would be
$e^{-2\lambda}$. Chord diagrams were also used in a:2018kvh  for computing
$1/N$ corrections in the SYK model.

In section 3.1 we review this step of the proof in more details since that
part of the proof will not change. Furthermore, we will also need to slightly
temper with it when computing the 2-pt function.

2) In step 2, one uses the Riordan and Touchard formulae
riordan1975distribution ; flajolet2000analytic  and the results of Ismail:1987
to show that (12) are the moments of the distribution (10) and further give an
explicit formulae for the moments as

$m_{L}={1\over(1-e^{-\lambda})^{L\over 2}}\sum_{j=-{L\over 2}}^{L\over
2}(-1)^{j}e^{-{j(j-1)\lambda\over 2}}{L\choose{L/2+j}}$ (13)

###  2.3 Summary of new results

In this paper we discuss a new proof for the value of $m_{L}$ and the energy
eigenvalue distribution of the spin glass. We use this to compute the exact
two point function for random operators, in the limit $\lambda$ fixed,$\
n\to\infty$. Denoting a new random operator by $M$, it has the form (1)-(2)
(with new randomly chosen coefficients, uncorrelated with those of the
Hamiltonian as mentioned in the beginning of section 2) but with a new
parameter length parameter $p^{\prime}\propto\sqrt{n}$.

More precisely we show that

$2^{-n}E\left[\mbox{Tr}\bigl{(}e^{-{\beta H\over 2}}M(t)e^{-{\beta H\over
2}}M(0)\bigr{)}\right]=\\\
{(q;q)_{\infty}^{2}({\tilde{q}}^{2};q)_{\infty}\over(2\pi)^{2}}\int_{0}^{\pi}d\theta_{1}d\theta_{2}e^{{2\cos(\theta_{1})(-{\beta\over
2}+it)\over\sqrt{(1-q})}}e^{{2\cos(\theta_{2})(-{\beta\over
2}-it)\over\sqrt{(1-q})}}\\\ \
\times{(e^{2i\theta_{1}},q)_{\infty}(e^{-2i\theta_{1}},q)_{\infty}(e^{2i\theta_{2}},q)_{\infty}(e^{-2i\theta_{2}},q)_{\infty}\over({\tilde{q}}e^{i(\theta_{1}+\theta_{2})},{\tilde{q}}e^{i(-\theta_{1}+\theta_{2})},{\tilde{q}}e^{i(\theta_{1}-\theta_{2})},{\tilde{q}}e^{i(-\theta_{1}-\theta_{2})};q)_{\infty}},\
\ \ $ (14)

where $\tilde{q}\equiv e^{-{4\over 3}{pp^{\prime}\over n}}$ and
$(a,q)_{\infty}$ is the q-Pochammer symbol (see (127)).

To prove this one evaluates

$m_{k_{1}k_{2}}=2^{n}E\left[\mbox{Tr}\bigl{(}MH^{k_{1}}MH^{k_{2}}\bigr{)}\right].$
(15)

We show that the relevant chord diagram which computes this two point function
is a chord diagram in which one of the lines is marked, and intersections with
this chord are assigned a different weight. An example of a marked chord
diagram is given in Figure 2. More precisely :

  * •

Given 2n+2 points on a circle, two specific points are connected. This is the
”marked” chord. The thick line in Figure 2 connecting the red dots represents
the marked chord.

  * •

Between the special points at the ends of the marked chord there are $k_{1}$
regular points on one side, and $k_{2}$ regular points on the other side
($k_{1}+k_{2}=2n$).

  * •

These remaining 2n points are paired. These will be called ”regular” chords.

  * •

Intersection between regular chords is assigned weight $q$, and the
intersection between the regular and marked chords is assigned weight
$\tilde{q}$.

  * •

The marked chord partition function is defined as a sum over pairings of the
$2n$ regular points, with $k_{1}$ and $k_{2}$ fixed and with weights as above,
i.e.,

$Z_{k_{1}k_{2}}=\sum_{\pi\in\text{\small{marked chord
diagram}}}q^{k_{\text{regular}}(\pi)}{\tilde{q}}^{k_{\text{marked}}(\pi)}$
(16)

where $k_{\text{regular}}$ ($k_{\text{marked}}$) is the number of regular-
regular (regular-marked) intersections. For example the 1-marked chord diagram
in Figure 2 contributes $q\tilde{q}$ to the $m_{1,5}$.

  * •

Similar to Erdos  we show that

$m_{k_{1}k_{2}}=Z_{k_{1}k_{2}}$ (17)

and evaluate the right hand side to obtain (14)

Figure 2: A sample marked Chord diagram.

The evaluation of the various chord partition functions in this work relies on
an auxiliary Hilbert space space where there is a natural Hamiltonian whose
action is equivalent, in a sense that will be made precise below, to the one
of the full Hamiltonian acting on the spin glass Hilbert space. We interpret
this auxiliary structure as the bulk dual to the spin glass. Furthermore, we
suggest how it reduces to the Schwarzian action in its Liouville form at low
energies555This new Hamiltonian is proportional to $a_{q}+a_{q}^{\dagger}$
where the latter are the creation and annihilation operators of the q-deformed
harmonic oscillator..

##  3 A new derivation of eigenvalue distribution

Given a random Hamiltonian as in (1), the authors in Erdos  compute the
asymptotic distribution of eigenvalues in the $\lambda$-limit (4), by
evaluating the moments

$m_{L}\equiv\lim_{n\to\infty,\lambda\text{
fixed}}\frac{1}{2^{n}}E\biggl{(}\mbox{Tr}H^{L}\biggr{)}$ (18)

and by finding the unique distribution compatible with them.
$E\bigl{(}\bigr{)}$ stands for an ensemble average. In section 3.1 we review
how Erdos  reduces the moments (18) to evaluating the chord partition function
(12). Erdos  then uses the results of Ismail:1987 , and also the formulae of
Touchard and Riordan riordan1975distribution ; flajolet2000analytic , to show
that the moments (12) arise from the distribution given in (10), and to give
the explicit formulae (13) for the moments. Our proof, in section 3.2,
replaces this second step, as well as generalizes it to other Chord diagrams,
as the one that will appear in the exact 2-pt function in section 6.

###  3.1 From spin glasses to chord diagrams

Given the Hamiltonian (2), the computation of the moments

$E\biggl{(}\mbox{Tr}(H^{L})\biggr{)}=\sum_{J_{1},..J_{L}}E\bigl{(}\alpha_{J_{1}}.....\alpha_{J_{L}}\bigr{)}\mbox{Tr}\bigl{(}\sigma_{J_{1}}...\sigma_{J_{L}}\bigr{)}$
(19)

proceeds by evaluating the ensemble average of the $\alpha_{J}$’s first. Any
non-vanishing contribution requires at least two insertions of each
$\alpha_{J}$. Moreover, Lemma (4) in Erdos  shows that the dominant
contribution, in the $\lambda$-scaling limit (4), is when $J_{1},...J_{L}$
appear exactly in pairs, with higher multiplicities being subleading in $n$ in
the large $n$ limit. This gives us the basic structure of chord diagrams where
pairing in the chord is defined by having the same $J$ on two different nodes
as in Figure 1. Summing over all the relevant values of $J$ amounts to summing
over all chord diagrams (i.e all possible pairings of $J$’s), and then sum
over all value of $J$ (i.e., both $\vec{e}$ and $\vec{a}$) for each chord.

Given a chord diagram we therefore need to evaluate what is the weight that is
associated with it, i.e.,

$\sum_{\text{paired}\
J^{\prime}s}\mbox{Tr}\bigl{(}\sigma_{J_{1}}...\sigma_{J_{L}}\bigr{)}$ (20)

where there are only $L/2$ independent $J$’s and the pairing is determined by
the chord diagram. The obstruction to immediate evaluation is that
$\sigma_{i}^{a}$ for the same site index $i$ can appear in different $J$’s.
However, Erdos  shows that with probability $1$, in the $\lambda$-scaling
limit, each node can appear in at most two of the chords, enabling the
evaluation of the weight.

More precisely, define the intersection of J’s by the intersection of the site
index, i.e.

$J_{i}\bigcap J_{j}=e_{i}\bigcap e_{j}.$ (21)

Erdos  shows that, for a given $J_{i}$ and $J_{j}$ the size of the overlap is
Poisson distributed, and that there is, with probability 1, no triple
intersections. I.e., we can assume

$J_{i}\bigcap J_{j}\bigcap J_{k}=0,\ \ i\not=j\not=k\ .$ (22)

This statement is summarized in lemma (9) there, and subsequent discussion.
Given two sets $a$ and $b$ of integers drawn out of the set $\\{1,2,...n\\}$
(without repetition in each set), we can think about it as $|a|$ independent
processes in which the overlap between the sets increasing by 1 with
probability ${|b|\over n}$ (in the limit $n\rightarrow\infty$). This is a
Poisson distribution with mean size of overlap ${3\lambda\over 4}={|a||b|\over
n}$. Recall that we scale the size of the set with $\sqrt{n}$ so this remains
finite in the limit $n\rightarrow\infty$. The average size of an overlap with
an additional index set - say $c$ \- is the latter times ${|c|\over
n}\rightarrow 0$, so with probability 1, triple overlaps are empty.

The interplay of chord intersections and overlap of the index sets is the key
for evaluating the weight of each chord diagram. As we sum over the $J$’s of
the chords, then if two chords do not intersect they will contribute

$\mbox{Tr}\bigl{(}\sigma_{J_{1}}\sigma_{J_{1}}\sigma_{J_{2}}\sigma_{J_{2}}\bigr{)}$
(23)

whereas if the chords intersect they give a factor proportional to

$\mbox{Tr}\bigl{(}\sigma_{J_{1}}\sigma_{J_{2}}\sigma_{J_{1}}\sigma_{J_{2}}\bigr{)}$
(24)

If there is a non-trivial overlap $J_{1}\bigcap J_{2}\not=0$, then these
factors will be different. So there is some ”penalty” that we pay for each
intersection.

More precisely, given a chord diagram (i.e. a pairing $\pi$), recall that
$k(\pi)$ is the number of pairwise chord intersection. Each chord intersection
has a Poisson distributed overlap of sites. Each overlap is independent of the
overlap of the other chord intersection. Each overlap of sites (for a given
intersection) comes with a factor

$3^{-2}\sum_{a,b=1}^{3}{1\over
2}\mbox{Tr}\bigl{(}\sigma^{(a)}\sigma^{(b)}\sigma^{(a)}\sigma^{(b)}\bigr{)}=-{1\over
3},$ (25)

relative to $1$ when the ordering is $(aabb)$ which originates from an overlap
in a pair of non-intersecting chords. Therefore, the size, $m$, of each
overlap is Poisson distributed with expectation value ${3\lambda\over 4}$ and
comes with a weight $(-{1\over 3})^{m}$. The expectation value of the weight
for each chord intersection is therefore $e^{-\lambda}$, and the total weight
associated with each chord diagram is $e^{-\lambda k(\pi)}$. Hence, one
finally obtains (12).

###  3.2 Evaluation of the Chord partition function

In this subsection, we will provide a alternative derivation of the chord
partition function reproducing the expression for $v(E|q)$ in (10). The proof
is rather compact, generalizes to more complicated chord partition functions,
such as the ones discussed in section 6666Which are not evaluated in the
mathematical literature, to the best of our knowledge. and suggests a bulk
interpretation that we develop in section 5. The evaluation of (12) is based
on a ”hopscotch” recursion relation satisfied by the concept of a partial, or
open, chord partition function as follows:

  * •

$2^{-n}\mbox{Tr}(H^{L})$ involves $L$ points in a chord diagram, as indicated
in figure 1. Choose one point, i.e. choose one of the $H$ factors, to be the
first and begin moving clockwise in the chord diagram. Each time one reaches
an extra point and hop over it, we shall refer to it as ”a step”. As we go
along, denote the number of such steps by $i$, i.e. the number of $H$ factors
that were hopped over. In step 1 we hopped over the factor of H that we chose
to be the first.

  * •

At the $i$’th step, a chord can end on the new $i$’th point, or a new chord
can emanate from it. The collection of these decisions defines a chord
history. Denote the number of chords that remain opened at this point (open
chords) by $l$. This number ranges between 0 and $L$.

