††institutetext: Department of Physics, Shinshu University,
3-1-1 Asahi, Matsumoto 390-8621, Japan

# Quenched free energy from spacetime D-branes

Kazumi Okuyama<EMAIL_ADDRESS>
###### Abstract

We propose a useful integral representation of the quenched free energy which
is applicable to any random systems. Our formula involves the generating
function of multi-boundary correlators, which can be interpreted on the bulk
gravity side as spacetime D-branes introduced by Marolf and Maxfield in
[arXiv:2002.08950]. As an example, we apply our formalism to the Airy limit of
the random matrix model and compute its quenched free energy under certain
approximations of the generating function of correlators. It turns out that
the resulting quenched free energy is a monotonically decreasing function of
the temperature, as expected.

##  1 Introduction

Recent works suggest that the gravitational path integral generically includes
the contribution of Euclidean wormholes and the result should be interpreted
as a certain ensemble average, at least for lower dimensional gravities in
spacetime dimensions $d\leq 3$ (see e.g. Saad:2018bqo ; Saad:2019lba ;
Marolf:2020xie ; Cotler:2020ugk ). The most well-known example is the
holographic duality between Sachdev-Ye-Kitaev (SYK) model Sachdev ;
kitaev2015simple  and Jackiw-Teitelboim (JT) gravity in two dimensions
Jackiw:1984je ; Teitelboim:1983ux . In a remarkable paper Saad:2019lba , it
was shown that the path integral of JT gravity is equivalent to a doubled
scaled random matrix model and the genus expansion of this matrix model
describes the splitting/joining of baby universes.

In a system with random Hamiltonian, one can define two kinds of free energy:
the quenched free energy $\langle\log Z\rangle$ and the annealed free energy
$\log\langle Z\rangle$, where $Z$ is the partition function with the inverse
temperature $\beta=1/T$

$\displaystyle Z\equiv Z(\beta)=\operatorname{Tr}e^{-\beta H},$ (1)

and the bracket $\langle\cdot\rangle$ denotes the ensemble average. In a
random system, thermodynamic quantities like the free energy $F$ and the
entropy $S$ should be defined via the quenched free energy 111In this paper,
we will loosely use the name “quenched free energy” for both $\langle\log
Z\rangle$ and $F=-T\langle\log Z\rangle$ depending on the context. We hope
this will not cause a confusion to the readers.

$\displaystyle F=-T\langle\log Z\rangle,\quad S=-\frac{\partial F}{\partial
T},$ (2)

which are expected to satisfy various thermodynamic inequalities. In
particular, the positivity of entropy $S\geq 0$ requires that the quenched
free energy is a monotonically decreasing function of temperature, while the
annealed free energy $F_{\text{ann}}=-T\log\langle Z\rangle$ is not
necessarily a monotonic function of temperature.

In a recent paper Engelhardt:2020qpv , the quenched free energy in JT gravity
is studied by the replica method

$\displaystyle\langle\log Z\rangle=\lim_{n\to 0}\frac{\langle
Z^{n}\rangle-1}{n},$ (3)

and it is argued that the wormholes connecting different replicas, so-called
replica wormholes, play an important role (see also Johnson:2020mwi ;
Alishahiha:2020jko ).222In a different context of the computation of von
Neumann entropy of Hawking radiation, replica wormholes also play an essential
role to recover the unitary Page curve Penington:2019kki ; Almheiri:2019qdq .
See also Almheiri:2020cfm  for a nice review on this subject. However, the
replica computation in Engelhardt:2020qpv  does not give rise to a well-
behaved monotonic free energy at low temperature due to the limitation of the
approximation used in Engelhardt:2020qpv . More importantly, as emphasized in
Engelhardt:2020qpv  there is a fundamental problem in the replica computation
of the quenched free energy since the analytic continuation of $\langle
Z^{n}\rangle$ from a positive integer $n$ to $n=0$ is highly non-unique.

The replica method (3) is just a convenient trick to compute the quenched free
energy and one can in principle compute it by directly taking the average of
the quantity $\log Z$. This is demonstrated in our previous paper
Okuyama:2020mhl  in a simple example of the Gaussian matrix model. We computed
the average $\langle\log Z\rangle$ directly by the matrix integral at finite
$N$ and found that the resulting quenched free energy is a monotonic function
of the temperature, as expected. It should be emphasized that if we take the
average $\langle\log Z\rangle$ directly without using the replica method (3),
the result is uniquely determined and there is no ambiguity in the
computation. However, the direct computation in Okuyama:2020mhl  relies on the
property of the finite $N$ matrix model and it is not straightforward to
generalize it to the double scaled matrix model of JT gravity. Thus, it is
desirable to find a general technique to compute the quenched free energy
without resorting to the replica trick (3).

In this paper, we propose a useful integral representation of the quenched
free energy

$\displaystyle\langle\log Z\rangle=\log\langle
Z\rangle-\int_{0}^{\infty}\frac{dx}{x}\Bigl{[}\big{\langle}e^{-Zx}\big{\rangle}-e^{-\langle
Z\rangle x}\Bigr{]},$ (4)

which can be applied to any random system. As far as we know, this expression
of quenched free energy (4) has not appeared in the literature before.333A
similar, but slightly different expression of $\langle\log Z\rangle$ is
discussed in svaiter2016distributional . Our formula (4) follows from the
simple identity (7) and the derivation of (4) does not rely on any
approximation. In section 2.2, we will argue that our formula (4) can be
obtained from the replica method (3) under a certain prescription of the
analytic continuation of $\langle Z^{n}\rangle$.

Our formula (4) might have an interesting bulk gravity interpretation. As
discussed in Marolf:2020xie , the insertion of $Z$ into the expectation value
$\langle\cdot\rangle$ corresponds to adding an extra boundary to the spacetime
in the bulk gravity picture. The insertion of $e^{-Zx}$ creates many
boundaries in spacetime and it is identified in Marolf:2020xie  as the
“spacetime D-brane” (or SD-brane for short). Based on this picture, our
formula (4) suggests that the bulk gravity dual of the quenched free energy
involves a superposition of SD-branes. Here, by “superposition” we mean that
we have to integrate over the parameter $x$ in the SD-brane operator
$e^{-Zx}$.

Introducing the generating function $\mathcal{Z}(x)$ of the connected
correlators $\langle Z^{n}\rangle_{c}$

$\displaystyle\mathcal{Z}(x)=-\sum_{n=1}^{\infty}\frac{(-x)^{n}}{n!}\langle
Z^{n}\rangle_{c},$ (5)

(4) is written as

$\displaystyle\langle\log Z\rangle=\log\langle
Z\rangle-\int_{0}^{\infty}\frac{dx}{x}\Bigl{[}e^{-\mathcal{Z}(x)}-e^{-\langle
Z\rangle x}\Bigr{]},$ (6)

where we used the relation $\langle e^{-Zx}\rangle=e^{-\mathcal{Z}(x)}$. Thus
the problem boils down to the computation of the generating function
$\mathcal{Z}(x)$ in (5). As a simple application of our formalism (6), we
consider the quenched free energy in the Airy limit of Gaussian matrix
model.444See e.g. Ginsparg:1993is  for a review of the Airy limit of matrix
model. We find that at genus-zero the generating function $\mathcal{Z}(x)$ can
be written in a closed form in terms of the Lambert function. If we plug this
genus-zero approximation of $\mathcal{Z}(x)$ into (6), we find a monotonically
decreasing behavior of the quenched free energy as a function of the
temperature. However, if we include higher genus corrections to
$\mathcal{Z}(x)$ in (6) it leads to an unphysical diverging behavior of free
energy as $T\to 0$. This simply means that the genus expansion is not a good
approximation at low temperature, and hence we will consider a different
approximation of $\mathcal{Z}(x)$ in (27) which is applicable at low
temperature. Then, we find a monotonic behavior of the quenched free energy in
this low temperature approximation (27).

This paper is organized as follows. In section 2, we explain the derivation of
the formula (4) for the quenched free energy. We also see that this expression
(4) can be obtained from the replica method (3). Then we briefly comment on
the possible bulk gravity interpretation of our formula (4). In section 3, we
apply our formalism (6) to the Airy limit of Gaussian matrix model. We find
that the genus expansion of $\mathcal{Z}(x)$ in the Airy limit is written in
terms of the Lambert function. The appearance of the Lambert function can be
understood from the string equation in a shifted background due to the
insertion of the operator $e^{-Zx}$. We find a monotonic behavior of the
quenched free energy in a certain approximation of $\mathcal{Z}(x)$. Finally,
we conclude in section 4 with some discussion for the future problems. In
appendix A, we compute the high temperature expansion of the quenched free
energy in the Airy limit.

