Late time behavior of n -point spectral form factors in Airy and JT gravities

: We study the late time behavior of n -point spectral form factors (SFFs) in two-dimensional Witten-Kontsevich topological gravity, which includes both Airy and JT gravities as special cases. This is conducted in the small (cid:126) expansion, where (cid:126) ∼ e − 1 /G N is the genus counting parameter and nonperturbative in Newton’s constant G N . For one-point SFF, we study its absolute square at two diﬀerent late times. We show that it decays by power law at t ∼ (cid:126) − 2 / 3 while it decays exponentially at t ∼ (cid:126) − 1 due to the higher order corrections in (cid:126) . We also study general n ( ≥ 2)-point SFFs at t ∼ (cid:126) − 1 in the leading order of the (cid:126) expansion. We ﬁnd that they are characterized by a single function, which is essentially the connected two-point SFF and is determined by the classical eigenvalue density ρ 0 ( E ) of the dual matrix integral. These studies suggest that qualitative behaviors of n -point SFFs are similar in both Airy and JT gravities, where our analysis in the former case is based on exact results.


Introduction
One of the goals in quantum gravity is to understand the microscopic structures of black hole spacetime.Given the AdS/CFT correspondence [1], in principle, one can understand the quantum spectrum of spacetime by directly solving the large N CFT.However, the large N CFTs in general are notoriously difficult to solve, and therefore it is in practice almost impossible to determine the microscopic states.
In recent years, as a simple toy model for AdS/CFT, the low-energy correspondence between a specific two-dimensional dilaton gravity, Jackiw-Teitelboim (JT) gravity [2,3], and the one-dimensional Majorana fermion model, Sachdev-Ye-Kitaev (SYK) model [4,5] has received a great deal of attention.Particularly noteworthy is that the low-energy reparametrization modes of both JT gravity and the SYK model are described by the same one-dimensional Schwarzian action [6,7], which shows the maximal chaos bound [8] expected from a black hole [9].
At the same time, there has been significant progress in understanding the effects of non-perturbative effects in Newton's constant G N , as exemplified by the finding of the island formula [10,11,12] and replica wormholes [13,14].The key point is that there are non-perturbative new saddle points in the gravitational path integral, which are irrelevant in the early time, but in the late time these new saddles become relevant and play a very important role for the entanglement entropy of the Hawking radiations showing the Page curve [15,16].Here early and late is in comparison with the Page time.
The reason why the late time is important must be clear: in the late time, the detailed energy eigenvalue distribution of microscopic theories plays an important role.Especially whether energy eigenvalues have correlations or not is essential: typically for chaotic systems, the energy eigenvalues repel [17], and are described by the "sine-kernel" [18,19].In fact, this property can be seen directly by the study of the two-point spectral form factor (SFF) in the SYK model [20], which is defined as the analytically continued correlator of partition functions where • denotes the disorder average for the SYK model.In the SYK model, this two-point SFF shows the decay (called slope) in the early time, but in the late time it shows the time-linear growth called ramp and the constant behavior called plateau.
The ramp and plateau are seen in the random matrix theory (RMT) as well and therefore are regarded as typical behaviors of chaotic systems.
Recently, in a beautiful paper [21] it has been shown that given the leading classical eigenvalue density ρ 0 (E) ∼ sinh √ E [22], there is an exact correspondence between JT gravity and RMT in the perturbation expansion; one in terms of e −S = e −1/G N , where S is the entropy, and the other in terms of 1/L, where L is the rank of RMT.This is based on the equivalence of the recursion relation between the Mirzakhani's one for Weil-Petersson volumes [23] and the topological recursion of RMT by Eynard and Orantin [24].This reveals that there are intertwined relationships between SYK models, RMT, and JT gravity.Furthermore, it has been pointed out in [25] that JT gravity is a special case of more general