  * •

Each open chord is assigned a vertical position relative to the other open
chords. Chords emanating from a point further from the left are higher than
chords emanating to their right (of course, all chords have emanated left of
where we are at right now in the diagram). This book keeping guarantees that
open chords have not intersected yet. However, since in previous steps chords
have already emanated and ended on various points, our procedure may have
taken us through chord configurations of many chords that have already closed
to the left of our current position.

  * •

Define $\Pi(i,l)$ as the set of chord histories ending with $l$ open chords at
step $i$, and define the partial, or open, chord partition function
$v^{(i)}_{l}$ as

$v^{(i)}_{l}=\sum_{\pi\in\Pi(i,l)}q^{-k_{p}(\pi)}\quad\text{with}\quad
v^{(0)}_{l}=\delta_{l,0}.$ (26)

Here $k_{p}(\pi)$ refers to the number of chord intersections to the left, in
the past of our ”hopscotch” process. It is convenient to think about $v^{(i)}$
as a column vector and $l$ as its index.

Given this set-up, one can write down a recursion relation for $v^{i}$. At
each step, one can either close a chord (as in Figure 4) or start a new one
(as in Figure 4) at the point one is hopping over.

Step $i$$\vdots$

Figure 3: Chord Diagram Recursions : A previous line closes

Step $i$$\vdots$

Figure 4: Chord Diagram Recursions : A new line begins

If one starts a new line, $l$ changes to $l+1$ and the new line enters at the
bottom. If one closes a line, it can be either of the $l$ open chords with
height between $1$ and $l$. If one closes the line at height $p$, it crosses
$(p-1)$ lines on its way down. This crossing generates a weight $q^{p-1}$ when
evaluating its contribution to the partial chord partition function.
Altogether, the vector of such partition functions satisfies the following
recursion relation

$v^{(i+1)}_{l}=v^{(i)}_{l-1}+(1+q+...+q^{l})v^{(i)}_{l+1}$ (27)

with initial condition $v^{(0)}_{l}=\delta_{l,0}$. The latter can be rewritten
in terms of an $(L+1)\times(L+1)$ transfer matrix $T_{(L)}$ propagating the
partial chord partition function forward

$v^{(i+1)}=T_{(L)}v^{(i)}\,,$ (28)

with matrix elements (indices running from $0$ to $L$, $l_{1}$ ($l_{2}$) is
the row (column) index)

$[\ T_{(L)}\ ]^{\
l_{2}}_{l_{1}}=\delta_{l_{1}-1}^{l_{2}}+\eta_{l_{1}}\delta_{l_{1}+1}^{l_{2}},\hskip
56.9055pt\eta_{l}=1+q+...+q^{l}={1-q^{l+1}\over 1-q}$ (29)

describing a matrix with 1’s and $\eta_{l}$’s on the diagonal below and above
the main diagonal, respectively, i.e.

$T_{(L)}=\begin{bmatrix}0&{1-q\over 1-q}&0&0&0&\dots\\\ 1&0&{1-q^{2}\over
1-q}&0&0&\dots\\\ 0&1&0&{1-q^{3}\over 1-q}&0&\dots\\\
\vdots&\ddots&\ddots&\ddots&\ddots&\ddots\\\ \end{bmatrix}_{(L+1)\times(L+1)}$
(30)

To compute the chord partition function (12), define the vector

$|0\rangle_{L}=v^{(0)}=(1,\\!\underbrace{0\dots,0}_{L\text{
entries}})^{\intercal}$ (31)

of length $L+1$, and then

$m_{L}={}_{L}\langle 0|T_{(L)}^{L}|0\rangle_{L}\ .$ (32)

The initial condition $v^{(0)}_{l}=\delta_{l,0}$ dictates the use of the
initial state $|0\rangle_{L}$. Ensuring our procedure counts only chord
diagrams that close by the time we reach the $L$-th point, such that we are
computing the usual chord partition function in which all lines are paired,
determines the final state.

Notice that we are computing the trace of $H^{L}$ in the original $2^{n}$
dimensional Hilbert space, using some auxiliary space based on partial chord
diagrams. We shall develop a ”bulk” interpretation for the latter in section
5.

Next, given some fixed $L$, one can always consider a larger L’-sized Hilbert
space $(L<L^{\prime})$ such that

$m_{L}={}_{L^{\prime}}\langle
0|T_{(L^{\prime})}^{L}|0\rangle_{L^{\prime}},\quad\quad L^{\prime}\geq L$ (33)

This allows us to take $L^{\prime}\to\infty$, keeping $L$ fixed. In this
infinite dimensional Hilbert space, one can define

$T\equiv\lim_{{L^{\prime}}\to\infty}T_{(L^{\prime})},\hskip
56.9055pt|0\rangle\equiv(1,0,0,....)^{\intercal}.$ (34)

Hence, $T$ is the infinite dimensional extension of (30). This provides an
auxiliary Hilbert space and a single matrix $T$ in which one can evaluate all
traces as

$m_{L}=\langle 0|T^{L}|0\rangle$ (35)

The problem of computing the moments (12) reduces to the problem of computing
the eigenvalues $\alpha$ of the operator $T$ and expanding the vector
$|0\rangle$ in terms of these eigenvectors $|\alpha\rangle$, in an expression

$m_{L}=\int_{\text{Spec}(T)}d\alpha\ \alpha^{L}\ \rho(\alpha)\
|\psi_{0}(\alpha)|^{2}\,,$ (36)

where $\text{Spec}(T)$ is the set of eigenvalues, $\rho(\alpha)$ is its
density and $\psi_{0}(\alpha)\equiv\langle 0|\alpha\rangle$ is the overlap of
$|0\rangle$ with the $|\alpha\rangle$ eigenvector of $T$. Fortunately,
$\text{Spec}(T)$ and the density are very easy to compute and the overlap is
given by specific q-Hermite polynomials, as we will see below.

In the notation of the spin glass model, comparing the $L$ dependence in the
original moment (18) with $\alpha^{L}$ in equation (36), suggests the
identification

$\alpha=E$ (37)

where $E$ is the energy of the system, properly interpreted. The asymptotic
distribution of the energies should then be identified as

$v(E|q)=\rho(E)|\psi_{0}(E)|^{2}\,.$ (38)

##### A short example.

It is worth while carrying out the procedure above in an explicit, low $L$
case, and compare the result with (13). For example $m_{4}(q)=2+q$, which can
obtained also from the three chord diagrams in Figure 5. In our approach we
start with $v^{(0)}$, act on it 4 times with $T_{(4)}$ (or $T$), and project
on $v^{(0)}$. Keeping track of chord histories give the following partial
chord partition functions:

$v^{(0)}=\begin{pmatrix}1\\\ 0\\\ 0\\\ 0\\\ 0\\\ \end{pmatrix}\hskip
14.22636ptv^{(1)}=\begin{pmatrix}0\\\ 1\\\ 0\\\ 0\\\ 0\\\ \end{pmatrix}\hskip
14.22636ptv^{(2)}=\begin{pmatrix}1\\\ 0\\\ 1\\\ 0\\\ 0\\\ \end{pmatrix}\hskip
14.22636ptv^{(3)}=\begin{pmatrix}0\\\ 2+q\\\ 0\\\ 1\\\ 0\\\
\end{pmatrix}\hskip 14.22636ptv^{(4)}=\begin{pmatrix}2+q\\\ 0\\\ 3+2q+q^{2}\\\
0\\\ 1\\\ \end{pmatrix}$ (39)

Figure 5: Chord Diagram for $L=4$ case

##### The symmetric form of the transfer matrix T.

The matrix $T$ in (30) is not Hermitian, but one can conjugate it to a
symmetric version by defining a new matrix ${\hat{T}}$

$\hat{T}\equiv PTP^{-1}$ (40)

where $P$ is a diagonal matrix with entries $(P_{0},P_{1},P_{2}\dots)$
satisfying

$P_{l}=\prod_{i=0}^{l-1}\sqrt{\eta_{i}}={\sqrt{(q;q)_{l}}\over(1-q)^{l\over
2}}\,,\quad l\neq 0\quad\quad P_{0}=1\,,$ (41)

where $(a;q)_{l}$ is the q-Pochammer symbol (see (126)). $\hat{T}$ has matrix
elements

$(\hat{T})^{\
l_{2}}_{l_{1}}=\sqrt{\eta_{l_{2}}}\delta^{l_{2}}_{l_{1}-1}+{\sqrt{\eta_{l_{1}}}}\delta^{l_{2}}_{l_{1}+1}.$
(42)

Thus, it is manifestly symmetric,

$\hat{T}=\begin{bmatrix}0&1&0&0&0&\dots\\\ 1&0&\sqrt{\eta_{1}}&0&0&\dots\\\
0&\sqrt{\eta_{1}}&0&\sqrt{\eta_{2}}&0&\dots\\\
\vdots&\ddots&\ddots&\ddots&\ddots&\ddots\\\ \end{bmatrix}$ (43)

and has the same original moments (35) since

$m_{L}=\langle 0|T^{L}|0\rangle=\langle 0|P^{-1}\hat{T}^{L}P|0\rangle=\langle
0|\hat{T}^{L}|0\rangle\,.$ (44)

We will switch between the two transfer matrix descriptions depending on which
is more convenient at each stage.

####  3.2.1 The spectrum of T

Obtaining the spectrum of T is straightforward. The matrix T asymptotes, down
the diagonal, to a matrix with 1 one diagonal below the main diagonal and
$\eta_{\infty}={1\over 1-q}$ one diagonal above the main diagonal. i.e.

$T_{\text{asymp}}\equiv\begin{bmatrix}0&{1\over 1-q}&0&0&0&\dots\\\
1&0&{1\over 1-q}&0&0&\dots\\\ 0&1&0&{1\over 1-q}&0&\dots\\\
\vdots&\ddots&\ddots&\ddots&\ddots&\ddots\\\ \end{bmatrix},\ \ \ \delta
T\equiv T-T_{\text{asymp}}$ (45)

We can think about the eigenvalue problem of T

$T\psi=E\psi$ (46)

as a scattering problem with the distance along the diagonal playing the role
of position. In this interpretation, infinity is captured by the asymptotic
form of the operator $T$ far down the diagonal. Hence, this is a scattering
problem on the half line with $\delta T$ acting as a scatterer close to the
origin. Indeed, up to an overall rescaling, by conjugating the matrix
$T_{asymp}$, and adding the identity matrix with an appropriate weight, we can
bring it to the form with -2 on the diagonal and 1 on the diagonals below and
above the main. It is then an approximation to the 2nd derivative operator,
making the asymptotic behaviour more familiar in the continuum limit. This
interpretation is elaborated in section 5, where the connection between this
eigenvalue problem and the Liouville equation is described.