##  2 General formula for the quenched free energy

###  2.1 Direct derivation of (4)

Let us prove our general formula (4) for the quenched free energy. (4) is
based on the following simple relation

$\displaystyle-\int_{0}^{\infty}\frac{dx}{x}(e^{-ax}-e^{-bx})=\log\frac{a}{b}.$
(7)

Note that, if we take the first term of (7) only, then the integral is
logarithmically divergent

$\displaystyle-\int_{\varepsilon}^{\infty}\frac{dx}{x}e^{-ax}=\log(a\varepsilon)+\gamma+\mathcal{O}(\varepsilon),$
(8)

where $\varepsilon$ is a small regularization parameter and $\gamma$ is the
Euler-Mascheroni constant. However, if we compute the difference of two such
integrals as in (7) the divergence from $x=0$ is canceled and we get the
finite result (7). Setting $a=Z$ and $b=\langle Z\rangle$ in (7) we obtain

$\displaystyle-\int_{0}^{\infty}\frac{dx}{x}\Big{(}e^{-Zx}-e^{-\langle
Z\rangle x}\Big{)}=\log\frac{Z}{\langle Z\rangle}.$ (9)

Finally, taking the expectation value of the both sides of (9) and using the
relation $\langle 1\rangle=1$, we find our desired result (4). We emphasize
that our formula (4) is an exact relation and we did not use any approximation
in deriving (4).

Let us consider an alternative derivation of (4) by expanding the quenched
free energy around the annealed free energy.

$\displaystyle\langle\log Z\rangle$
$\displaystyle=\big{\langle}\log\big{(}\langle Z\rangle+Z-\langle
Z\rangle\big{)}\big{\rangle}$ (10) $\displaystyle=\log\langle
Z\rangle+\Biggl{\langle}\log\Biggl{(}1+\frac{Z-\langle Z\rangle}{\langle
Z\rangle}\Biggr{)}\Biggr{\rangle}$ $\displaystyle=\log\langle
Z\rangle-\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}\frac{\big{\langle}\big{(}Z-\langle
Z\rangle\big{)}^{k}\big{\rangle}}{\langle Z\rangle^{k}}.$

Then using the relation

$\displaystyle\int_{0}^{\infty}\frac{dy}{y}e^{-y}y^{k}=(k-1)!,\qquad(k\geq
1),$ (11)

we rewrite the summation in (10) as

$\displaystyle-\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}\frac{\big{\langle}\big{(}Z-\langle
Z\rangle\big{)}^{k}\big{\rangle}}{\langle Z\rangle^{k}}$
$\displaystyle=-\int_{0}^{\infty}\frac{dy}{y}e^{-y}\sum_{k=1}^{\infty}\frac{(-y)^{k}}{k!}\frac{\big{\langle}\big{(}Z-\langle
Z\rangle\big{)}^{k}\big{\rangle}}{\langle Z\rangle^{k}}$ (12)
$\displaystyle=-\int_{0}^{\infty}\frac{dy}{y}e^{-y}\left[\Biggl{\langle}\exp\Biggl{(}-y\frac{Z-\langle
Z\rangle}{\langle Z\rangle}\Biggr{)}\Biggr{\rangle}-1\right]$
$\displaystyle=-\int_{0}^{\infty}\frac{dy}{y}\left[\Biggl{\langle}\exp\Biggl{(}-y\frac{Z}{\langle
Z\rangle}\Biggr{)}\Biggr{\rangle}-e^{-y}\right]$

Finally, by rescaling the integration variable $y=\langle Z\rangle x$ we
arrive at our formula (4).

###  2.2 Relation to the replica method

Let us consider a derivation of our formula (4) from the replica method (3).
In the replica method, we need to analytically continue the $n$-point function
$\langle Z^{n}\rangle$ from a positive integer $n$ to $n=0$. As pointed out in
Engelhardt:2020qpv , the analytic continuation of $\langle Z^{n}\rangle$ is
not unique and there might be some ambiguity in the computation of the right
hand side of (3). It turns out that there is a natural definition of the
analytic continuation of $\langle Z^{n}\rangle$ which leads to our formula
(4).

In general, the correlator $\langle Z^{n}\rangle$ is expanded as a combination
of the connected correlators $\langle Z^{k}\rangle_{c}$. If we order this
expansion by the number of connected components, the first term is the totally
disconnected part $\langle Z\rangle^{n}$. The next term is $\langle
Z\rangle^{n-2}\langle Z^{2}\rangle_{c}$ and the coefficient of this term is
$\binom{n}{2}=\frac{1}{2}n(n-1)$ which is the number of ways to choose two
boundaries out of $n$ boundaries. Then $\langle Z^{n}\rangle$ is expanded as

$\displaystyle\langle Z^{n}\rangle$ $\displaystyle=\langle
Z\rangle^{n}+\frac{1}{2}n(n-1)\langle Z\rangle^{n-2}\langle
Z^{2}\rangle_{c}+\cdots$ (13) $\displaystyle=\langle
Z\rangle^{n}\left[1+\frac{1}{2}n(n-1)\frac{\langle Z^{2}\rangle_{c}}{\langle
Z\rangle^{2}}+\cdots\right].$

Taking the $n\to 0$ limit in (3), we obtain

$\displaystyle\langle\log Z\rangle=\log\langle
Z\rangle-\frac{1}{2}\frac{\langle Z^{2}\rangle_{c}}{\langle
Z\rangle^{2}}+\cdots.$ (14)

In this computation, up to the overall factor $\langle Z\rangle^{n}$ the
$n$-dependence in (13) is essentially a polynomial in $n$ and hence the
analytic continuation can be defined unambiguously.

To see the general structure, it is convenient to define

$\displaystyle w_{n}=\frac{\langle Z^{n}\rangle_{c}}{\langle Z\rangle^{n}}.$
(15)

Then the expansion (13) is generalized as (see e.g. Okuyama:2019xvg )

$\displaystyle\langle Z^{n}\rangle=\langle Z\rangle^{n}\sum_{j_{i}\geq
0~{}(i=2,3,\cdots)}\frac{n!}{(n-\sum_{l\geq 2}lj_{l})!}\prod_{k\geq
2}\frac{1}{j_{k}!}\left(\frac{w_{k}}{k!}\right)^{j_{k}}.$ (16)

Now using the relation

$\displaystyle\lim_{n\to 0}\frac{1}{n}\frac{n!}{(n-m)!}=(-1)^{m-1}(m-1)!,$
(17)

the quenched free energy becomes

$\displaystyle\langle\log Z\rangle=\log\langle
Z\rangle+\sum_{\begin{subarray}{c}j_{i}\geq 0~{}(i=2,3\cdots)\\\
(j_{2},j_{3},\cdots)\neq(0,0,\cdots)\end{subarray}}(-1)^{\sum_{l\geq
2}lj_{l}-1}\Bigl{(}\sum_{l\geq 2}lj_{l}-1\Bigr{)}!\prod_{k\geq
2}\frac{1}{j_{k}!}\left(\frac{w_{k}}{k!}\right)^{j_{k}}.$ (18)

Again using the trick (11) this is rewritten as

$\displaystyle\langle\log Z\rangle$ $\displaystyle=\log\langle
Z\rangle-\sum_{\begin{subarray}{c}j_{i}\geq 0~{}(i=2,3,\cdots)\\\
(j_{2},j_{3},\cdots)\neq(0,0,\cdots)\end{subarray}}\int_{0}^{\infty}\frac{dy}{y}e^{-y}(-y)^{\sum_{l\geq
2}lj_{l}}\prod_{k\geq 2}\frac{1}{j_{k}!}\left(\frac{w_{k}}{k!}\right)^{j_{k}}$
(19) $\displaystyle=\log\langle
Z\rangle-\int_{0}^{\infty}\frac{dy}{y}e^{-y}\left[\exp\Biggl{(}\sum_{k=2}^{\infty}(-y)^{k}\frac{w_{k}}{k!}\Biggr{)}-1\right]$
$\displaystyle=\log\langle
Z\rangle-\int_{0}^{\infty}\frac{dx}{x}\Biggl{[}\exp\Biggl{(}\sum_{k=1}^{\infty}\frac{(-x)^{k}}{k!}\langle
Z^{k}\rangle_{c}\Biggr{)}-e^{-\langle Z\rangle x}\Biggr{]},$

where in the last step we changed the integration variable $y=\langle Z\rangle
x$. One can see that the first term of the integrand in (19) is
$e^{-\mathcal{Z}(x)}$ defined in (5), and hence (19) agrees with our formula
(6).