two-dimensional Witten-Kontsevich (WK) topological gravity.WK topological gravity contains infinitely many couplings t k , and by tuning all of t k to specific values, it reproduces the classical eigenvalue density ρ 0 (E) ∼ sinh √ E of JT gravity.Historically two-dimensional topological gravity, where observables are intersection numbers in the moduli space of Riemann surfaces, has been extensively studied from 80's [26,27,28,29].It was conjectured by Witten [30] and proved by Kontsevich [31] that this topological gravity is equivalent to the double-scaled one matrix model which counts triangulations of surfaces and similarly contains infinitely many couplings associated to the matrix potential V .Based on this equivalence, systematic studies on multi-boundary partition functions in two-dimensional topological gravity, which encompasses JT gravity, have been done in [32].The key points are that the generating function for the intersection numbers of topological gravity is known to obey the Korteweg-De Vries (KdV) equation and can be obtained by genus expansion.Once the multi-boundary partition functions in two-dimensional topological gravity are obtained, by the analytic continuation β i → β i + it i one can also obtain generic n-point SFFs in topological gravity.The case of n = 2 especially corresponds to the conventional SFF studied before [20].It is a natural question whether the full behavior of the SFF seen in SYK and RMT, especially the ramp and plateau can be reproduced from JT gravity or more generically topological WK gravity.Recently it has been shown in JT gravity that by taking an appropriate rescaling for the late time, one can conduct the summation of the Weil-Petersson volumes for arbitrary genus g and see not only the ramp, but also its transition to plateau as well [33,34,35].
Given the success of n = 2 SFF, in this paper we study more generic n-point SFFs in late time for generic WK topological gravity which includes JT gravity.In RMT, n-point generalizations of 2-point SFF in RMT for n even were studied in [36,37].In this paper, we study general n-point SFFs in topological gravity, i.e. after the double scaling limit of RMT.Furthermore, in our studies n can be odd, in fact we study the n = 1 case as well.By examining the n-point SFF with analytically continued time in the late time limit, we attempt to understand its universal behavior in topological gravity.
Especially we elucidate how to take the late time limit of the one-point SFF in generic two-dimensional WK topological gravity in the small expansion, where is the genus-counting parameter.Note that corrections are non-perturbative effects in G N .We analyze two different late times, one in t ∼ −2/3 and the other in t ∼ −1 in the → 0 limit.We will see that at relatively late time t ∼ −2/3 and very late time t ∼ −1 , the behavior of the one-point SFF is different.
We also present a systematic analysis of n-point SFF at t ∼ −1 .We find that they are characterized by a single function, which is essentially the connected two-point SFF and is determined by the classical eigenvalue density ρ 0 (E) of the dual matrix integral.Furthermore, not only generic topological WK gravity, we also consider Airy and JT gravities as concrete examples, both of which are obtained by tuning the infinitely many couplings t k in WK topological gravity.We see that both Airy and JT gravities behave in a qualitatively very similar way, which we can see from generic topological WK gravity.From these, we conjecture that the qualitative behavior of the n-point SFF for generic topological gravity including JT gravity is very similar to that of Airy gravity.In other words, if one ask how special Airy or JT gravity is, our temporal answer is that most probably Airy and JT gravities are quite typical in generic topological WK gravity parameter range.
The organization of this paper is as follows; in §2 we study the SFF for Airy gravity where the exact full n-point function is known.In §3 we study the generic behavior of the one-point SFF in general WK topological gravity, including JT gravity as one of the concrete examples.In §4 we study n-point SFF in general WK topological gravity at very late times t ∼ −1 .We also study the results of JT gravity n-point SFF as a concrete example.We end with conclusions and discussion at §5.
2 n-point spectral form factor in Airy gravity