However, as far as the spectrum and its density is concerned, the details of
the scatterer are not important as both can be read from the behaviour at
infinity777For the wave functions, or form factors, we will be more specific
below. Also, reading the spectrum and density from infinity also assumes that
there are no bound states near the origin.. So the spectrum of $T$ is the same
as that of $T_{\text{asymp}}$ which is a Toeplitz tridiagonal matrix (i.e with
constant elements one diagonal above and below the main diagonal NLA:NLA1811
), for which there is a simple formula for the eigenvalues, which in this case
is

$-\frac{2}{\sqrt{1-q}}\cdot\cos{s\pi\over L^{\prime}+1},\ s=1,\dots
L^{\prime}\ \ \
\xrightarrow{L^{\prime}\to\infty}\bigl{(}\frac{-2}{\sqrt{1-q}},\frac{2}{\sqrt{1-q}}\bigr{)}$
(47)

as was found in Erdos . This formula gives us both the spectrum of the $T$ and
the density of states on it. Denote $\theta={s\pi\over n+1}$ , then in the
limit $n\rightarrow\infty$ it covers the interval $[0,\pi]$ with uniform
distribution, i.e., inserting a complete set of energy eigenstates is simply
done by the replacement

$\sum_{E}\to\int_{0}^{\pi}d\theta$ (48)

####  3.2.2 Eigensystem of $T$ matrix

The previous asymptotic discussion suggests to parametrise the eigenvalues of
the matrix $T$ as $E(\mu)\equiv{2\mu\over\sqrt{1-q}}$. Let $v^{(\mu)}$ be the
corresponding eigenvector888$v^{(\mu)}\propto\psi(\alpha)$ of section 3. For
now the normalization is different though.. This allows us to write (28)
together with the recursion relation (27) as

$T\cdot v^{(\mu)}={2\mu\over\sqrt{1-q}}v^{(\mu)}\ \rightarrow\
{2\mu\over\sqrt{1-q}}v^{(\mu)}_{l}=v^{(\mu)}_{l-1}+{(1-q^{l+1})\over(1-q)}v^{(\mu)}_{l+1}\,,\quad
v_{0}^{(\mu)}=1$ (49)

Just for this subsection, we will allow $l=-1$ and define $v_{-1}^{(\mu)}=0$.
The recursion relation is simplified by working with the new vectors
$u_{l}^{(\mu)}$

$v_{l}^{(\mu)}={(1-q)^{l\over 2}\over(q;q)_{l}}u_{l}^{(\mu)}$ (50)

so that (49) becomes

$2\mu u_{l}^{(\mu)}=(1-q^{l})u_{l-1}^{(\mu)}+u_{l+1}^{(\mu)}\hskip
56.9055ptu_{-1}^{(\mu)}=0,u_{0}^{(\mu)}=1$ (51)

Comparing with the recursion relations satisfied by continuous q-Hermite
polynomials given in (133), we can identify $u_{l}^{(\mu)}$ with a q-Hermite
polynomial $H_{l}(\mu|q)$ and $\mu$ with $\cos\theta$, with
$\theta\in[0,\pi]$. Hence, the eigenvector of the transfer matrix $T$ equals

$v_{l}^{(\mu)}={(1-q)^{l\over 2}\over(q;q)_{l}}H_{l}(\mu|q)\hskip
85.35826pt-1\leq\mu\equiv\cos(\theta)\leq 1$ (52)

Using the full range $\theta$ is dictated by the the discussion of the
spectrum in section 3.2.1.

At this point, it is more useful to switch to $\hat{T}$, since we need to
conjugate the form factor. The eigenvectors of symmetric transfer matrix
$\hat{T}$ are just $Pv^{\mu}$. In components, the eigenvectors are

$\hat{\psi}_{l}(\theta)=N(\mu,q)\ P_{l}v^{(\mu)}_{l}=N(\mu,q)\
{H_{l}(\mu|q)\over\sqrt{(q;q)_{l}}}.\hskip 56.9055pt\mu=\cos\theta$ (53)

where $N(\mu,q)$ is a normalization which is fixed by the requirement that the
states are delta function normalized in $\theta$, which gives (see Appendix B)
$N(\mu,q)={\sqrt{(q;q)_{\infty}}|(e^{2i\theta};q)_{\infty}|\over\sqrt{2\pi}}$.
With this one can easily write down the matrix element 999Recall that the
density of states $\rho(\theta)$ is uniform.

$\langle l|\hat{T}^{L}|m\rangle=\int_{0}^{\pi}d\theta\
\hat{\psi}_{l}(\theta)\hat{\psi}_{m}(\theta)E(\theta)^{L}$ (54)

The moments of the distribution eq(36) can then be computed to be

$m_{L}(q)=\int_{0}^{\pi}d\theta{(q;q)_{\infty}|(e^{2i\theta};q)_{\infty}|^{2}\over
2\pi}E(\theta)^{L}=\int_{0}^{\pi}d\theta\ \Psi(\theta,q)E(\theta)^{L}$ (55)

where we have defined the distribution

$\Psi(\theta,q)\equiv|\hat{\psi}_{0}(\theta)|^{2}={(q;q)_{\infty}|(e^{2i\theta};q)_{\infty}|^{2}\over
2\pi}$ (56)

Below we will show that this formula is the same distribution given in (10).

##### Matching to the result in Erdos .

Recall that Erdos  obtained the moments $m_{L}(q)$ as moments of distribution
$v(E|q)$ given in (10), i.e

$m_{L}=\int_{-{2\over\sqrt{1-q}}}^{2\over\sqrt{1-q}}dE\ v(E|q)E^{k}$ (57)

Switching to angular variables via
$E=E(\theta)\equiv{2\cos\theta\over\sqrt{1-q}}$, we write $v(E|q)$ as

$\begin{split}v(E(\theta)|q)&={\sqrt{1-q}\over\pi\sin\theta}\prod_{k=0}^{\infty}{(1-q^{2k+2})\over(1-q^{2k+1})(1+q^{k})^{2}}\left\\{(1-e^{2i\theta}q^{k})(1-e^{-2i\theta}q^{k})\right\\}\\\
&={\sqrt{1-q}\over
4\pi\sin\theta}(q;q)_{\infty}|(e^{2i\theta};q)_{\infty}|^{2}\end{split}$ (58)

followed by a change of variables as

$d\theta\ \Psi(\theta,q)=dE\ v(E(\theta)|q)$ (59)

to obtain that (57) matches our result (55).

##  4 The $q\to 1$ limit of the distribution

Our interest in approximately factorized correlators, suggests to work in the
regime $\lambda\to 0$ or $q\to 1$. The analysis is most easily done by
arranging the Pochhammer symbols into Jacobi Theta functions and performing
modular transformations. The results in this section are similar to Appendix B
in Cotler:2016fpe  and Garcia-Garcia:2017pzl  after a suitable substitution
that takes us between our model and the SYK model in the same scaling as
above.

To study the $\lambda\to 0$ limit of the distribution $\Psi(\theta,q)$ (56),
it is convenient to rewrite it in terms of Jacobi Theta functions. Using
(129),

$\Psi(\theta,q)={\sin\theta\ \vartheta_{1}({\theta\over\pi}|{i\lambda\over
2\pi})\over\pi q^{1\over 8}}$ (60)

Using the modular transformation (130) we rewrite it as

$\vartheta_{1}({\theta\over\pi}|{i\lambda\over
2\pi})=\vartheta_{1}({2i\theta\over\lambda}|{2\pi
i\over\lambda})e^{-{2\theta^{2}\over\lambda}}{1\over
i}\sqrt{2\pi\over\lambda}\,.$ (61)

and the $\lambda\to 0$ limit becomes

$\begin{split}\vartheta_{1}({2i\theta\over\lambda}|{2\pi
i\over\lambda})&=2e^{-{\pi^{2}\over 2\lambda}}\sin({2\pi
i\theta\over\lambda})\prod_{m=1}^{\infty}(1-e^{-{4m\pi^{2}\over\lambda}})(1-2\cos({4\pi
i\theta\over\lambda})e^{-{4m\pi^{2}\over\lambda}}+e^{-{8m\pi^{2}\over\lambda}})\\\
&\xrightarrow{\lambda\to 0}2ie^{-{\pi^{2}\over
2\lambda}}\sinh({2\pi\theta\over\lambda})\prod_{m=1}^{\infty}\left(1-2\cosh({4\pi\theta\over\lambda})e^{-4\pi^{2}m\over\lambda}\right)\\\
&\hskip 14.22636pt=2ie^{-{\pi^{2}\over
2\lambda}}\sinh({2\pi\theta\over\lambda})\left(1-2\cosh({4\pi\theta\over\lambda})e^{-4\pi^{2}\over\lambda}\right)\
.\end{split}$ (62)

The last equality follows since the exponential overcomes the hyperbolic
cosine factor for $m\geq 1$ given that $\theta\leq\pi$. Plugging the above
$\lambda\to 0$ expansion in (61), one gets

$\begin{split}\vartheta_{1}({\theta\over\pi}|{i\lambda\over
2\pi})&=2\sqrt{2\pi\over\lambda}e^{-{2\theta^{2}\over\lambda}-{\pi^{2}\over
2\lambda}}\sinh({2\pi\theta\over\lambda})\left(1-2\cosh({4\pi\theta\over\lambda})e^{-4\pi^{2}\over\lambda}\right)\\\
&\approx 4\sqrt{2\pi\over\lambda}e^{-{2\pi^{2}\over\lambda}}\ \
e^{-{2\over\lambda}(\theta-{\pi\over
2})^{2}}\sinh({2\pi\theta\over\lambda})\sinh({2\pi(\pi-\theta)\over\lambda})\end{split}$
(63)

where in the last step we used $\theta\geq 0$. This determines the dominant
contribution to the distribution (56) to be

$\Psi(\theta,q)\approx
4\sqrt{2\over\pi\lambda}e^{-{2\pi^{2}\over\lambda}}\,e^{-{2\over\lambda}(\theta-{\pi\over
2})^{2}}\sin(\theta)\sinh(\frac{2\pi\theta}{\lambda})\sinh({2\pi(\pi-\theta)\over\lambda})$
(64)

Notice this function is symmetric under $E\to-E$, and vanishes at the edges
$E_{max}=-E_{min}={2\over\sqrt{\lambda}}$, which correspond to $\theta=0$ and
$\theta=\pi$, respectively, since $E=\frac{2\cos(\theta)}{\sqrt{1-q}}$.

The distribution (64) has several interesting regimes:

  * •

The $\lambda\to 0$ with $E$ fixed regime. As highlighted in Erdos , pointwise,

$\Psi(\theta,q)\propto e^{-{E^{2}\over
2}}\,,\,\,\,\,E={-2\over\sqrt{\lambda}}\bigl{(}\theta-{\pi\over 2}\bigr{)}\,,$
(65)

which is the Gaussian limit of the distribution (64).

  * •

The other interesting behaviour is close to edges. Setting
$\varphi=\pi-\theta$, we begin with the limit

$\varphi=\pi-\theta\propto\lambda$ (66)

(where by this we also include ${\varphi\over\lambda}\gg 1\ \text{fixed},\
\lambda\rightarrow 0$). In this case the distribution becomes

$\displaystyle\Psi(\theta,q)$
$\displaystyle=2\sqrt{2\over\pi\lambda}e^{-{2\over\lambda}({\pi\over
2}-\varphi)^{2}-{2\pi\varphi\over\lambda}}\sin(\varphi)\sinh({2\pi\varphi\over\lambda})$
(67) $\displaystyle=2\sqrt{2\over\pi\lambda}e^{-{\pi^{2}\over
2\lambda}-{2\varphi^{2}\over\lambda}}\sin(\varphi)\sinh({2\pi\varphi\over\lambda})\equiv\Psi(\varphi;q)\,,$

The quadratic term in the exponential can also be neglected in this regime,
giving rise to the density of states of the Schwarzian theory
($\propto\sinh(2\pi\sqrt{(E-E_{\text{min}})/\lambda^{3\over 2}})$) after
recalling that near the edge

$E-E_{\text{min}}={\varphi^{2}\over\sqrt{\lambda}}$ (68)

  * •

Note that we can actually neglect the quadratic term in the exponential
already at $\lambda\ll\varphi\ll\sqrt{\lambda}$, and extend the Schwarzian
regime. This points to another simplification of the spectrum which actually
covers the bulk of the spectrum at $\lambda\ll\varphi\ll\pi-\lambda$. In this
range we can also expand the second sinh, and obtain that the distribution is
just a gaussian in $(\varphi-\pi/2)$. The center of this range includes the
Gaussian-in-energy distribution, and its edges overlaps with the Schwarzian
distribution. It would be interesting to find a symmetry argument for this
entire range.

###  4.1 The canonical ensemble in the $q\to 1$ limit

Similarly one can analyze the canonical partition function in the limit
$q\rightarrow 1$. Using the variable $\varphi\equiv\pi-\theta$ as before

$Z(\beta)=\int_{0}^{\pi}d\varphi\,e^{2\beta\cos\varphi\over\sqrt{\lambda}}\Psi(\varphi;q)\
.$ (69)

We can treat most of the spectrum using the discussion in bullet 3 above,
leaving out only a very low temperature regime where $\phi\sim\lambda$. We
will prefer however to split the discussion according to first two bullets,
i.e., to a high temperature phase and a Schwarzian phase which then splits
into a low temperature and a very low temperature phase. Both of the latter
are obtained from the Schwarzian density of states and go smoothly into each
other, and we make this division mainly for the sake of the discussion of the
2-pt function in section 6, for which the difference between these regimes is
more meaningful.