To summarize, we have derived our formula (4) from the replica method by using
the prescription (17) for the analytic continuation. For instance, the example
in (14) of this computation corresponds to the $m=2$ case of (17)

$\displaystyle\lim_{n\to 0}\frac{1}{n}\frac{n!}{(n-2)!}=\lim_{n\to
0}\frac{1}{n}n(n-1)=-1.$ (20)

At fixed $m$, the factor $\frac{n!}{(n-m)!}$ in (17) is a polynomial in $n$

$\displaystyle\frac{n!}{(n-m)!}=n(n-1)\cdots(n-m+1),$ (21)

and hence we believe that our prescription (17) of the analytic continuation
is a natural choice.

###  2.3 Possible interpretation of (4)

Our formula (4) is very suggestive for a possible interpretation of the bulk
spacetime picture of the quenched free energy. In Marolf:2020xie , it is
discussed that the operator $e^{-Zx}$ corresponds to the so-called “spacetime
D-brane” (SD-brane). Our formula (4) suggests that the bulk spacetime picture
of quenched free energy involves a superposition of SD-branes. As mentioned in
section 1, in order to obtain the quenched free energy, we have to integrate
over the parameter $x$ in the SD-brane operator $e^{-Zx}$.

Recently, in Penington:2019kki ; Almheiri:2019qdq  the von Neumann entropy of
Hawking radiation is computed by the replica method and it is argued that the
so-called replica wormholes are essential for recovering the unitary Page
curve. In the saddle point approximation of gravitational path integral, the
dominant contribution comes from either maximally connected or maximally
disconnected spacetimes and these contributions exchange dominance around the
Page time. On the other hand, in our formula (4) no such saddle point
approximation is made. In our derivation of (4) from the replica method, all
terms in the decomposition of $\langle Z^{n}\rangle$ in (16) contribute to the
quenched free energy and they add up to the generating function $\langle
e^{-Zx}\rangle$. The first term $\log\langle Z\rangle$ in (4) comes from the
maximally disconnected part $\langle Z\rangle^{n}$, while the remaining term
in (4) is a contribution of SD-brane $e^{-Zx}$ which does not have a simple
interpretation as the contribution of maximally connected part. As we will see
in section 3.3, the insertion of the operator $e^{-Zx}$ can be interpreted as
a shift of the background couplings, or a deformation of the matrix model
potential.

It is argued in Engelhardt:2020qpv  that the contribution of the so-called
replica wormholes should be included in the replica computation of the
quenched free energy. The second term of (4) represents the deviation from the
annealed free energy, which might be interpreted as the contribution of
replica wormholes. It is interesting that even after taking the limit $n\to 0$
(3), infinite number of boundaries contributes to $\langle
e^{-Zx}\rangle=e^{-\mathcal{Z}(x)}$ in (5). In other words, we need to include
a coherent superposition $\mathcal{Z}(x)$ of the infinite number of boundaries
for the computation of quenched free energy.

In general, the large $N$ limit serves as a semi-classical approximation in
the random matrix model. One might think that we can see an exchange of
dominance between disconnected and connected contributions in the large $N$
limit. However, if we naively take the large $N$ limit at fixed temperature,
the disconnected correlator always dominates over the connected correlators.
Around the temperature $T\sim N^{-2/3}$, the disconnected and connected
contributions become comparable Okuyama:2019xvg , but this temperature is
pushed to zero as $N\to\infty$ which implies that the disconnected part is
always dominant at finite temperature. As discussed in Engelhardt:2020qpv ;
Johnson:2020mwi , to focus on this temperature scale $T\sim N^{-2/3}$ one can
take a certain double scaling limit, called the Airy limit of random matrix
model, which we will consider in the next section.

##  3 Quenched free energy in the Airy limit

In this section we apply our general formula (4) to the Airy limit of Gaussian
random matrix model. In this case, the connected part of the $n$-point
correlator $\langle Z^{n}\rangle_{c}$ is known in the integral representation
okounkov2002generating . For instance, $\langle Z^{n}\rangle_{c}$ for
$n=1,2,3$ can be written in the closed form okounkov2002generating ;
Beccaria:2020ykg

$\displaystyle\langle Z(\beta)\rangle$
$\displaystyle=\frac{e^{\frac{\xi}{12}}}{\sqrt{4\pi\xi}},$ (22)
$\displaystyle\langle Z(\beta)^{2}\rangle_{c}$ $\displaystyle=\langle
Z(2\beta)\rangle\text{Erf}\Biggl{(}\sqrt{\frac{\xi}{2}}\Biggr{)},$
$\displaystyle\langle Z(\beta)^{3}\rangle_{c}$ $\displaystyle=\langle
Z(3\beta)\rangle\Biggl{[}1-12T\Bigl{(}\sqrt{3\xi},\frac{1}{\sqrt{3}}\Bigr{)}\Biggr{]},$

where $\text{Erf}(z)$ is the error function and $T(z,a)$ denotes the Owen’s
$T$-function defined by

$\displaystyle
T(z,a)=\frac{1}{2\pi}\int_{0}^{a}dt\frac{e^{-\frac{z^{2}}{2}(1+t^{2})}}{1+t^{2}}.$
(23)

The parameter $\xi$ in (22) is given by

$\displaystyle\xi=\hbar^{2}\beta^{3},$ (24)

where $\hbar$ is the genus-counting parameter. In the Airy limit, $\hbar$ and
$\beta$ appear in $\langle Z^{n}\rangle_{c}$ only through the combination
$\xi$.

In principle one can compute $\langle Z^{n}\rangle_{c}$ using the integral
representation of okounkov2002generating , but it is not straightforward to
find the generating function $\mathcal{Z}(x)$ of the connected correlators
since the integral in okounkov2002generating  is difficult to compute in a
closed form for general $n$. In fact, the closed form expression of $\langle
Z^{n}\rangle_{c}$ is not known for $n\geq 4$. Instead, here we will consider
the genus expansion of the generating function $\mathcal{Z}(x)$

$\displaystyle\mathcal{Z}(x)=\sum_{g=0}^{\infty}\xi^{g-1}\mathcal{Z}_{g}(x),$
(25)

where $\mathcal{Z}_{g}(x)$ is the generating function of the genus-$g$ part of
the connected correlators. It turns out that $\mathcal{Z}_{g}(x)$ is written
as a combination of the Lambert $W$-function. If we use the genus-zero
approximation $\mathcal{Z}(x)\approx\xi^{-1}\mathcal{Z}_{0}(x)$ in (6), we
find that the quenched free energy $F=-T\langle\log Z\rangle$ decreases
monotonically as a function of temperature. However, the genus expansion is
not a good approximation at low temperature since the expansion parameter
$\xi$ in (25) becomes large when $T\ll\hbar^{2/3}$.

At low temperature we can use the result of Okuyama:2019xvg  that the
connected part of the $n$-point correlator is approximated by the one-point
function555Strictly speaking, this result (26) is shown in the Gaussian matrix
model Okuyama:2019xvg , but we expect that (26) holds in the Airy limit as
well.

$\displaystyle\langle Z(\beta)^{n}\rangle_{c}\approx\langle
Z(n\beta)\rangle,\quad(T\lesssim\hbar^{2/3}).$ (26)

One can see this behavior explicitly from the exact result of $\langle
Z(\beta)^{n}\rangle_{c}$ for $n=2,3$ in (22). We emphasize that (26) is the
general property of the connected correlator for any $n$. Then
$\mathcal{Z}(x)$ at low temperature is approximated as

$\displaystyle\mathcal{Z}(x)\approx-\sum_{n=1}^{\infty}\frac{(-x)^{n}}{n!}\langle
Z(n\beta)\rangle.$ (27)

The right hand side of this equation can be computed explicitly since the
exact form of the one-point function $\langle Z(n\beta)\rangle$ is known in
the Airy limit. Plugging (27) into (6), we find the monotonic behavior of the
quenched free energy even at low temperature $T\lesssim\hbar^{2/3}$.