Airy gravity overview
Exact general n-point partition functions are difficult to obtain in generic WK topological gravity.Therefore we are obliged to employ some sort of expansions [32].However, there is an exception; which is Airy gravity.Airy gravity corresponds to the one matrix model which is given by Gaussian potential, therefore it is solvable.In fact, general n-point function is given exactly in the integral form, which was worked out first by Okounkov [38]. 1  The Airy gravity is obtained by zooming in on the edge of the Wigner semi-circle: its classical eigenvalue density is given by2 ρ 0 (E) = 2 √ E. (2.1) The double-scaled wave function ψ(E), which is a Baker-Akhiezer function of the KdV hierarchy, obeys the Schrödinger equation: Here x and E are continuous parameters obtained by the double-scaled limit of polynomial index n and matrix eigenvalue λ respectively.We also introduce the notation where |x is the coordinate eigenstate and |E is the energy eigenstate satisfying Q|E = −E|E .The Baker-Akhiezer function in the present case is written in terms of the Airy function Given the Baker-Akhiezer function ψ(E) ≡ x|E , one can obtain the full eigenvalue density as where ζ = − − 2 3 E and the Airy function satisfies Ai ζ = ζAi ζ .This defines a non-perturbative completion of the classical eigenvalue density (2.1).In fact, using the asymptotic formula for the Airy function one can see that in the → 0 limit, ρ Airy (E) given by eq.(2.5) reduces to the classical ρ 0 (E) given by eq.(2.1).The one-point function of the macroscopic loop operator is given by (2.7) In the planar limit, this one-point function behaves as (2.9) Here the subscript 0 represents the genus zero planar limit.

One point function: n = 1 SFF
We start our analysis from the one-point function of analytically continued partition function in Airy gravity.For that purpose, in eq.(2.7), we analytically continue β → β + it and obtain Furthermore, by normalizing it at the value of t = 0, the exact one-point function becomes (2.12) On the other hand, in the planar limit, where we keep only the leading order part in the → 0 limit, we have where Z(β) 0 is given by eq.(2.9) and we implicitly assumed that 2 (β + it) 3  1 . (2.14) To get rid of the phase factor, let us compare the planar contribution Z(β+it) 0 Z(β) 0 with the full contribution Z(β+it) by taking their absolute square.This yields the disconnected two-point SFF (2.15) Since the planar part is the difference between the planar and the full contribution is whether the exponential factor can be neglected or not, : 1.
(2.17) Thus, higher order corrections in become important at the time-scale of where At t = O(t higher genus ), higher genus corrections become important (see Figure 1).This time-scale t higher genus is called t plateau in [25], the time-scale where ramp changes into plateau.

Connected two-point spectral form factor
Next we consider the two-point SFF for Airy gravity, which is obtained by analytically continuing the connected correlator of two macroscopic loops where Π is the projector The full connected two-point function is written in terms of the error function as By analytically continuing the arguments as and normalizing by the one point function at t = 0, the full two-point function is Note that this does not go to zero at t = 0, since we have normalized the connected function by the disconnected one.In Figures 2 and 3, we plot the connected SFF given in eq.(2.24) and connected plus disconnected SFF for β = 2, = 0.01/ √ 2. In the planar limit → 0, we have   2, with more focus on the ramp region t 100.One observes the t-linear growth, which is due to the eigenvalue repulsion.
Therefore the planar contribution to the above connected two-point function is  2.24) and its planar limit given by eq.(2.27).The deviation starts at t higher genus due to higher order corrections, which change ramp into plateau.We take β = 2, = 0.01/ √ 2, so that t higher genus = ( √ β) −1 = 100.and at t β, its planar contribution is proportional to time t, At late times t β, the argument of the error function becomes of order one βt 2 = O(1) ⇔ t = O (t higher genus ) . (2.28) Therefore higher order corrections become important at t higher genus , given in eq.(2.19) (see Figure 4).Note that this typical time-scale t higher genus where higher genus contribution becomes important is the same for both one-point and connected two-point functions.
At t = O(t higher genus ), time-dependence changes; for one-point function, the way of decay changes from power law decay to exponential decay due to higher genus corrections and for two-point function, higher genus corrections change the ramp into plateau.Note also that the signature of ramp and plateau is common for both Airy and JT gravities [20].