  * •

High Temperature phase (when $\lambda^{-{1\over 2}}\gg\beta$) : localizing in
the region $|\varphi-{\pi\over 2}|\ll 1$, reduces the partition function to a
Gaussian around $E=0$, and the partition function can then be written as

$Z(\beta)=\sqrt{2\over\pi\lambda}\int_{0}^{\pi}d\varphi
e^{2\beta\cos\varphi\over\sqrt{\lambda}}e^{-{2\over\lambda}({\pi\over
2}-\varphi)^{2}}\sin\varphi$ (70)

It is clear that the gaussian cuts off the integral if $\varphi$ deviates from
$\pi\over 2$. To evaluate $Z(\beta)$, we set ${\pi\over 2}-\varphi\equiv x$.
The limit above translates to $x\ll 1$, in which case we approximate the
cosine by $x^{2}$ to obtain $Z(\beta)\sim e^{\beta^{2}\over 2}$ and the
integral is supported at $x={\beta\sqrt{\lambda}\over 2}$ with a width of
order $\sqrt{\lambda}$.

  * •

Low Temperature phase (when $\lambda^{-{3\over 2}}\gg\beta\gg\lambda^{-{1\over
2}}$) : generically, one expects $\varphi\ll 1$, but the thermodynamic
behaviour of the system is sensitive to how small $\varphi$ is compared to
$\lambda$, due to the argument in the sinh factor in (67). Consider the regime
$\lambda\ll\varphi\ll 1$, where the distribution is approximated by

$\Psi(\varphi;q)\approx\sqrt{2\over\pi\lambda}\,e^{-{2\over\lambda}({\pi\over
2}-\varphi)^{2}}\varphi$ (71)

Expanding the Boltzmann factor, the partition function reduces to101010The
term $-{2\varphi^{2}\over\lambda}$ in the exponent is negligible compared to
$-\beta\varphi^{2}\over\sqrt{\lambda}$ due to $\beta\sqrt{\lambda}\gg 1$.

$\displaystyle Z(\beta)$
$\displaystyle=\sqrt{2\over\pi\lambda}\int_{0}^{\pi}d\varphi\,e^{2\beta\cos\varphi\over\sqrt{\lambda}}e^{-{2\over\lambda}({\pi\over
2}-\varphi)^{2}}\varphi$ (72)
$\displaystyle\approx\sqrt{2\over\pi\lambda}e^{{2\beta\over\sqrt{\lambda}}-{\pi^{2}\over
2\lambda}}\int_{0}^{\pi}d\varphi\,e^{-{\beta\varphi^{2}\over\sqrt{\lambda}}+{2\pi\varphi\over\lambda}}\varphi\approx{\sqrt{2}\pi\over\beta^{3\over
2}\lambda^{3\over 4}}e^{{2\beta\over\sqrt{\lambda}}-{\pi^{2}\over
2\lambda}+{\pi^{2}\over\beta\lambda^{3\over 2}}}$

Notice the integral is mainly supported near
$\varphi={\pi\over\beta\sqrt{\lambda}}$ with a width of $\lambda^{1\over
4}\over\sqrt{\beta}$. The consistency with the assumption
$\lambda\ll\varphi\ll 1$ requires $\beta\ll\lambda^{-{3\over 2}}$. This
regime, along with the next one, are part of the conformal low energy limit of
the theory.

  * •

Very Low Temperature phase (when $\beta\gg\lambda^{-{3\over 2}}$) : consider
the regime $\varphi\ll\lambda$. After linearising both the $\sin$ and $\sinh$
factors, the distribution (67) simplifies to

$\Psi(\varphi;q)\approx{4\over\lambda}\sqrt{2\pi\over\lambda}\
e^{-{\pi^{2}\over 2\lambda}-{2\varphi^{2}\over\lambda}}\varphi^{2}$ (73)

The Boltzmann factor in the partition function cuts off the integral around
$\varphi\sim{\lambda^{1\over 4}\over\sqrt{\beta}}$. This is consistent with
our regime $\varphi\ll\lambda$, since $\beta\gg\lambda^{-{3\over 2}}$. The
partition function can then be evaluated as

$Z(\beta)={4\over\lambda}\sqrt{2\pi\over\lambda}e^{{2\beta\over\sqrt{\lambda}}-{\pi^{2}\over
2\lambda}}\int_{0}^{\pi}d\varphi\,e^{-{\beta\varphi^{2}\over\sqrt{\lambda}}}\varphi^{2}\approx\frac{\pi\sqrt{2}}{\beta^{3\over
2}\lambda^{3\over 4}}\,e^{{2\beta\over\sqrt{\lambda}}-{\pi^{2}\over
2\lambda}}$ (74)

where in evaluating the integral, we have replaced the upper limit by
$\infty$.

##### Relation to previous work.

The low energy behaviour identified in (67) is the one discussed in appendix B
in Cotler:2016fpe  and in Garcia-Garcia:2017pzl . To make the comparison with
Cotler:2016fpe  easier, notice the density of states (58) can be written as

$v(E(\mu)|q)={\cal N}\
{1\over\sqrt{1-\mu^{2}}}\prod_{k=0}^{\infty}\left(1-{\mu^{2}\over\cosh^{2}({k\lambda\over
2})}\right)$ (75)

where $\mu=\cos\theta$ and ${\cal
N}={\sqrt{1-q}\over\pi}\prod_{k=0}^{\infty}{1-q^{2k+2}\over 1-q^{2k+1}}$. This
matches equation (81) in Cotler:2016fpe  by identifying their parameters
$a,\lambda_{s},{\cal J}$ as

$a=\mu\,,\quad{\cal
J}=\sqrt{\lambda}\,e^{\lambda/8}\,,\quad\lambda_{s}=\frac{\lambda}{2}\,$ (76)

whereas both energies are the same111111Our normalizations are different from
Cotler:2016fpe  since their distribution $\rho_{s}(E)$ integrates to
$2^{N\over 2}$ whereas our $v(E|q)$ integrates to $1$. The second matching
follows from the observation that our variances equal unity, as in Erdos ,
whereas our normalisation was $\text{Tr}H^{2}/\text{Tr}\mathbb{I}=1$ (see
equation (80) in Cotler:2016fpe ). The third matching is due to the Majorana
nature of the fermions in the SYK model.

Thus the density of states in Erdos  is exactly the same as the doubled scaled
SYK, up to these identifications. The further triple scaled limit,

$\lambda\to 0\hskip 28.45274pt{2(E-E_{\text{min}})\over\lambda^{3\over
2}}\mbox{ fixed}\,,$ (77)

isolating the Schwarzian action in SYK, corresponds to the low energy
behaviour captured by the density of states (67) in our set-up.

##  5 Bulk reconstruction

In section 3 we presented a new derivation of the density of energies in
$v(E|q)$ in the $\lambda$-scaling limit (4)

$2^{-n}E\biggl{(}\mbox{Tr}(e^{-\beta
H})\biggr{)}=\int_{E_{min}}^{E_{max}}dEv(E|q)e^{-\beta E}$ (78)

keeping $\beta$ finite and where $E()$ on the left hand side is the average
over the ensemble of Hamiltonians. The range of integration on the right hand
side is the spectrum of the random Hamiltonian, and its ”randomness” now hides
in the $1/n$ corrections which are neglected in this limit. Loosely, one can
hope that for a specific realization of the Hamiltonian H, one can write

$\mbox{Tr}(e^{-\beta H})=\int_{E_{min}}^{E_{max}}dEv(E|q)e^{-\beta
E}\bigl{(}1+{\cal O}({1\over n})\bigr{)}\ ,$ (79)

with probability 1 (or $1-{\cal O}(1/n)$) on the space of random Hamiltonians,
in the large n limit. In this case one is dealing with a specific Hamiltonian
on the left hand side. This single Hamiltonian realization corresponds to the
boundary field theory Hamiltonian in the AdS/CFT written in terms of the
fundamental field theory objects - in our case the spin operators. The
operator is random and only in the $n\to\infty$ limit its spectrum converges
to anything universal.

In this section, we suggest that the operator $T$ (or ${\hat{T}}$) is the bulk
Hamiltonian, i.e. the analogue of the bulk Hamiltonian for the near-AdS
background - whose low energy limit is given by the Schwarzian action - but
extended to the full model. Recall that the parameter $E$ appearing in the
right hand side of (78) and (79) can be reinterpreted as the energy of the
field theory Hamiltonian, but it is also the eigenvalue of the operator $T$
(or ${\hat{T}}$) which acts on the (altogether different) Hilbert space of
weights of open chord lines. Whereas the spectrum of $H$ changes from
realization to realization, the matrix ${\hat{T}}$ is fixed. There is no
contradiction since we work in the limit $n\to\infty$, fixed $\lambda$ limit,
where the spectrum of $H$ is universal.

Furthermore, the operator ${\hat{T}}$ can be used as the Hamiltonian not only
for the partition function, but for a much broader set of computations. It
should be clear that the insertion of any finite polynomial of $H$ in
expectation values involving density matrices of the form

$\sum_{eigenvalues\ E}|E\rangle f(E)\langle E|$ (80)

for any analytic weight function $f(E)$, can be turned into the insertion of
the same polynomial with ${\hat{T}}$ as its argument, following the procedure
described in section 3. In other words, the insertion of $e^{-itH}$ in
expectation values involving (80) can be exactly replaced by $e^{-it\hat{T}}$,
while the density matrix itself is mapped into the density matrix (as an
operator in the Hilbert space defined on the chord diagram side)

$f(\hat{T})|v_{0}\rangle\langle v_{0}|.$ (81)

Having two different Hamiltonians, acting on different Hilbert spaces but
propagating the system in exactly the same way, supports the dual
interpretation we suggest for ${\hat{T}}$.

This means that we can access a large set of weights on the energy eigenstates
as long as the function is smoother than the energy spacing (actually smoother
than ${1\over n}$ for the entire energy band). This is not in contradiction
with what we know about the bulk Hamiltonian (anything which extends the low
energy effective action), since it is not clear that it should be able to
capture states whose support on close by energy states is rapidly
varying121212Unless, for example, one believes in the microstate program in
its strongest form where one can choose a specific energy eigenstate in the
most extreme case..

Phrased differently we regard $E$, when used as the eigenvalue of ${\hat{T}}$,
as a parameter which scans over the allowed energy range only after taking the
limit $n\to\infty$. It is not the discrete spectrum of energies of the finite
$n$ system. It should be viewed as a coarse grained version of the latter,
very much like the energy measured in gravity is a coarse grained version of
the discrete set of energies of the field theory (when defined on a compact
space). Going from the eigenvalues of $\hat{T}$ to the eigenvalues of $H$ at
finite $n$ is an interesting problem, and it is similar to seeing - in General
Relativity - the discreteness in energies of a black hole.

The above discussion, together with the behaviour of the partition function in
the low temperature regime, suggests the low energy physics for $q\to 1$
should be governed by the Schwarzian action (in the gravity dual), as in the
SYK model. In the following, we derive this connection by matching the
continuum limit of the equation determining the spectrum of $\hat{T}$ with
Liouville quantum mechanics131313We would like to thank D. Bagrets for a
discussion of this point., which can be written as the Schwarzian action, as
discussed in Bagrets:2016cdf ; Bagrets:2017pwq 141414See Mertens:2017mtv  for
a 2d CFT perspective on this matter..

To take the continuum limit, it is convenient to define the matrix
$\tilde{T}\equiv S\hat{T}S^{-1}$ where $S$ is a diagonal matrix with entries
$S_{ii}=(-1)^{i}$. Notice that solving for the eigenvalues of the $\tilde{T}$
matrix still resembles a scattering problem on the half line, with the index
$i$ of the vector measuring the distance from the origin, just like it did for
the $\hat{T},T$ matrices. The asymptotic form of the $\tilde{T}$ matrix is

$\tilde{T}=\frac{1}{\sqrt{1-q}}\begin{bmatrix}0&-1&0&0&0&\dots\\\
-1&0&-1&0&0&\dots\\\ 0&-1&0&-1&0&\dots\\\
\vdots&\ddots&\ddots&\ddots&\ddots&\ddots\\\ \end{bmatrix}$ (82)

In the continuum limit, the above matrix includes the second derivative
operator. To make this more precise, define

$\phi=\log(q)i$ (83)

Using the form of the $\hat{T}$ operator in (42), its continuum limit equals

${\tilde{T}}\to\frac{1}{\sqrt{1-q}}\biggl{(}-2-(\log
q)^{2}\partial_{\phi}^{2}+{q\over 2}e^{\phi}\biggr{)}$ (84)

Notice the potential term comes from the expansion $\sqrt{1-q^{i+1}\over
1-q}={1\over\sqrt{1-q}}(1-{q^{i+1}\over 2}+\dots)$ in $\eta_{i}$, as defined
in (29), which is accurate since $i$ is large and $q\to 1$, from below.