###  3.1 Quenched free energy at genus-zero

Let us consider the genus-zero part of the expansion of $\mathcal{Z}(x)$ in
(25). The connected correlator $\langle Z^{n}\rangle_{c}$ is obtained by acing
the “boundary creation operator” $\widehat{Z}$ to the free energy
$\mathcal{F}(\\{t_{k}\\})$ of 2d topological gravity Moore:1991ir

$\displaystyle\langle
Z^{n}\rangle_{c}=\big{(}\widehat{Z}\big{)}^{n}\mathcal{F},$ (28)

where $\widehat{Z}$ is given by

$\displaystyle\widehat{Z}=g_{s}\sqrt{\frac{\beta}{2\pi}}\sum_{k=0}^{\infty}\beta^{k}\partial_{k}$
(29)

with $\partial_{k}\equiv\frac{\partial}{\partial t_{k}}$. Then the generating
function $\mathcal{Z}(x)$ is written as

$\displaystyle\mathcal{Z}(x)=-\sum_{n=1}^{\infty}\frac{(-x)^{n}}{n!}\big{(}\widehat{Z}\big{)}^{n}\mathcal{F}=\mathcal{F}-e^{-\widehat{Z}x}\mathcal{F}.$
(30)

$g_{s}$ in (29) is the natural genus-counting parameter in the topological
gravity

$\displaystyle\mathcal{F}=\sum_{g=0}^{\infty}g_{s}^{2g-2}\mathcal{F}_{g},$
(31)

and $g_{s}$ is related to $\hbar$ in (24) by

$\displaystyle g_{s}=\sqrt{2}\hbar.$ (32)

The Airy limit corresponds to the trivial background $t_{n}=0~{}(n\geq 0)$. As
demonstrated in Zograf:2008wbe ; Okuyama:2019xbv ; Okuyama:2020ncd , in order
to compute the connected correlators (28) one can utilize the KdV equation
obeyed by the specific heat $u=g_{s}^{2}\partial_{0}^{2}\mathcal{F}$
Witten:1990hr ; Kontsevich:1992ti . In the KdV approach of the computation of
$\langle Z^{n}\rangle_{c}$, it is convenient to turn on $t_{0},t_{1}$

$\displaystyle t_{0},t_{1}\neq 0,\quad t_{n}=0~{}~{}(n\geq 2),$ (33)

and set $t_{0}=t_{1}=0$ at the end of the calculation. One can show that in
this subspace (33) $u$ is given by

$\displaystyle u=\frac{t_{0}}{1-t_{1}},$ (34)

without any higher genus corrections.

At genus-zero, the general expression of the connected correlator (28) reduces
to Moore:1991ir

$\displaystyle\langle
Z^{n}\rangle_{c}^{g=0}=\left(\frac{\beta}{2\pi}\right)^{\frac{n}{2}}(g_{s}\partial_{0})^{n-2}\frac{e^{n\beta
u_{0}}}{n\beta},$ (35)

where $u_{0}$ is determined by the genus-zero string equation Itzykson:1992ya
666The all-genus string equation is given by  $\displaystyle
u=\sum_{k=0}^{\infty}t_{k}\mathcal{R}_{k},$ (36) where
$\mathcal{R}_{k}=\frac{u^{k}}{k!}+\mathcal{O}(\partial_{0})$ is the Gelfand-
Dikii differential polynomial determined by the recursion relation
$\displaystyle(2k+1)\partial_{0}\mathcal{R}_{k+1}=\frac{\hbar^{2}}{2}\partial_{0}^{3}\mathcal{R}_{k}+2u\partial_{0}\mathcal{R}_{k}+(\partial_{0}u)\mathcal{R}_{k}.$
(37)

$\displaystyle u_{0}=\sum_{k=0}^{\infty}t_{k}\frac{u_{0}^{k}}{k!}.$ (38)

In our case (33), $u_{0}$ is equal to $u$ in (34)

$\displaystyle u_{0}=\partial_{0}^{2}\mathcal{F}_{0}=\frac{t_{0}}{1-t_{1}}.$
(39)

Plugging this $u_{0}$ into (35) the genus-zero part of the $n$-point
correlator $\langle Z^{n}\rangle_{c}^{g=0}$ becomes

$\displaystyle\langle
Z^{n}\rangle_{c}^{g=0}=\frac{(1-t_{1})^{2-n}}{2\xi}n^{n-3}\left(\frac{\xi}{\pi}\right)^{\frac{n}{2}}e^{\frac{n\beta
t_{0}}{1-t_{1}}},$ (40)

where $\xi$ is defined in (24). Then, by setting $t_{0}=t_{1}=0$ we find
$\langle Z^{n}\rangle_{c}^{g=0}$ in the Airy limit

$\displaystyle\langle
Z^{n}\rangle_{c}^{g=0}=\frac{1}{2\xi}n^{n-3}\left(\frac{\xi}{\pi}\right)^{\frac{n}{2}}.$
(41)

Now let us consider the generating function of the genus-zero correlators in
(41)

$\displaystyle\xi^{-1}\mathcal{Z}_{0}(x)=-\sum_{n=1}^{\infty}\frac{(-x)^{n}}{n!}\langle
Z^{n}\rangle_{c}^{g=0}=\frac{1}{2\xi}\sum_{n=1}^{\infty}\frac{(-1)^{n-1}n^{n-3}}{n!}\left(x\sqrt{\frac{\xi}{\pi}}\right)^{n}.$
(42)

It turns out that this summation is closely related to the Lambert function
$W(z)$

$\displaystyle W(z)=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}n^{n-1}}{n!}z^{n},$
(43)

which satisfies the relation

$\displaystyle z=W(z)e^{W(z)}.$ (44)

From (42) and (43), one can see that $\mathcal{Z}_{0}$ obeys

$\displaystyle(z\partial_{z})^{2}\mathcal{Z}_{0}=\frac{1}{2}W(z),$ (45)

where $z$ is related to $x$ by

$\displaystyle z=x\sqrt{\frac{\xi}{\pi}}.$ (46)

Using the property of the Lambert function

$\displaystyle z\partial_{z}W(z)=\frac{W(z)}{1+W(z)},$ (47)

(45) is integrated as

$\displaystyle\mathcal{Z}_{0}=\frac{1}{2}\Biggl{(}\frac{1}{6}W^{3}+\frac{3}{4}W^{2}+W\Biggr{)}.$
(48)

Here we have suppressed the argument of $W(z)$ for brevity.

Then at genus-zero the quenched free energy (6) is written as

$\displaystyle\langle\log Z\rangle^{g=0}=\log\langle
Z\rangle^{g=0}-\int_{0}^{\infty}\frac{dx}{x}\Bigl{[}e^{-\xi^{-1}\mathcal{Z}_{0}(x)}-e^{-\langle
Z\rangle^{g=0}x}\Bigr{]},$ (49)

where $\langle Z\rangle^{g=0}=\frac{1}{\sqrt{4\pi\xi}}$ is the genus-zero one-
point function. Using the fact that $z\in[0,\infty]$ is mapped to
$W\in[0,\infty]$, we can take $W$ as the integration variable in (49). From
(47), (49) is rewritten as

$\displaystyle\langle\log
Z\rangle^{g=0}=-\frac{1}{2}\log(4\pi\xi)-\int_{0}^{\infty}\frac{dW}{W}(1+W)\Bigl{[}e^{-\frac{1}{2\xi}\left(\frac{1}{6}W^{3}+\frac{3}{4}W^{2}+W\right)}-e^{-\frac{1}{2\xi}We^{W}}\Bigr{]}.$
(50)

In Fig. 1 we show the plot of rescaled free energy $F^{\prime}=\hbar^{-2/3}F$
of the Airy case at genus-zero. In other words, $F^{\prime}$ is defined by

$\displaystyle F^{\prime}=-T^{\prime}\langle\log Z\rangle,\quad
T^{\prime}=\hbar^{-2/3}T,\quad\xi=T^{\prime-3}.$ (51)

From Fig. 1 one can see that the quenched free energy at genus-zero (blue
solid curve) is a monotonic function of the temperature. As a comparison, we
have also plotted the annealed free energy at genus-zero (orange dashed curve
in Fig. 1)

$\displaystyle F_{\text{ann}}^{\prime}=-T^{\prime}\log\langle
Z\rangle^{g=0}=\frac{1}{2}T^{\prime}\log(4\pi T^{\prime-3}),$ (52)

which is not a monotonic function of the temperature.

Figure 1:  Plot of the genus-zero free energy for the Airy case. The vertical
and the horizontal axes are the rescaled free energy
$F^{\prime}=\hbar^{-2/3}F$ and the rescaled temperature
$T^{\prime}=\hbar^{-2/3}T$, respectively. The blue solid curve represents the
quenched free energy in (50) while the orange dashed curve is the annealed
free energy in (52).