Dip time t dip vs t higher genus
Before we proceed to the three point function, we comment on the dip time t dip .Dip time is calculated as the time where the magnitude of the connected two-point SFF becomes of the same order as the disconnected contribution.Generically t dip t plateau as we will see.
Since t plateau is the timescale where higher genus corrections become important, if t dip t plateau , at the dip time, all exponential factor can be approximated as one, in other words, the lowest genus contribution dominates due to the definition of t plateau .The disconnected part of the two point function in the planar limit is given by eq.(2.16).Equating this with eq.(2.27), we have From this, we obtain the dip time If the temperature is of order one, i.e., β = O(1), in the limit of → 0, the dip time and higher genus time are (2.31) and there is a large hierarchy between t dip and t higher genus , As we will analyze in detail later, the timescale t ∼ −1 is where higher genus effects are important and simultaneously one can analyze the SFF using what is called the τ -scaling limit.These results in Airy gravity show that the behavior of SFF in Airy gravity is very similar to that of JT gravity [32,33,34,35].The SFF in Airy gravity shows ramp and plateau just as in JT gravity and the hierarchy between t dip and t higher genus is with ∼ e −S , which is the same as seen in both JT gravity and RMT in [41,20].

Three-point spectral form factor
Even though the general n-point function of Z(β)'s in Airy gravity was obtained by Okounkov in the form of an n-dimensional integral [38], this integral is still very complicated and the closed form expression of the n-point function for n ≥ 4 is not known in the literature, as far as we are aware of.However for n = 3, the closed form expression was obtained in [42] as where T (z, a) denotes the Owen's T -function As before, let us consider the analytic continuation of β i to a complex value, We can focus on the late time behavior of the three-point function (2.35) around the time scale of plateau by taking the τ -scaling limit It turns out that the late time behavior is quite different for i τ i = 0 and i τ i = 0. First, let us consider the case i τ i = 0. Without loss of generality, we can assume Using the property of the Owen's T -function, one can show that in the τ -scaling limit the three-point function becomes where β tot = i β i and Z(β) 0 is the planar limit of one-point function given by eq.(2.9).This late time approximation (2.40) is also written as where F (τ ) is given by As we will see in §4, eq.(2.41) is a special case of the general result of n-point function in the τ -scaling limit.In Figure 5, we show the plot of the three-point function as a function of τ 1 with fixed τ 2 .One can see that eq.(2.40) is a good approximation of the exact three-point function eq.(2.35) at late times.
Next, let us consider the case where i τ i = 0.In this case, it is difficult to find the analytic form of the late time behavior of (2.35).Instead, we can study the behavior of (2.35) numerically.As an example, let us consider the case where all β i 's are equal Z(β + iτ −1 ) 3 conn . (2.43) As we can see from Figure 6, both three-point function in eq.(2.43) and two point function eq.(2.22) with β 1 = β 2 = β + iτ −1 decay exponentially as a function of τ .This is in contrast to the one-point function which decays ∼ e −τ 2 rather than ∼ e −τ .
3 One point SFF in general topological gravity

Preliminaries
In this section we consider general WK topological gravity one point function.Especially we consider the absolute square of the one point function, which is equivalent to a disconnected two-point function.When the general couplings t k are turned on, the correlation functions can no longer be expressed in a closed form.In certain specific regimes of the parameters, however, one can still study them analytically by series expansion.We briefly recall how the one point function is expanded [25,32].The correlation functions for general topological gravity are neatly expressed in terms of the Itzykson-Zuber variables [43] Here −u 0 = E 0 is the threshold energy.It is zero for the Airy and JT gravity cases, but for the general case it is given by a formal power series in t k determined by the genus zero string equation In what follows let us present three different regimes where exact expansion is available.First, let us consider the regime In this regime the one-point function can be expanded as [25,32] where Next, let us consider what we call the low-temperature limit → 0, β → ∞ with h := β 3/2 fixed.(3.6)In this regime Z(β) can be expanded as [25] Z where T = β −1 and Third, we can also consider what we call the 't Hooft limit In this regime the one point function can be expanded as [42] Z where By using these results, we will evaluate the one point function at two different time scales in the following subsections.