The eigenvalue problem then reduces to the quantum mechanical eigenvalue
problem

$\left(-(\log\,q)^{2}\partial^{2}_{\phi}+\frac{q}{2}e^{\phi}\right)\Psi=\sqrt{1-q}(E-E_{0})\Psi\,.$
(85)

This is equivalent to the Liouville form of the Schwarzian action in equation
32 in Bagrets:2017pwq , given by

$H=-{\partial^{2}_{\phi}\over 2M}+\gamma e^{\phi}$ (86)

after a constant shift of $\phi$. In Bagrets:2017pwq  $M$ was the scale
$M=\frac{N\log N}{64J\sqrt{\pi}}$ (for the SYK model with quartic
interactions). For us it is set by $|\log(q)|^{-2}\sim\lambda^{-2}$.

The prescription in Bagrets:2016cdf ; Bagrets:2017pwq  (and in Mertens:2017mtv
for 2D case) requires that, in the path integral, we sum over trajectories
that begin and end in the strong coupling region $\phi\to\infty$. This is in
qualitative agreement with our prescription since we place the state $v_{0}$
as initial and final states. Recall that $v_{0}=(1,0,0,0....)$, i.e., only the
$i=0$ term is turned on, which where the term $q^{i}$ is the largest. In terms
of $\phi$, $e^{\phi}$ is largest which is indeed the analogue of the Liouville
strong coupling region. The models are of course not exactly the same since
the model in Bagrets:2016cdf  captures the low energy and the ${\hat{T}}$
matrix captures the full dynamics.

This also gives an interpretation of the index $i$ via its relation to $\phi$.
$\phi(t)$ measures where the AdS2 space is glued to whatever non-universal UV
we have (the leading effect being the Schwarzian action), i.e., $\phi(t)$
parametrizes the length of the AdS2 throat. We see that in the full model the
size of $AdS_{2}$ is actually quantized, giving rise to a minimal size $AdS$
which corresponds to the state $v_{0}$.

It is worth reiterating that the density of states for the $H$ Hamiltonian is
different than the density of states for the matrix $\hat{T}$, even in the
large $n$ limit. Rather the density of states in the former is related to the
density of states in the latter $H$ by equation (38) or, equivalently, by

$E\biggl{(}Tr_{H}(e^{-\beta H})\biggr{)}=v_{0}^{\dagger}e^{-\beta T}v_{0}$
(87)

which means that we have to put a specific initial and final states for
$\hat{T}$ in order to compute the partition function. It is tempting to
interpret this in Minkowski space as a computation with initial and final
states at the past and future singularities of the black hole.

##  6 The two point function

###  6.1 The exact 2-pt function

As explained in section 2, we want to compute correlators of random operators
$M$ taken from the same universality class as the Hamiltonian (2). Hence,
these are defined by

$M=3^{-p_{m}/2}{n\choose p_{m}}^{-1/2}\sum_{J}m_{J}\sigma_{J}\,,$ (88)

where $J$ is now a string of $p_{m}$ distinct sites and Pauli matrices. The
sum runs over all such possible $J$’s, and $m_{J}$ are independent Gaussian
variables with zero mean and unit standard deviation (in particular they are
also independent of the coefficients $\alpha_{J}$ in H).

There are two relevant parameters that we will keep fixed in the limit
$n\to\infty$. The first is the analogue of $\lambda$ (see (3)) for the random
operator (88)

$\lambda_{m}=\frac{3}{4}\frac{p_{m}^{2}}{n},\quad\quad
q_{m}=e^{-\lambda_{m}}\,.$ (89)

The second is

${\tilde{\lambda}}=\sqrt{\lambda\lambda_{m}}=\frac{3}{4}\frac{pp_{m}}{n},\quad\quad{\tilde{q}}=e^{-\tilde{\lambda}}\,.$
(90)

We want to evaluate the exact thermal 2-pt function for the random operator M

$2^{-n}\cdot E\left[\mbox{Tr}(e^{-\beta H}M(t)M(0))\right],\ \ \text{or}\ \
2^{-n}\cdot E\left[Tr(e^{-{\beta H\over 2}}M(t)e^{-{\beta H\over
2}}M(0))\right]$ (91)

for any value of $\beta$ and $t$. The formalism developed below, based on the
set-up in section 3, proceeds by evaluating, and then resumming, expressions
of the form

$2^{-n}\cdot E\left[Tr(H^{k_{1}}MH^{k_{2}}M)\right]\,.$ (92)

This formalism can be extended to compute any n-pt function Vladi .

The strategy is to reduce the computation to some relevant chord partition
function, and then to evaluate it. The identification of the relevant
partition function follows the discussion in section 3. The Gaussian
integration over the random coefficients of the operators still pairs them.
Hence, one can still think in terms of chord diagrams. The only difference is
that the Gaussian integral over $m_{J}$’s pairs the two $M$ insertions,
whereas the Gaussian integral over $\alpha_{J}$’s pairs the $k_{1}+k_{2}$
insertions of $H$. To take into account the index sets of the different chords
and their intersections, one must evaluate the trace over the Pauli matrices.
The arguments leading to only pairwise intersection and to the length of the
intersections being Poisson distributed remain the same. The only difference
is that intersections between two $H$-lines are determined by the length
parameter of the $H$ operator, giving a factor of $q$ to their intersection,
whereas the intersection of an $H$-line with $M$-line is determined by both
the length of $M$ and the length of $H$, giving a factor of ${\tilde{q}}$ to
their intersection. The net result is that one is left with a marked chord
diagram where one chord is distinguished - an example is given in figure 2 \-
and the partition function that we are interested in is the marked chord
partition function, as promised in section 2.3.

To evaluate the marked chord diagram, we need to modify the ”hopscotch”
procedure described in section 3.2 to include the marked chord. Given the
distinguished nature of the pair of $M$ insertions, it is convenient to choose
where to open the circular chord diagram in such a way that one $M$ appears to
the rightmost of the line, and the other $M$ somewhere in the interior, as in
figure 6. The two operators $M$ are denoted by red dots and are paired by the
bold faced line. The procedure now consists in pairing the remaining $H$’s
starting with the rightmost insertion of $H$. Propagating the system through
the first $k_{2}$ steps, i.e the $k_{2}$ $H$’s between the two $M$ operators,
gives the same contribution as before. However, when we hop over the second
insertion of $M$, the open $H$-lines cross the $M$-line picking up an
additional factor of ${\tilde{q}}^{\text{no.\ of\ lines}}$. The last step is
to propagate for the remaining $k_{1}$ steps. The expression for the marked
chord diagram, or the 2-pt function of M, is therefore

$\dots$$\vdots$ Figure 6: Marked chord diagram for a two point function.

$2^{-n}\cdot
E\left[\mbox{Tr}\bigl{(}H^{k_{1}}MH^{k_{2}}M\bigr{)}\right]=\langle
0|T^{k_{1}}W({\tilde{q}})T^{k_{2}}|0\rangle$ (93)

where

$W({\tilde{q}})=\text{Diag}(1,{\tilde{q}},{\tilde{q}}^{2},....)$ (94)

encodes the intersection of an $H$-line with an $M$-line, when the former hops
over the latter. Recalling that $T=P^{-1}\hat{T}P$ and noticing that
$[P,W]=0$, we can also write the two point function as

$2^{-n}\cdot
E\left[\mbox{Tr}\bigl{(}H^{k_{1}}MH^{k_{2}}M\bigr{)}\right]=\langle
0|\hat{T}^{k_{1}}W({\tilde{q}})\hat{T}^{k_{2}}|0\rangle$ (95)

The 2-pt function can be written in terms of the matrix elements of $\hat{T}$
defined in (54) by inserting a complete set of states

$\begin{split}\langle
0|\hat{T}^{k_{1}}W({\tilde{q}})\hat{T}^{k_{2}}|0\rangle&=\sum_{l=0}^{\infty}\langle
0|{\hat{T}}^{k_{1}}|l\rangle{\tilde{q}}^{l}\langle
l|{\hat{T}}^{k_{2}}|0\rangle\\\ &=\int
d\theta_{1}d\theta_{2}\hat{\psi}_{0}(\theta_{1})\hat{\psi}_{0}(\theta_{2})E(\theta_{1})^{k_{1}}E(\theta_{2})^{k_{2}}\sum_{l}\hat{\psi}_{l}(\theta_{1}){\tilde{q}}^{l}{\hat{\psi}}_{l}(\theta_{2})\end{split}$
(96)

where $|l\rangle$ stands for a vector having 1 in the $l$’th place and
$E(\theta)={2\cos\theta\over\sqrt{1-q}}$. Using the $\hat{\psi}$ wavefunctions
in (142), the infinite sum (96) reduces to

$\displaystyle\sum_{l}\hat{\psi}_{l}(\theta_{1}){\tilde{q}}^{l}{\hat{\psi}}_{l}(\theta_{2})$
$\displaystyle=$
$\displaystyle\hat{\psi}_{0}(\theta_{1}){\hat{\psi}}_{0}(\theta_{2})\sum_{l}H_{l}(\cos(\theta_{1})|q)H_{l}(\cos(\theta_{2})|q)\frac{{\tilde{q}}^{l}}{(q;q)_{l}}$
$\displaystyle=$
$\displaystyle\hat{\psi}_{0}(\theta_{1}){\hat{\psi}}_{0}(\theta_{2})\frac{({\tilde{q}}^{2};q)_{\infty}}{({\tilde{q}}e^{i(\theta_{1}+\theta_{2})},{\tilde{q}}e^{i(-\theta_{1}+\theta_{2})},{\tilde{q}}e^{i(\theta_{1}-\theta_{2})},{\tilde{q}}e^{i(-\theta_{1}-\theta_{2})};q)_{\infty}}$

where we used the identity (135).

To evaluate $\mbox{Tr}[e^{-{\beta H\over 2}}M(t)e^{-{\beta H\over 2}}M(0)]$,
we expand the exponentials insert (95), and resum the power of the eigenvalues
${\hat{T}}$ matrices (which now appear twice) into an exponential. One then
gets

$E[\mbox{Tr}\left(e^{-{\beta H\over 2}}M(t)e^{-{\beta H\over
2}}M(0)\right)]\\\
={(q;q)_{\infty}^{2}({\tilde{q}}^{2};q)_{\infty}\over(2\pi)^{2}}\int_{0}^{\pi}d\theta_{1}d\theta_{2}e^{{2cos(\theta_{1})(-{\beta\over
2}+it)\over\sqrt{(1-q})}}e^{{2cos(\theta_{2})(-{\beta\over
2}-it)\over\sqrt{(1-q})}}\\\ \
\times{(e^{2i\theta_{1}},q)_{\infty}(e^{-2i\theta_{1}},q)_{\infty}(e^{2i\theta_{2}},q)_{\infty}(e^{-2i\theta_{2}},q)_{\infty}\over({\tilde{q}}e^{i(\theta_{1}+\theta_{2})},{\tilde{q}}e^{i(-\theta_{1}+\theta_{2})},{\tilde{q}}e^{i(\theta_{1}-\theta_{2})},{\tilde{q}}e^{i(-\theta_{1}-\theta_{2})};q)_{\infty}}$
(98)

where the value (142) of ${\hat{\psi}}_{0}(\theta)^{2}$ was inserted.

In the next subsection we will evaluate this expression for a special case of
$q$ and $\tilde{q}$. But before we do that, we will perform a quick check on
our results above.