###  3.2 Higher genus corrections

Next let us consider the higher genus corrections $\mathcal{Z}_{g}$ in (25).
As we will see below, $\mathcal{Z}_{g}$ can be systematically obtained from
the KdV equation. Our starting point is the KdV equation for $u$

$\displaystyle\partial_{1}u=u\partial_{0}u+\frac{\hbar^{2}}{6}\partial_{0}^{3}u.$
(53)

Introducing the potential $\phi$ by

$\displaystyle u=\partial_{0}\phi,$ (54)

(53) is integrated once as

$\displaystyle\partial_{1}\phi=\frac{1}{2}(\partial_{0}\phi)^{2}+\frac{\hbar^{2}}{6}\partial_{0}^{3}\phi.$
(55)

Note that $\phi$ is written as

$\displaystyle\phi=g_{s}^{2}\partial_{0}\mathcal{F}=2\hbar^{2}\partial_{0}\mathcal{F}.$
(56)

We can derive an equation obeyed by $\mathcal{Z}(x)$ by acting the operator
$e^{-\widehat{Z}x}$ on both sides of (55). From (30) and (56),
$e^{-\widehat{Z}x}\phi$ is written as

$\displaystyle
e^{-\widehat{Z}x}\phi=2\hbar^{2}\partial_{0}e^{-\widehat{Z}x}\mathcal{F}=2\hbar^{2}\partial_{0}(\mathcal{F}-\mathcal{Z})=\phi-2\hbar\mathcal{W},$
(57)

where we defined

$\displaystyle\mathcal{W}=\hbar\partial_{0}\mathcal{Z}.$ (58)

Acting $e^{-\widehat{Z}x}$ on both sides of (55), we find

$\displaystyle\partial_{1}(\phi-2\hbar\mathcal{W})=\frac{1}{2}(\partial_{0}\phi-2\hbar\partial_{0}\mathcal{W})^{2}+\frac{\hbar^{2}}{6}\partial_{0}^{3}(\phi-2\hbar\mathcal{W}).$
(59)

From (55) and (59), we arrive at the equation for $\mathcal{W}$ 777This
equation was originally found by Kazuhiro Sakai. The author would like to
thank him for sharing his unpublished note.

$\displaystyle\partial_{1}\mathcal{W}=u\partial_{0}\mathcal{W}-\hbar(\partial_{0}\mathcal{W})^{2}+\frac{\hbar^{2}}{6}\partial_{0}^{3}\mathcal{W}.$
(60)

Let us consider the genus expansion of $\mathcal{W}$

$\displaystyle\mathcal{W}=\beta^{-\frac{1}{2}}\sum_{g=0}^{\infty}\xi^{g-\frac{1}{2}}\mathcal{W}_{g}.$
(61)

It turns out that $\mathcal{W}_{g}$ is a rational function of $W\equiv W(z)$
with $z$ given by (46). To compute $\mathcal{W}_{g}$ recursively from (60), it
is useful to work on the subspace (33). On this space (33), one can easily
show that $z$ in (46) is generalized as (see (40))

$\displaystyle z=\frac{1}{1-t_{1}}xe^{\frac{\beta
t_{0}}{1-t_{1}}}\sqrt{\frac{\xi}{\pi}}.$ (62)

Then the derivative $\partial_{0,1}$ appearing in (60) is written in terms of
$D=z\partial_{z}$. We also note that $\hbar$ and $t_{1}$ appear only via the
combination

$\displaystyle\hat{\hbar}\equiv\frac{\hbar}{1-t_{1}}.$ (63)

With these remarks in mind, $\partial_{0}\mathcal{W}$ and
$\partial_{1}\mathcal{W}$ at $t_{0}=t_{1}=0$ are given by

$\displaystyle\partial_{0}\mathcal{W}\Big{|}_{t_{0}=t_{1}=0}$
$\displaystyle=\beta D\mathcal{W},$ (64)
$\displaystyle\partial_{1}\mathcal{W}\Big{|}_{t_{0}=t_{1}=0}$
$\displaystyle=(D+\hbar\partial_{\hbar})\mathcal{W}.$

Then after setting $t_{0}=t_{1}=0$ (60) reduces to

$\displaystyle(D+2g-1)\mathcal{W}_{g}=\frac{1}{6}D^{3}\mathcal{W}_{g-1}-\sum_{h=0}^{g}D\mathcal{W}_{h}D\mathcal{W}_{g-h}.$
(65)

From the definition of $\mathcal{W}$ (58), one can easily show that
$\mathcal{Z}_{g}$ and $\mathcal{W}_{g}$ are related by

$\displaystyle\mathcal{W}_{g}=D\mathcal{Z}_{g}.$ (66)

To solve (65) recursively, we rewrite (65) as

$\displaystyle\Bigl{[}(1+2D\mathcal{W}_{0})D+2g-1\Bigr{]}\mathcal{W}_{g}=\frac{1}{6}D^{3}\mathcal{W}_{g-1}-\sum_{h=1}^{g-1}D\mathcal{W}_{h}D\mathcal{W}_{g-h}.$
(67)

Note that, from (47) $D=z\partial_{z}$ is written as

$\displaystyle D=\frac{W}{1+W}\partial_{W},$ (68)

and from (45) $D\mathcal{W}_{0}$ is given by

$\displaystyle D\mathcal{W}_{0}=D^{2}\mathcal{Z}_{0}=\frac{1}{2}W.$ (69)

Finally we arrive at the recursion relation for $\mathcal{W}_{g}$

$\displaystyle(W\partial_{W}+2g-1)\mathcal{W}_{g}=\frac{1}{6}D^{3}\mathcal{W}_{g-1}-\sum_{h=1}^{g-1}D\mathcal{W}_{h}D\mathcal{W}_{g-h}.$
(70)

One can easily solve this relation recursively starting from
$D\mathcal{W}_{0}$ in (69). More explicitly, (70) is solved as

$\displaystyle\mathcal{Z}_{g}=\int dW(1+W)W^{-2g}\int
dWW^{2g-2}\left[\frac{1}{6}D^{3}\mathcal{W}_{g-1}-\sum_{h=1}^{g-1}D\mathcal{W}_{h}D\mathcal{W}_{g-h}\right].$
(71)

Using this formalism, we can compute $\mathcal{Z}_{g}$ up to any desired
order. For instance, the first few terms are given by

$\displaystyle\mathcal{Z}_{1}$ $\displaystyle=\frac{1}{24}\log(1+W),$ (72)
$\displaystyle\mathcal{Z}_{2}$
$\displaystyle=\frac{5W-19W^{2}+4W^{3}}{2880(1+W)^{5}},$
$\displaystyle\mathcal{Z}_{3}$
$\displaystyle=\frac{35W-1977W^{2}+12393W^{3}-15919W^{4}+3960W^{5}-16W^{6}}{725760(1+W)^{10}}.$

Using this result (72), we can study the behavior of free energy including the
higher genus corrections up to genus-$g$

$\displaystyle\langle\log Z\rangle_{[g]}=\log\langle
Z\rangle_{[g]}-\int_{0}^{\infty}\frac{dW}{W}(1+W)\left[e^{-\mathcal{Z}_{[g]}}-e^{-\langle
Z\rangle_{[g]}\sqrt{\frac{\pi}{\xi}}We^{W}}\right],$ (73)

where we defined $\langle Z\rangle_{[g]}$ and $\mathcal{Z}_{[g]}$ by

$\displaystyle\langle Z\rangle_{[g]}$
$\displaystyle=\frac{1}{\sqrt{4\pi\xi}}\sum_{n=0}^{g}\frac{\xi^{n}}{12^{n}n!},$
(74) $\displaystyle\mathcal{Z}_{[g]}$
$\displaystyle=\sum_{h=0}^{g}\xi^{h-1}\mathcal{Z}_{h}.$

In Fig. 2(a), 2(b) and 2(c), we show the plot of quenched free energy
including the higher genus corrections up to $g=1$, $g=2$ and $g=3$,
respectively (see blue solid curves in Fig. 2). For $g=2$ and $g=3$, we see
that the quenched free energy diverges as $T\to 0$. This diverging behavior at
low temperature comes from the fact that $\mathcal{Z}_{2}$ and
$\mathcal{Z}_{3}$ in (72) become negative at some value of $W$. This implies
$\mathcal{Z}_{[g]}$ in (74) diverges in the negative direction at some value
of $W$ as $T\to 0$. As $\mathcal{Z}_{[g]}\to-\infty$, the factor
$e^{-\mathcal{Z}_{[g]}}$ in (73) grows exponentially and it leads to the
diverging behavior of the quenched free energy in Fig. 2(b) and 2(c).