One point function at t ∼ −2/3
Let us first evaluate the one-point function in the regime t ∼ −2/3 .This corresponds to the low-temperature limit and we can use the expansion (3.7).As in the Airy case, we consider the absolute square corresponding to the disconnected two-point function (3.12) Given the low-temperature expression eq.(3.7) we replace the variables as Here β and τ are supposed to be of the order of 0 .We see that Substituting these into (3.7)we have This expression is valid as long as τ ∼ 0 .We see that at the leading order of expansion, the one point function shows a power law scaling decay in τ as Naively, one can see from eq. (3.15) that the exponential decay, as a function of τ , becomes relevant when τ ∼ −1/3 β −1/2 .In terms of the original time t = τ / 2/3 , this means t ∼ −1 β −1/2 ∼ t higher genus .This is seemingly in accordance with the analysis in the Airy case on eq.(2.19).However, careful analysis is required because for τ ∼ −1/3 the parameter h actually diverges as h 2 ∼ −1 and the expansion in eq.(3.7) no longer makes sense.We will therefore study this time scale using a different expansion in the next subsection.

Airy and JT gravity one point function at t ∼ −2/3
Let us look into some concrete examples.
1.In the Airy case, where t k = 0 for all k, we have (3.17) 2. In the JT gravity case, we have which means Therefore, in both cases eq.(3.16) becomes Of course, this is consistent with the Airy gravity case, where we have eq.(2.11) with the low-temperature scaling given by eq.(3.13).The difference between Airy gravity and JT gravity starts appearing only at the higher orders in .

One point function at t ∼ −1
Let us next evaluate (3.12) at the time scale t ∼ −1 , which is known as the τ -scaling limit [34,33,44].This is done by substituting into the 't Hooft expansion eq.(3.10) and then re-expanding it in .We obtain where Therefore we see that the one point function shows an exponential decay at t ∼ −1 , controlled by the function E τ .This exponential decay is in contrast with the power law decay we have seen in eq.(3.16) at t ∼ −2/3 .This E τ ubiquitously appears in the study of the spectral form factor in the τ -scaling limit [35].

Late time behavior in Airy and JT gravity
Here we look into concrete examples.
1.In the Airy case, we have and eq.(3.22) becomes This is exactly what we can obtain from the exact formula eq.(2.10) with τ -scaling limit t = τ / β. 2. In the JT gravity case, we have and eq.(3.22) becomes For small τ , there is not much difference between Airy gravity and JT gravity since arcsinh(τ ) = τ + O(τ 3 ).However at late time τ 1, the difference appears more significantly, see Figure 7.In the Airy case, the one point function decays exponentially.However in the JT gravity case, since arcsinh(τ ) does not grow much and arcsinh(τ ) ∼ log τ at large τ , their exponential decay becomes much milder compared with the Airy case.
4 n-point SFF in general topological gravity at t ∼ −1

Integral expression of n-point function
In this section we consider the n-point function at the time scale t ∼ −1 .As we saw in the last section, we can make use of the results of the 't Hooft expansion [32] to study the correlation functions at this time scale.To do this, let us first recall that the n-point function can be expressed as [45] Z and u = u(t k ; ) is the KdV potential specified by the initial condition u(t k ; 0) = u 0 (t k ) (see [32] for the details).The symbol 1) means that we extract the terms linear in all ξ i .More explicitly, (4.1) is written for n = 1, 2, 3 as From the above expression we see that Z n is a sum of the trace Tr(e Note that β i here is in general not identical to the original β i appearing in (4.1) but could be a sum of them.(−Q) is viewed as a Hamiltonian and let |E denote the energy eigenstate with eigenvalue E. By inserting 1 = dE i |E i E i | in front of each Π, the above trace is rewritten as Here is the Christoffel-Darboux (CD) kernel.At the leading order of the small expansion, it is approximated by the sine kernel4 Here ρ 0 (E) is the genus zero part of the eigenvalue density.For general topological gravity, it is given by Note that E 0 = −u 0 .