##### A check.

Before evaluating (98) for a special case of $q$ and $\tilde{q}$, one can
perform a check by taking the $\tilde{q}\to 1$ limit. There should be no cost
for the $H$ lines crossing the $M$ lines in this limit. Hence, it must be that

$E\left[Tr(H^{k_{1}}MH^{k_{2}}M)\right]=E\left[Tr(H^{k_{1}+k_{2}})\right]\quad\text{when}\quad\tilde{q}\to
1$ (99)

To check (98) is compatible with this behaviour, notice that near
$\tilde{q}\to 1$, $(\tilde{q}^{2};q)_{\infty}\to 0$, due to the first term in
the product. Hence (98) vanishes, unless $\theta_{1}\to\theta_{2}$, since an
additional zero in the denominator occurs then151515Another zero may appear in
the denominator when $\theta_{1}+\theta_{2}=\pi$ but this appears in a co-
dimension 2 in the range of integration.. Hence, the integrand in (98) behaves
like a delta function whose strength is given by

${(\tilde{q}^{2};q)_{\infty}\over|(\tilde{q}e^{i(\theta_{1}-\theta_{2})};q)_{\infty}|^{2}}={2(1-\tilde{q})\over(1-\tilde{q})^{2}+(\theta_{1}-\theta_{2})^{2}}\times{(q;q)_{\infty}\over(q;q)_{\infty}^{2}}=2\pi\delta(\theta_{1}-\theta_{2})\times{1\over(q;q)_{\infty}}$
(100)

Thus, in the $\tilde{q}\to 1$ limit, the correlator (98) equals

${(q;q)_{\infty}\over(2\pi)}\ \int d\theta_{1}E^{k_{1}+k_{2}}(\theta_{1})\
|(e^{2i\theta_{1}};q)_{\infty}|^{2}=\int d\theta
E^{k_{1}+k_{2}}(\theta)\Psi(\theta,q)=m_{k_{1}+k_{2}}(q)\,,$ (101)

where we used (55) in the last step, in agreement with (99).

###  6.2 The $q\to 1$ limit with ${\tilde{q}}=q^{m}$

The exact 2-pt function of $M$ (98) holds for all ranges of time (which are
held fixed in the $n\to\infty$ limit). In the remainder of this section we
will compute the formula in a specific case, which is the low energy regime
where conformal symmetry is expected to appear, as discussed in section 4 and
in Appendix B of Cotler:2016fpe .

We will work in the limit $q,\ {\tilde{q}}\to 1$ since we want to work in the
limit in which the correlators of each operator separately approximately
factorize. However, more and more terms contribute in the Pochammer symbols in
this limit, similar to what we had for the partition function, and hence it is
important how we take this limit. Since in gravity non-factorization of
correlation functions for different operators is governed by the same
parameters (e.g. the same 1/N), then the rates of $q\to 1$ and ${\tilde{q}}\to
1$ should be related.

A particularly simple case to analyze is ${\tilde{q}}=q^{m}$ with $m$ an
integer. This has technical advantages, but it is also physically interesting
because it corresponds to

$p_{m}=m\ p\,.$ (102)

That is, if the Hamiltonian is made out of a sum of strings of p spin
operators (with random coefficients), then the random operator M is made out
of a string of $m\cdot p$ spin operators. This is reminiscent of the statement
that, say for $4D,\ {\cal N}=4$ SYM, the Hamiltonian is a descendant of
$\text{Tr}(X^{2})$, yet we can probe the system with low energy fields, which
correspond to single trace operators of the form $\text{Tr}(X^{n})$, $n>2$,
and their conformal descendants.

As discussed in Cotler:2016fpe  and section 4, our model has a conformal low
energy limit. Hence, conformal symmetry should assign a dimension one to the
Hamiltonian. If the fundamental fields (in this case the spin operators) can
be assigned a specific conformal dimension, and if this conformal dimension is
additive in composite operators - as in the SYK model on both counts - then
one expects the conformal dimension of $M$ to be $m$. We will see how our
exact formula matches this, up to the existence of mixing with operators of
lower dimension when we work at finite temperature. Despite this, our exact
formula always has an overlap with an operator of the right dimension.

Before doing the computation we would like to recall an additional formula to
which we will compare our result. We will actually be computing the “two sided
correlator”

$2^{-n}E\left[Tr\bigl{(}e^{-{\beta H\over 2}}M(t)e^{-{\beta H\over
2}}M(0)\bigr{)}\right].$ (103)

This computation is slightly easier than the ordinary thermal correlator. We
refer to this as the two sided correlator since, in an eternal black hole in
$AdS$, it is the relevant correlator when there is one operator on each of the
boundaries. For an particle of mass $M$ in the BTZ black hole this correlator
is (see for example Shenker:2013pqa  in the Eikonal approximation, with a
shock wave there)

$\propto{1\over\cosh\bigl{(}\pi\beta(t_{R}-t_{L})\bigr{)}^{2Ml}}$ (104)

where $l$ is the $AdS_{3}$ radius, $M$ is the mass of the particle and $Ml$ is
the conformal dimension of the associated operator. This is what is expected
from conformal invariance. In the single sided correlator the cosh is replaced
by a sinh, to obtain the expected short distance behaviour of $1/t^{2Ml}$, and
the correct Euclidean time periodicity.

####  6.2.1 The reduced formula

When $\tilde{q}=q^{m}$, the identity

$|(\tilde{q}e^{i\phi};q)_{\infty}|^{2}={|(e^{i\phi};q)_{\infty}|^{2}\over
4\sin^{2}{\phi\over 2}\prod_{l=1}^{m-1}|1-e^{i\phi}q^{l}|^{2}}\,,$ (105)

allows to write the 2-pt function (98) as

$\begin{split}16(q^{2m};q)_{\infty}&{(q;q)_{\infty}^{2}\over(2\pi)^{2}}\times\int_{0}^{\pi}d\theta_{1}d\theta_{2}e^{{2cos(\theta_{1})(-{\beta\over
2}+it)\over\sqrt{(1-q})}}e^{{2cos(\theta_{2})(-{\beta\over
2}-it)\over\sqrt{(1-q})}}\sin^{2}({\theta_{1}+\theta_{2}\over
2})\sin^{2}({\theta_{1}-\theta_{2}\over 2})\\\
&\times\prod_{l=1}^{m-1}|(1-q^{l}e^{i(\theta_{1}+\theta_{2})})|^{2}|(1-q^{l}e^{i(\theta_{1}-\theta_{2})})|^{2}\times{|(e^{2i\theta_{1}},q)_{\infty}|^{2}|(e^{2i\theta_{2}},q)_{\infty}^{2}\over|(e^{i(\theta_{1}+\theta_{2})};q)_{\infty}|^{2}|(e^{i(\theta_{1}-\theta_{2})};q)_{\infty}|^{2}}\end{split}$
(106)

Notice the finite product can be rewritten, within the integral, as

$\begin{split}{\cal
D}_{m}(t,\beta)\equiv&\prod_{l=1}^{m-1}|(1-q^{l}e^{i(\theta_{1}+\theta_{2})})|^{2}|(1-q^{l}e^{i(\theta_{1}-\theta_{2})})|^{2}\\\
=&\prod_{l=1}^{m-1}\bigg{[}(1-q^{2l})^{2}+q^{l}\left({1+q^{l}\over
2}\right)^{2}(1-q)(-i\partial_{t})^{2}-q^{l}(1-q^{l})^{2}(1-q)(\partial_{\beta})^{2}\bigg{]}\\\
\equiv&\prod_{l=1}^{m-1}D_{l}(t,\beta)\xrightarrow{q\to
1}(1-q)^{m-1}(-i\partial_{t})^{2m-2}\ .\end{split}$ (107)

Taking the derivatives outside of the integral, allows to write the integrand
in terms of $\vartheta$ functions (see (129)) depending only on $q$,

$\begin{split}16(q^{2m};q)_{\infty}&{(q;q)_{\infty}^{2}\over(2\pi)^{2}}\ \
{\cal
D}_{m}(t,\beta)\int_{0}^{\pi}d\theta_{1}d\theta_{2}e^{{2cos(\theta_{1})(-{\beta\over
2}+it)\over\sqrt{(1-q})}}e^{{2cos(\theta_{2})(-{\beta\over
2}-it)\over\sqrt{(1-q})}}\\\ &\times\sin({\theta_{1}+\theta_{2}\over
2})\sin({\theta_{1}-\theta_{2}\over 2})\sin\theta_{1}\sin\theta_{2}\ \
{\vartheta_{1}({\theta_{1}\over\pi}|{i\lambda\over
2\pi})\vartheta_{1}({\theta_{2}\over\pi}|{i\lambda\over
2\pi})\over\vartheta_{1}({\theta_{1}+\theta_{2}\over 2\pi}|{i\lambda\over
2\pi})\vartheta_{1}({\theta_{1}-\theta_{2}\over 2\pi}|{i\lambda\over
2\pi})}\end{split}$ (108)

The $q\to 1$ limit of the Jacobi Theta functions is evaluated as in (63),
bringing the 2-pt function to the form

$\begin{split}32(q^{2m};q)_{\infty}&{(q;q)_{\infty}^{2}\over(2\pi)^{2}}\ \
{\cal
D}_{m}(t,\beta)\int_{0}^{\pi}d\theta_{1}\int_{0}^{\pi}d\theta_{2}e^{{2cos(\theta_{1})(-{\beta\over
2}+it)\over\sqrt{\lambda}}}e^{{2cos(\theta_{2})(-{\beta\over
2}-it)\over\sqrt{\lambda}}}\\\ &\times\sin({\theta_{1}+\theta_{2}\over
2})\sin({\theta_{1}-\theta_{2}\over 2})\sin\theta_{1}\sin\theta_{2}\ \
e^{-{3\pi^{2}\over{2}\lambda}-{1\over\lambda}{(\theta_{1}-{\pi\over
2})^{2}}-{1\over\lambda}{(\theta_{2}-{\pi\over 2})^{2}}}\\\
&\times{\sinh({2\pi\theta_{1}\over\lambda})\sinh({2\pi(\pi-\theta_{1})\over\lambda})\over\sinh({2\pi({\theta_{1}+\theta_{2}\over
2})\over\lambda})\sinh({2\pi(\pi-{\theta_{1}+\theta_{2}\over
2})\over\lambda})}\
{\sinh({2\pi\theta_{2}\over\lambda})\sinh({2\pi(\pi-\theta_{2})\over\lambda})\over\sinh({2\pi({\theta_{1}-\theta_{2}\over
2})\over\lambda})(1-2\cosh({2\pi(\theta_{1}-\theta_{2})\over\lambda})e^{-4\pi^{2}/\lambda})}\\\
&\equiv{32}(q^{2m};q)_{\infty}{(q;q)_{\infty}^{2}\over(2\pi)^{2}}\ \ {\cal
D}_{m}(t,\beta)\cdot I(\beta,t,q)\end{split}$ (109)

This is the exact 2-pt function for $q=\tilde{q}^{m}$ in the limit $\lambda\to
0$. In the next subsections, we study the function $I(\beta,t,q)$, from which
all $m>1$ correlators can be extracted, in the low temperature and very low
temperature regimes (or long time, and very long time regimes).