However, this is not the expected behavior for the generating function
$\mathcal{Z}(x)$. In fact, since $Z$ and $x$ are positive quantities, the
generating function $\langle e^{-Zx}\rangle$ satisfies the inequality

$\displaystyle e^{-\mathcal{Z}(x)}=\langle e^{-Zx}\rangle<1,$ (75)

which implies that $\mathcal{Z}(x)$ is always positive. The appearance of the
negative coefficients in (72) simply means that the genus expansion is not a
good approximation of $\mathcal{Z}(x)$ at low temperature since the expansion
parameter $\xi$ is not small at low temperature $T\ll\hbar^{2/3}$. In other
words, the genus expansion of $\mathcal{Z}(x)$ breaks down at low temperature.

To study the low temperature behavior of $\mathcal{Z}(x)$ we need a different
approximation of $\mathcal{Z}(x)$, which we will consider in section 3.4.

(a) Free energy up to $g=1$ (b) Free energy up to $g=2$

(c) Free energy up to $g=3$

Figure 2:  Plot of free energy including the higher genus corrections up to
2(a) $g=1$, 2(b) $g=2$ and 2(c) $g=3$. The blue solid curves are the quenched
free energy in (73) while the orange dashed curves represent the annealed free
energy $-T^{\prime}\log\langle Z\rangle_{[g]}$.

###  3.3 $\mathcal{Z}_{g}$ from constitutive relation

Before discussing the low temperature behavior of $\mathcal{Z}(x)$ in section
3.4, let us consider an alternative method for the computation of
$\mathcal{Z}_{g}$. This method clarifies the appearance of the Lambert
function in this problem. Using the boundary creation operator $\widehat{Z}$
in (29), the generating function $\mathcal{Z}(x)$ is obtained by acting
$e^{-\widehat{Z}x}$ to the topological free energy (30)

$\displaystyle\mathcal{Z}(x)$
$\displaystyle=\mathcal{F}(\\{t_{k}\\})-e^{-\widehat{Z}x}\mathcal{F}(\\{t_{k}\\})=\mathcal{F}(\\{t_{k}\\})-\mathcal{F}(\\{t_{k}^{\prime}\\}),$
(76)

where $t_{k}^{\prime}$ is given by

$\displaystyle t_{k}^{\prime}$
$\displaystyle=t_{k}-x\hbar\sqrt{\frac{\beta}{\pi}}\beta^{k}.$ (77)

In the Airy case $t_{k}=0~{}(k\geq 0)$, the first term of (76) vanishes and
hence we have

$\displaystyle\mathcal{Z}(x)=-\mathcal{F}\Bigl{(}\Bigl{\\{}-x\hbar\sqrt{\frac{\beta}{\pi}}\beta^{k}\Bigr{\\}}\Bigr{)}.$
(78)

To see the structure of the derivatives of $\mathcal{Z}_{g}$ with respect to
$t_{0},t_{1}$, it is convenient to work in the subspace (33) with non-zero
$t_{0},t_{1}$. On this subspace (33), the genus-zero string equation (38) for
the shifted coupling $t_{k}^{\prime}$ (77) becomes

$\displaystyle
v=\sum_{k=0}^{\infty}t_{k}^{\prime}\frac{v^{k}}{k!}=t_{0}+t_{1}v-x\hbar\sqrt{\frac{\beta}{\pi}}e^{\beta
v}.$ (79)

Here we denote $u_{0}$ for the background $t_{k}^{\prime}$ (77) as $v$ in
order to distinguish it from $u_{0}$ in (34) for the original background (33).
Now we see that the string equation (79) for the background (77) can be
written in the form of the Lambert function

$\displaystyle z=We^{W},$ (80)

where $z$ is given by (62) and $W$ is defined by

$\displaystyle W$ $\displaystyle=\frac{\beta t_{0}}{1-t_{1}}-\beta v.$ (81)

Namely, the Lambert function naturally appears from the genus-zero string
equation for the shifted background (77).

As shown in Eguchi:1994cx , the genus-$g$ free energy $\mathcal{F}_{g}$ in
(31) is written as a combination of $v_{m}=\partial_{0}^{m}v$, which is known
as the constitutive relation Dijkgraaf:1990nc . From (80) and (81), one can
show that $v_{m}$ at $t_{0}=t_{1}=0$ is given by

$\displaystyle v_{m}=\delta_{m,1}-\beta^{m-1}D^{m}W,$ (82)

where $D$ is defined in (68). In particular $v_{1}$ is given by

$\displaystyle v_{1}=1-DW=\frac{1}{1+W}.$ (83)

As discussed in Eguchi:1994cx , if we assign the weight $m-1$ to $v_{m}$, the
genus-$g$ free energy $\mathcal{F}_{g}$ has the weight $3g-3$. From (82), this
implies that $\mathcal{F}_{g}$ is written as

$\displaystyle\mathcal{F}_{g}=\beta^{3g-3}\widehat{\mathcal{F}}_{g},$ (84)

where $\widehat{\mathcal{F}}_{g}$ is a function of $W$ only. Then, in the Airy
limit $t_{0}=t_{1}=0$ (78) becomes

$\displaystyle\mathcal{Z}(x)=-\sum_{g=0}^{\infty}(\sqrt{2}\hbar)^{2g-2}\beta^{3g-3}\widehat{\mathcal{F}}_{g}=\sum_{g=0}^{\infty}\xi^{g-1}\mathcal{Z}_{g},$
(85)

which implies

$\displaystyle\mathcal{Z}_{g}=-2^{g-1}\widehat{\mathcal{F}}_{g}.$ (86)

For instance, using the expression of $\mathcal{F}_{g}$ in terms of $v_{m}$
Eguchi:1994cx  and plugging $v_{m}$ in (82) into $\mathcal{F}_{g}$, we find

$\displaystyle\mathcal{Z}_{1}$
$\displaystyle=-\widehat{\mathcal{F}}_{1}=-\frac{1}{24}\log
v_{1}=\frac{1}{24}\log(1+W),$ (87) $\displaystyle\mathcal{Z}_{2}$
$\displaystyle=-2\widehat{\mathcal{F}}_{2}=-2\beta^{-3}\left(\frac{v_{2}^{3}}{360v_{1}^{4}}-\frac{7v_{2}v_{3}}{1920v_{1}^{3}}+\frac{v_{4}}{1152v_{1}^{2}}\right)=\frac{5W-19W^{2}+4W^{3}}{2880(1+W)^{5}}.$

As expected, this agrees with the result (72) obtained from the recursion
relation. We have also checked that $\mathcal{Z}_{3}$ in (72) is correctly
reproduced from $\mathcal{F}_{3}$ in Eguchi:1994cx .

We note in passing that our $\mathcal{Z}_{g}$ is related to the generating
function of the Hodge integrals studied in dubrovin2017classical .888It is
well-known that the Lambert function also appears in the problem of Hurwitz
numbers Bouchard:2007hi ; Eynard:2009xe ; mulase2009polynomial . As pointed
out in dubrovin2017classical , the generating function of Hurwitz numbers is a
special case of the generating function of the Hodge integrals (see the last
page of dubrovin2017classical  for details). For instance, $\mathcal{Z}_{2}$
is expanded as

$\displaystyle\mathcal{Z}_{2}=-\frac{7}{720(1+W)^{5}}+\frac{11}{576(1+W)^{4}}-\frac{31}{2880(1+W)^{3}}+\frac{1}{720(1+W)^{2}}.$
(88)

This agrees with $-2\mathcal{H}_{2}^{\text{Hodge}}$ in eq.(87) of
dubrovin2017classical  after setting $x_{i}=0$ an $T=-W$. From the result in
dubrovin2017classical , we find that $\mathcal{Z}_{g}$ is expanded as

$\displaystyle\mathcal{Z}_{g}=\frac{1}{2}(1+W)^{2-2g}\sum_{n=1}^{3g-3}\kappa_{g,n}\left(\frac{W}{1+W}\right)^{n},$
(89)

where

$\displaystyle\kappa_{g,1}=\frac{1}{12^{g}g!},\quad\kappa_{g,3g-3}=\frac{(-1)^{g}c_{g}}{12^{g}(5g-5)(5g-3)}.$
(90)

As discussed in dubrovin2017classical , $c_{g}$ is given by the coefficient of
the formal power series solution of the Painlevé I equation

$\displaystyle
U=\sum_{g=0}^{\infty}c_{g}X^{\frac{1}{2}(1-5g)},\quad\frac{d^{2}U}{dX^{2}}+\frac{1}{16}U^{2}-\frac{1}{16}X=0.$
(91)