###  6.3 Low and very low temperature regimes

Since the integral $I(\beta,t,q)$ localizes near the edges at low energies, we
define $\phi_{i}=\pi-\theta_{i}$. Expanding the integral near $\phi_{i}\sim
0$,

$\begin{split}I(\beta,t,q)=&\ {1\over
8}e^{2\beta\over\sqrt{\lambda}}\int_{0}^{\pi}d\phi_{1}\int_{0}^{\pi}d\phi_{2}\,e^{{\phi_{1}^{2}(-{\beta\over
2}+it-{1\over\sqrt{\lambda}})\over\sqrt{\lambda}}}e^{{\phi_{2}^{2}(-{\beta\over
2}-it-{1\over\sqrt{\lambda}})\over\sqrt{\lambda}}}\\\
&\times(\phi_{1}+\phi_{2})(\phi_{2}-\phi_{1})\phi_{1}\phi_{2}\times{\sinh({2\pi\phi_{1}\over\lambda})\over\sinh({2\pi({\phi_{1}+\phi_{2}\over
2})\over\lambda})}{\sinh({2\pi\phi_{2}\over\lambda})\over\sinh({2\pi({\phi_{2}-\phi_{1}\over
2})\over\lambda})}\\\ \end{split}$ (110)

As explained in section 4.1, the low energy (and very low energy) regime
satisfies $\beta\,\sqrt{\lambda}\gg 1$. To study the behaviour of the Gaussian
factors in the above integral, it is convenient to rescale the integration
variable $\varphi_{i}\equiv{\phi_{i}\over\lambda}$ together with the time and
temperature parameters ${\tilde{\beta}}\equiv\lambda^{3/2}\beta,\
{\tilde{t}}\equiv\lambda^{3/2}t$. The low energy regime is equivalently
described by

$\beta\,\sqrt{\lambda}\gg 1\quad\Leftrightarrow\quad{\tilde{\beta}}\gg\lambda$
(111)

allowing to approximate (110) by

$\begin{split}I(\beta,t,q)=&{\lambda^{6}\over
8}e^{2\beta\over\sqrt{\lambda}}\int_{0}^{\pi/\lambda}d\varphi_{1}\int_{0}^{\pi/\lambda}d\varphi_{2}\,e^{{\varphi_{1}^{2}(-{{\tilde{\beta}}\over
2}+i{\tilde{t}})}}e^{{\varphi_{2}^{2}(-{{\tilde{\beta}}\over
2}-i{\tilde{t}})}}\\\
&\times(\varphi_{1}+\varphi_{2})(\varphi_{2}-\varphi_{1})\varphi_{1}\varphi_{2}\times{\sinh({2\pi\varphi_{1}})\over\sinh({2\pi({\varphi_{1}+\varphi_{2}\over
2})})}{\sinh({2\pi\varphi_{2}})\over\sinh({2\pi({\varphi_{2}-\varphi_{1}\over
2})})}\\\ \end{split}$ (112)

This integral has two regimes, following a similar discussion for the
partition function in section 4.1 :

  * •

The low energy-long time regime characterised by
${\tilde{\beta}},\,{\tilde{t}}\ll 1$, where the integral receives
contributions from the range $\varphi_{i}\gg 1$.

  * •

The very low temperature regime, or very long time scale regime, characterised
by ${\tilde{\beta}}\gg 1$ or ${\tilde{t}}\gg 1$, where the integral receives
contributions primarily from $\varphi_{i}\ll 1$.

####  6.3.1 Low temperature regime

The low energy-long time regime ${\tilde{\beta}},{\tilde{t}}\ll 1$ allows to
extend the range of integration to $\infty$ since the gaussian in the
integrand cuts off the integral well before the limits in (112). Notice also
the integral is supported at large values of $\varphi_{1},\varphi_{2}$,
allowing us to approximate three of the $\sinh$ functions by their larger
exponentials

$\begin{split}I(\beta,t,q)&={\lambda^{6}e^{2\beta\over\sqrt{\lambda}}\over
2}\int_{0}^{\infty}d\varphi\int_{-\varphi}^{\varphi}d\sigma
e^{-{\tilde{\beta}}\varphi^{2}-{\tilde{\beta}}\sigma^{2}-4i{\tilde{t}}\varphi\sigma}\varphi\sigma(\varphi^{2}-\sigma^{2}){e^{2\pi\varphi}\over\sinh{2\pi\sigma}}\end{split}$
(113)

where we changed variables to $\varphi={\varphi_{1}+\varphi_{2}\over 2},\
\sigma={\varphi_{2}-\varphi_{1}\over 2}$. Due to the
${1\over\sinh(2\pi\sigma)}$ term, the $\sigma$ integral receives contributions
from finite $\sigma$, whereas its limit of integration is $\pm\varphi$, much
larger quantities. This means we can trade the $\sigma$ limits by $\pm\infty$.
Furthermore, we can also neglect the $e^{-{\tilde{\beta}}\sigma^{2}}$ term and
the $\sigma^{2}$ term relative to $\varphi^{2}$ in the
$(\varphi^{2}-\sigma^{2})$ term. After these approximations, our integral
reduces to

$\begin{split}I={\lambda^{6}e^{2\beta\over\sqrt{\lambda}}\over
2}\int_{0}^{\infty}d\varphi
e^{-{\tilde{\beta}}\varphi^{2}}\varphi^{3}e^{2\pi\varphi}\int_{-\infty}^{\infty}d\sigma
e^{-4i{\tilde{t}}\varphi\sigma}{\sigma\over\sinh{2\pi\sigma}}\end{split}$
(114)

Using the identity

$\int_{-\infty}^{\infty}d\sigma
e^{-{4i{\tilde{t}}\sigma\varphi}}\frac{\sigma}{\sinh(2\pi\sigma)}=\frac{1}{8\cosh^{2}({\tilde{t}}\varphi)},$
(115)

and introducing a further variable of integration
$\varphi\equiv{\varphi_{s}\over\sqrt{\tilde{\beta}}}+{\pi\over\tilde{\beta}}$,
we finally get

$\displaystyle I(\beta,t,q)$
$\displaystyle={\lambda^{6}e^{2\beta\over\sqrt{\lambda}}\over
16}\int_{0}^{\infty}d\varphi{\varphi^{3}e^{-{\tilde{\beta}}\varphi^{2}+2\pi\varphi}\over\cosh^{2}{({\tilde{t}}\varphi)}}$
(116)
$\displaystyle={\pi^{3}\lambda^{6}e^{{2\beta\over\sqrt{\lambda}}+{\pi^{2}\over\tilde{\beta}}}\over
16\tilde{\beta}^{3}\sqrt{\tilde{\beta}}}\int_{-{\pi\over\sqrt{\tilde{\beta}}}}^{\infty}d\varphi_{s}e^{-\varphi_{s}^{2}}{(1+{\sqrt{\tilde{\beta}}\varphi_{s}\over\pi})^{3}\over\cosh^{2}{[{{\pi\tilde{t}}\over\tilde{\beta}}(1+{\sqrt{\tilde{\beta}}\varphi_{s}\over\pi})]}}$

Since ${\tilde{\beta}}\ll 1$ we can in any case neglect the $\varphi_{s}$
dependence in the numerator. The integral shows different behaviours depending
on the scaling of ${\tilde{t}}$ :

  * •

When ${\tilde{t}\over\sqrt{\tilde{\beta}}}\ll 1$, the $\varphi_{s}$ dependence
in the denominator can be neglected and, to leading order in $\tilde{\beta}$,
the result is

$I(\beta,t,q)={\lambda^{3\over 4}\over 16}\left({\pi\over\beta}\right)^{7\over
2}e^{{2\beta\over\sqrt{\lambda}}+{\pi^{2}\over\beta\lambda^{3\over 2}}}\
{1\over\cosh^{2}\left(\pi t\over\beta\right)}\,,\quad\quad
1\gg\sqrt{\tilde{\beta}}\gg\tilde{t}$ (117)

  * •

When $\tilde{t}\gg\sqrt{\tilde{\beta}}(\gg{\tilde{\beta}})$, the cosh in the
denominator contributes. Keeping only the larger exponential due to the
latter, the integral becomes

$\displaystyle I(\beta,t,q)$
$\displaystyle={\lambda^{6}e^{2\beta\over\sqrt{\lambda}}\over
4}\int_{0}^{\infty}d\varphi
e^{-{\tilde{\beta}}\varphi^{2}+2\pi\varphi-2{\tilde{t}}\varphi}\varphi^{3}$
(118) $\displaystyle={\lambda^{6}e^{2\beta\over\sqrt{\lambda}}\over
4}e^{{(\pi-{\tilde{t}})^{2}\over{\tilde{\beta}}}}{(\pi-{\tilde{t}})^{3}\over{\tilde{\beta}}^{3}\sqrt{\tilde{\beta}}}\int_{-{\pi-\tilde{t}\over\sqrt{\tilde{\beta}}}}^{\infty}d\varphi_{s}\,e^{-\varphi_{s}^{2}}\bigg{(}1+{\sqrt{\tilde{\beta}}\varphi_{s}\over\pi-\tilde{t}}\bigg{)}^{3}$
$\displaystyle\sim\frac{\lambda^{3/4}}{16}\left(\frac{\pi}{\beta}\right)^{7\over
2}e^{2\beta\over\sqrt{\lambda}}\,4e^{\frac{(\pi-{\tilde{t}})^{2}}{\tilde{\beta}}}\,,\quad\quad
1\gg\tilde{t}\gg\sqrt{\tilde{\beta}}$

where we changed the integration variable to
$\varphi\equiv{\varphi_{s}\over\sqrt{\tilde{\beta}}}+{\pi-{\tilde{t}}\over\tilde{\beta}}$
in the second step. Due to the large $t/\beta$, or
${\tilde{t}}/{\tilde{\beta}}$, (118) differs from (117) by an additional
$e^{{\tilde{t}}^{2}\over{\tilde{\beta}}}$.

####  6.3.2 Very low temperature

When $\tilde{\beta}=\beta\lambda^{3/2}\gg 1$, the angles $\varphi_{1}$ and
$\varphi_{2}$ are localized to a range much smaller than $1$. This allows to
expand the $\sinh$ functions in (112) to obtain

$I(\beta,t,q)={\lambda^{6}\over
2}e^{2\beta\over\sqrt{\lambda}}\int_{0}^{\infty}d\varphi_{1}\int_{0}^{\infty}d\varphi_{2}\,e^{\varphi_{1}^{2}(-{\tilde{\beta}\over
2}+i{\tilde{t}})+\varphi_{2}^{2}(-{\tilde{\beta}\over
2}-i{\tilde{t}})}\varphi_{1}^{2}\varphi_{2}^{2}={\pi\lambda^{6}e^{2\beta\over\sqrt{\lambda}}\over
4(\tilde{\beta}^{2}+4\tilde{t}^{2})^{3\over 2}}$ (119)

where we traded the upper limit with $\infty$. This may have the following
interpretation. This quantity equals

$\int
dE_{1}dE_{2}\rho(E_{1})\rho(E_{2})e^{-\beta_{1}E_{1}-\beta_{2}E_{2}}E\bigg{(}|\langle
E_{2}|M|E_{1}\rangle|^{2}\bigg{)}$ (120)

where $E()$ is the statistical average and $\beta_{1},\beta_{2}$ are related
to $\beta,t$. This means that

$E\bigg{(}|\langle
E_{2}|M|E_{1}\rangle|^{2}\bigg{)}\sim\varphi_{1}^{2}\varphi_{2}^{2}\sim
E_{1}E_{2}$ (121)

where $E_{i}$ measures the energy of the state above the ground state. We can
interpret this as if the operator $M$ acts as an underlying gaussian random
matrix which couples to low energy states with form factors $\varphi^{2}$,
i.e. consider a set of random vectors

$|v_{\alpha}\rangle=\sum_{i}\sqrt{E_{i}}c_{i,\alpha}|E_{i}\rangle,$ (122)

where the sum is up to some energy higher than the scale set by the very low
temperature, and $c_{i,\alpha}$ are independent complex Gaussian random
variables with mean zero and unit standard deviation. Take $M$ to be a random
Gaussian Hermitian matrix in terms of these variables

$M=\sum_{\alpha,\beta}|v_{\alpha}\rangle{\hat{M}}_{\alpha,\beta}\langle
v_{\beta}|$ (123)

where and ${\hat{M}}$ are independent complex Gaussian variables with mean
zero and standard deviation 1. In this case (121) is satisfied.

###  6.4  The final correlator

The evaluation of the exact 2-pt function (109) requires to compute the action
of the operator ${\cal D}_{m}(t,\beta)$ in (107) on $I(\beta,t,q)$ and
interpret the result. It is easy to read the results without actually having
to worry about the details of ${\cal D}_{m}(t,\beta)$.

  * •

The low temperature regime corresponds to the conformal regime, when the
fluctuations of the pseudo-Goldstone modes are still small. Assigning the
Hamiltonian $H$ the conformal dimension 1, one would expect an operator made
of $m$ spin operators to have dimension $m$.

This is exactly what happens in our formulas. For $m=1$, the operator ${\cal
D}_{1}(t,\beta)$ reduces to the identity. Hence, our result (117) is the
correlator for an operator of dimension 1 i.e., $\sim{1\over\cosh^{2}({\pi
t\over\beta})}$.

For $m>1$, there exists operator mixing, but we can extract the operator
content from the correlator as follows. To isolate the conformal dimensions of
the participating operators, first insert the operators on the same side, or
equivalently take $t=i\beta/2+t^{\prime}$. This turns the cosh into a sinh.
Second, take the limit $t^{\prime}\ll\beta$. In this case the leading
contribution161616This can be seen from (107). More precisely $\partial_{t}$
appears as $\lambda\partial_{t}^{2}$ and $\partial_{\beta}$ appears as
$\lambda^{3}\partial_{\beta}^{2}$. Acting with them on a function of the form
${\beta^{A}\over t^{B}}$ we obtain an expression
$({\lambda^{3}\over\beta^{2}})^{n_{\beta}}({\lambda\over
t^{2}})^{n_{t}}{\beta^{A}\over t^{B}}$. For a fixed power of $t$, an addition
derivative with respect to $\beta$ adds a power of
${\lambda^{3}\over\beta^{2}}\ll 1$.  in ${\cal D}_{m}(t,\beta)$ acts on it
with $\partial_{t}^{2m-2}$ turning the correlator into $1/t^{2m}$ which is the
2-pt function for an operator of dimension m.

  * •

For the very low temperature/long time regime we can compare (119) with
equation (67) in Bagrets:2017pwq . Although their discussion is for SYK model
with quartic interactions, it is within the Liouville description of the
Schwarzian action. Since our spin glass model reproduces the latter in this
very low temperature regime, both results should be similar. The finite
temperature 2-pt function of a pair of Majorana fermions in the SYK model at
long times/low temperatures (in the conventions used in Bagrets:2017pwq )
equals

$G(\tau)\sim-\frac{M^{2}\beta^{1/2}}{\sqrt{J}}\,\frac{\text{sgn}(\tau)}{\tau^{3/2}(\beta-\tau)^{3/2}}\,,\quad\quad\tau\gg
M\equiv\frac{N\log N}{64\sqrt{\pi}\,J}$ (124)

For a 2-pt function of higher dimension operators, the time dependence (at
long time and low temperature) remains with the same power, except that the
coefficient of power of $M$ in front of the expression increases.

To match with (119), one needs to work with Lorentzian time, $\tau=it$ and to
shift the time imaginary axis by $t\to t-i\frac{\beta}{2}$. Altogether,

$\tau^{3/2}(\beta-\tau)^{3/2}\to\left(\frac{\beta^{2}}{4}-\tau^{2}\right)^{3/2}\to\left(\frac{\beta^{2}}{4}+t^{2}\right)^{3/2}\,.$

where we analytically continued back to lorentzian time in the last step.

This is to be compared with our expression following from (119)

$G^{(m)}(t)=-{2\over\pi}(q^{2m};q)_{\infty}(q;q)^{2}_{\infty}{\cal
D}_{m}(t,\beta)\frac{\lambda^{3\over
2}\,e^{2\beta\over\sqrt{\lambda}}}{4\left(\frac{\beta^{2}}{4}+t^{2}\right)^{3/2}}\,.$
(125)

in the limit ${\beta^{2}\over 4}+t^{2}\to\infty$. For $m=1$, the $t$
dependence agrees. For $m>1$, the leading long time behaviour also agrees, and
arises from the terms in ${\cal D}_{m}(t,\beta)$ which either have no
derivatives, or have the $\partial_{\beta}^{2}$ terms acting on the
$e^{2\beta/\sqrt{\lambda}}$. Note, however, that these are not strictly
reliable results as this expression is multiplied by $\lambda^{2(m-1)}$, and
we dropped terms of similar order throughout our discussion. However, within
the terms that we kept, the leading very low temperature/very long time
expressions agrees with that of Bagrets:2017pwq .

### Acknowledgements

We would like to thank O. Aharony, A. Altland, D. Bagrets, M. Isachenkov, N.
Itzhaki, D. Kutasov, V. Narovlansky, M. Rozali, S. Shenker and G. Torrents for
useful discussions. The work of MB is supported by an ISF center of excellence
grant (1989/14). The work of JS is supported by the Science and Technology
Facilities Council (STFC) [grant number ST/L000458/1]. PN is grateful for
support provided by International Centre for Theorectical Sciences, India
where part of this work was done. MB holds the Charles and David Wolfson
Professorial chair of Theoretical Physics.

##  Appendix A q-Pochammer symbols and Jacobi $\vartheta$ functions

The q-Pochammer symbol is defined as

$(a;q)_{n}\equiv\prod_{k=0}^{n-1}(1-a\,q^{k})\,.$ (126)

It allows an extension to an infinite product

$(a;q)_{\infty}\equiv\prod_{k=0}^{\infty}(1-a\,q^{k})\,.$ (127)

The Jacobi $\vartheta_{1}$ function is defined as (see equation (8.A.2) in
GSW-2 )

$\vartheta_{1}(\theta|\tau)\equiv 2\bar{q}^{1\over
4}\sin\pi\theta\prod_{m=1}^{\infty}[(1-\bar{q}^{2m})(1-2\cos(2\pi\theta)\bar{q}^{2m}+\bar{q}^{4m})]\,,\quad\text{with}\quad\bar{q}=e^{i\pi\tau}$
(128)

It is convenient for our discussion in section 4 to write products of
q-Pochammer symbols in terms of the Jacobi $\vartheta_{1}$ function. To do
this, note that with $q=e^{-\lambda}$,

$\begin{split}\vartheta_{1}({\theta\over\pi}|{i\lambda\over 2\pi})&=2q^{1\over
8}\ \sin\theta\
(q;q)_{\infty}(e^{2i\theta}q;q)_{\infty}(e^{-2i\theta}q;q)_{\infty}\\\
&={q^{1\over 8}\over
2\sin\theta}(q;q)_{\infty}(e^{2i\theta};q)_{\infty}(e^{-2i\theta};q)_{\infty}\end{split}$
(129)

##### Modular Transformations of Jacobi Theta functions.

$\vartheta$ functions obey modular transformation properties. The one relevant
for our purposes is (see 8.A.20 in GSW-2 )

$\vartheta_{1}(z|\tau)=\vartheta_{1}(-{z\over\tau}|-{1\over\tau})\frac{1}{\eta\,(\tau)^{1\over
2}}e^{-i\pi z^{2}/\tau}\,,\quad\text{with}\quad\eta=e^{i\pi/4}\,.$ (130)

##  Appendix B Some properties of Continuous q-hermite polynomials

Continuous q-hermite polynomials are defined as

$H_{n}(x|q)\equiv\sum_{k=0}^{n}{(q;q)_{n}\over(q;q)_{k}(q;q)_{n-k}}e^{i(n-k)\theta}\,,\hskip
56.9055ptx=\cos\theta$ (131)

They can equivalently be defined using the generating function171717
$(a_{1},a_{2},\dots;q)_{\infty}\equiv(a_{1};q)_{\infty}(a_{2};q)_{\infty}\dots$.

$\sum_{n=0}^{\infty}H_{n}(x|q){t^{n}\over(q;q)_{n}}={1\over(te^{i\theta},te^{-i\theta};q)_{\infty}}$
(132)

$H_{n}(x|q)$ turns out to be a polynomial in $x,q$. We list its relevant
properties for our work below (see gasper2004basic  for example):

  * •

$H_{n}(x|q)$ satisfies a recursion relation (wikipedia)

$2xH_{n}(x|q)=H_{n+1}(x|q)+(1-q^{n})H_{n-1}(x|q)\hskip
28.45274ptH_{-1}(x|q)=0,H_{0}(x|q)=1,$ (133)

  * •

$H_{n}(x|q)$ satisfies x-orthogonality of the form

$\int_{0}^{\pi}H_{m}(\cos\theta|q)H_{n}(\cos\theta|q)|(e^{2i\theta};q)_{\infty}|^{2}d\theta=2\pi\
{(q;q)_{n}\over(q;q)_{\infty}}\delta_{mn}$ (134)

  * •

$H_{n}(x|q)$ satisfies n-orthogonality of the form

$\sum_{n=0}^{\infty}H_{n}(x|q)H_{n}(y|q){t^{n}\over(q;q)_{n}}={(t^{2};q)_{\infty}\over(te^{i(\theta+\phi)},te^{i(\theta-\phi)},te^{-i(\theta-\phi)},te^{-i(\theta+\phi)};q)_{\infty}}$
(135)

To normalize the eigenfunctions of $\hat{T}$ in section 3.2.2 (see (53)), we
need the $t\to 1$ limit of the above identity. In this limit, the right hand
side above can be expanded as

$\displaystyle\frac{(t^{2};q)_{\infty}}{|(te^{i(\theta+\phi)};q)_{\infty}|^{2}|(te^{i(\theta-\phi)};q)_{\infty}|^{2}}$
$\displaystyle=\frac{(1-t^{2})}{|1-te^{i(\theta+\phi)}|^{2}|1-te^{i(\theta-\phi)}|^{2}}$
(136)
$\displaystyle\cdot\frac{(qt^{2};q)_{\infty}}{|(qte^{i(\theta+\phi)};q)_{\infty}|^{2}|(qte^{i(\theta-\phi)};q)_{\infty}|^{2}}$

Since this expression vanishes for $t=1$ at generic values of $\theta,\phi$,
but blows up for $\theta=\pm\phi$, it is proportional to
$\delta(\theta\pm\phi)$. To determine the strength of the $\delta$ function,
one notes that if $t\equiv 1-\epsilon$, in the $\epsilon\to 0,\theta\to\phi$
limit the expression above becomes181818We need to use
$\delta(t)=\lim_{\epsilon\to
0}\frac{1}{\pi}\frac{\epsilon}{\epsilon^{2}+t^{2}}$ and the identity
$(q\,e^{2i\theta};q)_{\infty}=\frac{(e^{2i\theta};q)_{\infty}}{1-e^{2i\theta}}$

$\frac{2\epsilon}{(\epsilon^{2}+(\theta-\phi)^{2})}\frac{(q,q)_{\infty}}{|(e^{2i\theta};q)_{\infty}|^{2}|(q;q)_{\infty}|^{2}}=2\pi\delta(\theta-\phi)\frac{1}{|(e^{2i\theta};q)_{\infty}|^{2}|(q;q)_{\infty}|}\,,$
(137)

Hence, the $t=1$ limit of the formulae (135) reads

$\sum_{n}{H_{n}(x|q)H_{n}(y|q)\over(q;q)_{n}}=\frac{2\pi\left(\delta(\theta-\phi)+\delta(\theta+\phi)\right)}{|(e^{2i\theta};q)_{\infty}|^{2}(q;q)_{\infty}}$
(138)

Using these identities, one can fix the normalization constant in (53). Define

$\hat{\psi}_{l}(x|q)\equiv\sqrt{(q;q)_{\infty}}|(e^{2i\theta};q)_{\infty}|\ \
\frac{H_{l}(x|q)}{\sqrt{2\pi(q;q)_{l}}}$ (139)

The latter satisfies both the unit normalized n-orthogonality and
$x$-orthogonality relation191919The $n$-orthogonality relation also has a
$\delta(\theta+\phi)$ term, which we will neglect below since they give
negligible contribution in $\int d\theta d\phi$ compared to
$\delta(\theta-\phi)$.

$\displaystyle\sum_{n=0}^{\infty}\hat{\psi}_{n}(\cos\theta|q)\hat{\psi}_{n}(\cos\phi|q)$
$\displaystyle=$ $\displaystyle\delta(\theta-\phi)$ (140)
$\displaystyle\int_{0}^{\pi}\hat{\psi}_{m}(\cos\theta|q)\hat{\psi}_{n}(\cos\theta|q)d\theta$
$\displaystyle=$ $\displaystyle\delta_{mn}$ (141)

Finally, note that (139) can also be rewritten as

$\psi_{l}(x|q)=\psi_{0}(x|q)\frac{H_{l}(x|q)}{\sqrt{(q;q)_{l}}},\hskip
56.9055pt\psi_{0}(x|q)=\sqrt{(q;q)_{\infty}\over
2\pi}|(e^{2i\theta};q)_{\infty}|$ (142)

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