###  3.4 Low temperature resummation

As shown in Okuyama:2019xvg , at low temperature $\langle
Z(\beta)^{n}\rangle_{c}$ is approximated by the one-point function $\langle
Z(n\beta)\rangle$ (26) and hence the generating function $\mathcal{Z}(x)$ is
approximated by (27). The summation on the right hand side of (27) can be
performed by rewriting the one-point function $\langle Z(n\beta)\rangle$ in
terms of the eigenvalue density $\rho(E)$

$\displaystyle\langle
Z(n\beta)\rangle=\int_{-\infty}^{\infty}dE\rho(E)e^{-n\beta E}.$ (92)

For the Airy case, $\rho(E)$ is given by

$\displaystyle\rho(E)=\hbar^{-2/3}\left[\text{Ai}^{\prime}(-\hbar^{-2/3}E)^{2}-\text{Ai}(-\hbar^{-2/3}E)\text{Ai}^{\prime\prime}(-\hbar^{-2/3}E)\right],$
(93)

where $\text{Ai}(z)$ denotes the Airy function. We emphasize that this
$\rho(E)$ is the exact eigenvalue density including all-genus contributions.
Then the low temperature approximation of $\mathcal{Z}(x)$ in (27) becomes

$\displaystyle\mathcal{Z}(x)\approx-\sum_{n=1}^{\infty}\frac{(-x)^{n}}{n!}\int_{-\infty}^{\infty}dE\rho(E)e^{-n\beta
E}=\int_{-\infty}^{\infty}dE\rho(E)\Bigl{(}1-e^{-xe^{-\beta E}}\Bigr{)}.$ (94)

This expression of $\mathcal{Z}(x)$ is positive for $x>0$ since $\rho(E)$ is
positive definite. Thus this approximation of $\mathcal{Z}(x)$ in (94)
satisfies the necessary condition (75).

In Fig. 3 we show the plot of the quenched free energy using the low
temperature approximation (94). One can see that the quenched free energy is a
monotonic function of the temperature even in the low temperature regime.

Figure 3: Plot of the free energy in the low temperature approximation. The
blue solid curve is the quenched free energy in the low temperature
approximation while the orange dashed curve is the all-genus result of the
annealed free energy.

##  4 Conclusions and outlook

In this paper we have studied the quenched free energy in an ensemble average
of random systems. We found the general formula (4) for the quenched free
energy and argued that it can be derived by the replica method (3) under a
certain prescription of the analytic continuation of $\langle Z^{n}\rangle$.
We also discussed that the bulk gravity picture of our formula (4) involves
the so-called spacetime D-branes $e^{-Zx}$ introduced in Marolf:2020xie . Then
we have applied our general formula (4) to the Airy limit of the Gaussian
matrix model. We found that the genus-zero approximation of the generating
function $\mathcal{Z}(x)$ leads to a monotonic behavior of the quenched free
energy. The inclusion of higher genus corrections to $\mathcal{Z}(x)$ leads to
a diverging free energy at low temperature, but this is not a problem of our
formula (4); it simply means that the genus expansion is not a good
approximation at low temperature. Instead, we found that the alternative low
temperature approximation (27) gives rise to a well-behaved monotonic free
energy even at low temperature.

There are many interesting open problems. It would be interesting to
generalize our analysis of the Airy case to the matrix model of JT gravity
Saad:2019lba . It is speculated in Saad:2019lba  that the quenched free energy
of JT gravity exhibits a spin glass phase with the replica symmetry breaking
at low temperature. Based on the result in Okuyama:2019xbv ; Okuyama:2020ncd ,
it is argued in Engelhardt:2020qpv  that in the low temperature scaling limit

$\displaystyle\beta\to\infty,\quad S_{0}\to\infty\quad\text{with}\quad\beta
e^{2S_{0}/3}:~{}\text{fixed},$ (95)

the quenched free energy of JT gravity reduces to that of the Airy case. Thus,
according to the argument in Engelhardt:2020qpv , we would be able to see the
replica symmetry breaking and spin glass phase in the quenched free energy of
the Airy case studied in our paper. However, in our computation in section 3,
it is not obvious whether the replica symmetry breaking occurs in the Airy
case or not. Instead, our analysis is based on the spacetime D-branes
$e^{-Zx}$. It would be interesting to clarify the relation between these two
pictures.

Our analysis of the Airy case is limited to the genus expansion of
$\mathcal{Z}(x)$ in (25) and the low temperature approximation of
$\mathcal{Z}(x)$ in (27). It would be nice if we can find a closed form
expression of the generating function of the integral representation of
$\langle Z^{n}\rangle_{c}$ given in okounkov2002generating . We leave this as
an interesting future problem.

In section 3.3, we have seen that $\mathcal{Z}(x)$ in the Airy limit is
closely related to the generating function of the Hodge integrals studied in
dubrovin2017classical . However, the quenched free energy is not the
generating function of the Hodge integrals itself, but some integral
transformation of it. By introducing the chemical potential $\mu=\log x$, (6)
is written as

$\displaystyle\langle\log Z\rangle=\log\langle
Z\rangle-\int_{-\infty}^{\infty}d\mu\Bigl{[}e^{-\mathcal{Z}(e^{\mu})}-e^{-\langle
Z\rangle e^{\mu}}\Bigr{]}.$ (96)

This reminds us of the frame change in the topological string/spectral theory
(TS/ST) correspondence (see e.g. Marino:2015nla  for a review). It would be
interesting to understand the mathematical meaning of this integral
transformation.

In our previous paper Okuyama:2020mhl , we find that the quenched free energy
of Gaussian matrix model has a finite limit at zero temperature

$\displaystyle\lim_{T\to 0}F=E_{0},$ (97)

and $E_{0}$ is given by the average of the minimal eigenvalue of the random
matrix. Our result in the Airy case suggests that the quenched free energy in
the Airy limit also has a finite limit at zero temperature (see Fig. 3). It
would be interesting to compute this zero temperature value of the free energy
and clarify its physical meaning.

Finally, let us comment on the status of our formula (6). We emphasize that
(6) is not just a formal expression, but it is useful in practice for the
actual computation of the quenched free energy in random matrix models. We
have explicitly demonstrated this computation for the Airy case in the genus
expansion and also in the low temperature approximation (27). For the JT
gravity case, as shown in Okuyama:2019xbv ; Okuyama:2020ncd  the deviation
from the Airy limit (95) is organized as a low temperature expansion and it
would be possible to compute $\mathcal{Z}(x)$ of JT gravity in this expansion.
In general, the summation (5) over the number of boundaries might be an
asymptotic series and one might worry that $\mathcal{Z}(x)$ is only defined
perturbatively in the small $x$ expansion. However, in principle
$\mathcal{Z}(x)$ can be defined non-perturbatively as the free energy of
random matrix model with a shifted background (76), as demonstrated in section
3.3 in the Airy case. In other words, $\mathcal{Z}(x)$ is the free energy of
random matrix model with a deformed matrix potential due to the insertion of
the SD-brane operator $e^{-Zx}$. This definition of $\mathcal{Z}(x)$ does not
rely on the small $x$ expansion in (5) and it may serve as a non-perturbative
definition of $\mathcal{Z}(x)$, which in turn leads to a non-perturbative
definition of $\langle\log Z\rangle$ via (6). Instead of computing the
connected correlators $\langle Z^{n}\rangle_{c}$ term by term in (5), we
should consider the expectation value of the SD-brane operator $\langle
e^{-Zx}\rangle$ as a whole. We stress that this viewpoint is not obvious from
the replica method (3) and it is one of the advantages of our formalism (6),
at least conceptually. One might think that the expansion (16) used in the
replica method breaks down when $w_{n}$’s in (15) become of order one. This is
indeed the case at low temperature (see (98)). However, the computation in
section 2.2 is just an illustration that our formula (4) can be obtained from
the replica method and the final result (4) makes sense without referring to
the replica method. Our formula (4) does not require $\langle\log Z\rangle$ to
be computed via the expansion (16); (4) is well-defined even at low
temperature once $\langle e^{-Zx}\rangle$ is defined non-perturbatively by a
shift of background (76). We hope that our expression of the quenched free
energy (4) in terms of the SD-brane operator $e^{-Zx}$ provides us with a new
perspective on our understanding of random systems and gravitational path
integrals.

###### Acknowledgements.

The author would like to thank Kazuhiro Sakai for useful discussions. This
work was supported in part by JSPS KAKENHI Grant No. 19K03845.

##  Appendix A High temperature expansion in the Airy limit

In this appendix we consider the high temperature expansion of the quenched
free energy in the Airy limit. The natural expansion parameter at high
temperature is $\xi=\hbar^{2}\beta^{3}$. For small $\xi$, $w_{n}$ defined in
(15) behaves as

$\displaystyle w_{n}=\mathcal{O}(\xi^{n-1}).$ (98)

Up to the order $\mathcal{O}(\xi^{2})$, the expansion (16) reads

$\displaystyle\frac{\langle Z^{n}\rangle}{\langle
Z\rangle^{n}}=1+\frac{1}{2}n(n-1)w_{2}+\frac{1}{6}n(n-1)(n-2)w_{3}+\frac{1}{8}n(n-1)(n-2)(n-3)w_{2}^{2}+\mathcal{O}(\xi^{3}).$
(99)

For the Airy case, $w_{2}$ and $w_{3}$ are known in the closed form
okounkov2002generating ; Beccaria:2020ykg . From (22) we find

$\displaystyle w_{2}$
$\displaystyle=\sqrt{\frac{\pi\xi}{2}}e^{\frac{1}{2}\xi}\text{Erf}\left(\sqrt{\frac{\xi}{2}}\right),$
(100) $\displaystyle w_{3}$
$\displaystyle=\frac{4\pi\xi}{3\sqrt{3}}e^{2\xi}\Biggl{[}1-12T\Bigl{(}\sqrt{3\xi},\frac{1}{\sqrt{3}}\Bigr{)}\Biggr{]}.$

Plugging (100) into (99), we find the small-$\xi$ expansion of $\langle
Z^{n}\rangle/\langle Z\rangle^{n}$ in the Airy limit

$\displaystyle\frac{\langle Z^{n}\rangle}{\langle
Z\rangle^{n}}=1+\frac{n(n-1)}{2}\xi+\frac{n(n-1)(3n-5)(n+2)}{24}\xi^{2}+\mathcal{O}(\xi^{3}).$
(101)

This agrees with the result eq.(1.17) in Beccaria:2020ykg . Taking the $n\to
0$ limit (3), we obtain the quenched free energy of the Airy case as a small
$\xi$ expansion

$\displaystyle\langle\log Z\rangle$ $\displaystyle=\log\langle
Z\rangle-\frac{1}{2}\xi+\frac{5}{12}\xi^{2}+\mathcal{O}(\xi^{3})$ (102)
$\displaystyle=-\frac{1}{2}\log(4\pi\xi)-\frac{5}{12}\xi+\frac{5}{12}\xi^{2}+\mathcal{O}(\xi^{3}),$

where we used the result of one-point function $\langle Z\rangle$ in (22). For
the higher $w_{n\geq 4}$ we do not know the closed form expression, but the
small $\xi$ expansion of $w_{n}$ is easily obtained from the KdV equation
(60), as we will see below.

Let us briefly explain the computation of $w_{n}$ in the small $\xi$
expansion. To do this, it is convenient to introduce the generating function
of the connected correlators $\widetilde{\mathcal{Z}}(x)$, which is related to
$\mathcal{Z}(x)$ in (5) by

$\displaystyle\widetilde{\mathcal{Z}}(x)=-\mathcal{Z}(-x)=\sum_{n=1}^{\infty}\frac{x^{n}}{n!}\langle
Z^{n}\rangle_{c}.$ (103)

Then one can show that
$\widetilde{\mathcal{W}}=\hbar\partial_{0}\widetilde{\mathcal{Z}}$ satisfies a
similar relation as (60)

$\displaystyle\partial_{1}\widetilde{\mathcal{W}}=u\partial_{0}\widetilde{\mathcal{W}}+\frac{\hbar^{2}}{6}\partial_{0}^{3}\widetilde{\mathcal{W}}+\hbar(\partial_{0}\widetilde{\mathcal{W}})^{2}.$
(104)

To solve this relation recursively, we work on the subspace (33). On this
subspace (33), the one-point function is given by

$\displaystyle\langle
Z(\beta)\rangle=\frac{1}{\sqrt{4\pi\hat{\xi}}}\exp\left(\frac{\hat{\xi}}{12}+\frac{\beta
t_{0}}{1-t_{1}}\right).$ (105)

Here $\hat{\xi}=\hat{\hbar}^{2}\beta^{3}$ and $\hat{\hbar}$ is defined in
(63). From the result of $\langle Z^{n}\rangle_{c}$ for small $n$ in (22), it
is natural to make an ansatz

$\displaystyle\langle Z(\beta)^{n}\rangle_{c}=\langle Z(n\beta)\rangle
C_{n}(\hat{\xi}).$ (106)

The important point is that $C_{n}$ is independent of $t_{0}$ and it depends
on $t_{1}$ only through the combination $\hat{\hbar}$ in (63). From (104),
$W_{n}=\hbar\partial_{0}\langle Z(\beta)^{n}\rangle_{c}$ satisfies

$\displaystyle\partial_{1}W_{n}$
$\displaystyle=u\partial_{0}W_{n}+\frac{\hbar^{2}}{6}\partial_{0}^{3}W_{n}$
(107)
$\displaystyle+\hbar\sum_{m=1}^{n-1}\binom{n}{m}\partial_{0}W_{m}\partial_{0}W_{n-m}.$

Using the fact that the one-point part $\langle Z(n\beta)\rangle$ of the
ansatz (106) satisfies the first line of (107), we find the equation for
$C_{n}(\hat{\xi})$

$\displaystyle(1-t_{1})\partial_{1}C_{n}=\sqrt{\frac{\hat{\xi}}{4\pi}}\sum_{m=1}^{n-1}\binom{n}{m}\sqrt{nm(n-m)}e^{-\frac{1}{4}nm(n-m)\hat{\xi}}C_{m}C_{n-m}.$
(108)

Since $C_{n}$ depends on $t_{1}$ only through the combination $\hat{\xi}$, the
left hand side of (108) is equal to $2\hat{\xi}\partial_{\hat{\xi}}C_{n}$.
Finally, after setting $t_{0}=t_{1}=0$, we find the recursion relation for
$C_{n}(\xi)$

$\displaystyle
2\xi\partial_{\xi}C_{n}=\sqrt{\frac{\xi}{4\pi}}\sum_{m=1}^{n-1}\binom{n}{m}\sqrt{nm(n-m)}e^{-\frac{1}{4}nm(n-m)\xi}C_{m}C_{n-m}.$
(109)

One can easily solve this equation recursively starting from $C_{1}=1$. For
instance, $C_{2}$ satisfies

$\displaystyle\partial_{\xi}C_{2}=\frac{e^{-\frac{1}{2}\xi}}{\sqrt{2\pi\xi}}.$
(110)

This is solved as

$\displaystyle C_{2}=\text{Erf}\left(\sqrt{\xi/2}\right),$ (111)

which agrees with the known result (22) of the two-point function in the Airy
limit. In a similar manner, for $n=3$ we can show that

$\displaystyle C_{3}=1-12T(\sqrt{3\xi},1/\sqrt{3})$ (112)

is the solution for the recursion relation (109).

For $n\geq 4$ we do not have a closed form solution for $C_{n}$, but it is not
difficult to solve the recursion relation (109) in the small $\xi$ regime and
find $C_{n}$ as a power series in $\xi$. Then $w_{n}=\langle
Z^{n}\rangle_{c}/\langle Z\rangle^{n}$ is also obtained as a power series in
$\xi$

$\displaystyle
w_{n}=n^{-\frac{3}{2}}(4\pi\xi)^{\frac{n-1}{2}}e^{\frac{n^{3}-n}{12}\xi}C_{n}(\xi).$
(113)

For instance, $w_{4}$ and $w_{5}$ are expanded as

$\displaystyle w_{4}$
$\displaystyle=32\xi^{3}+84\xi^{4}+\frac{1936\xi^{5}}{15}+\mathcal{O}(\xi^{6}),$
(114) $\displaystyle w_{5}$
$\displaystyle=400\xi^{4}+\frac{5776\xi^{5}}{3}+\mathcal{O}(\xi^{6}).$

In this way, we finally find the small $\xi$ expansion of the quenched free
energy in the Airy limit of Gaussian matrix model

$\displaystyle\langle\log
Z\rangle=-\frac{1}{2}\log(4\pi\xi)-\frac{5}{12}\xi+\frac{5}{12}\xi^{2}-\frac{133}{90}\xi^{3}+\frac{8053}{840}\xi^{4}-\frac{171781}{1890}\xi^{5}+\mathcal{O}(\xi^{6}).$
(115)

We have computed this expansion up to $\mathcal{O}(\xi^{50})$.999The data of
this expansion are available upon request to the author.

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