General prescription for n ≥ 2
We are interested in the n-point function with β i replaced by β i + iτ i −1 .For generic values of τ i , each component of the form (4.5) in the n-point function (4.1) scales differently and it is not straightforward to discuss the scaling behavior in a uniform way.As we will see, however, under the constraint the n-point function shows a universal leading order behavior.This can be thought of as a natural generalization of the ordinary two-point SFF to the n-point one and in what follows we assume that this constraint is imposed.
As we have seen above, the calculation of the n-point SFF boils down to evaluating the integral of the form where indices in the argument of I represent indices for β i + iτ i −1 .β i and τ i here are not always identical to the original ones but could be a sum of them.Correspondingly, we will use the abbreviation where the sum of β i + iτ i −1 is represented by the sum of indices in the argument of I.For example, With this notation two-and three-point SFFs, obtained from eq. ( 4.3) by analytic continuation β i → β i + iτ i −1 , are concisely expressed as Clearly, from the cyclic property of the trace we have We stress that τ i in eq.(4.10) obeys the constraint which is inherited from eq. (4.9).To evaluate the integral, we introduce the variables and rewrite the exponent as In the last equality we have used the constraint (4.14) and introduced the notations Using (4.16), let us evaluate the integral (4.10) for small in the leading order approximation.At this level, we can replace the CD kernel by the sine kernel (4.7) and thus the integral is approximated as . (4.18) The last integral is neatly evaluated by the method of Lagrange multipliers. 5By inserting the integral (4.18) is expressed as sin 1 2 ρ 0 (E)ω ω . (4.20) As in the case of the ordinary SFF [46,35], we can evaluate this sort of integral using the formula Here, θ(x) is the step function.The result is where we have formally introduced τ (0) := 0 for the sake of brevity.Observe that the arguments of the step functions satisfy (4.23) 5 We would like to thank the anonymous referee for suggesting this prescription.
Summing them up, one obtains where in the second equality we have used the fact that E τ in eq.(3.23) satisfies the equation [35] ρ 0 (E τ ) = τ. (4.27) We introduce the function and write the result as A few comments are in order.First, F (τ ) is an odd function: This immediately follows from the fact that E τ is an even function in τ , as given in (3.23).Second, using the fact that E τ satisfies the equation (4.27), we can change the integration variable and rewrite F (τ ) as6 From this expression it is clear that in the limit of τ → ∞, F (τ ) becomes This leads us to define the normalized function Clearly, f (τ ) satisfies As we will see, this function f (τ ) is essentially the connected two-point SFF.In terms of f (τ ) the final result (4.29) is expressed as As an illustration we write down explicit formulas for Airy and JT gravity cases.
• In the Airy case, we have Then f (τ ) is calculated as For the case of β 1 and β 2 given in eq.(2.23), we have β tot = 2β and Then, f (τ ) in eq.(4.37) matches with the connected two-point function given by eq.(2.24).
• In the JT gravity case, we have E τ = arcsinh(τ ) Then f (τ ) is obtained as In fact, this reproduces the known result of the two-point function [34].
To sum up, we arrive at the conclusion that the n-point function with the constraint (4.9) is given by, in the leading order approximation, a certain sum of the function (4.28).As an illustration we will present the explicit form of the sum for small n in the next subsection.

Several examples
In this subsection we will present the explicit form of the sum of f (τ ) for two-and three-point functions.We will also present the sum for four-point function in a particular case.
• For the two point function, without loss of generality we can assume in the leading order approximation the first term gives The second term is evaluated by using the result in the previous subsection.Obviously, we have τ (max) = τ (1) = τ 1 , τ (min) = 0, and thus In total, we obtain This manifestly shows that f (τ ) is essentially the connected two-point SFF.

.48)
The third equality we have used Using the second property of f (τ ) in eq. ( 4.34), one can see that the plateau is correctly reproduced lim Eq. (4.48) reproduces the three point result presented in eq.(6.9) of [33].
• Let us next consider the four-point function.In this paper we only consider the case

.51)
In this case, for every integral I in (4.50) one can take τ (i) > 0, so that τ (min) = 0 and τ (max) is unambiguously determined.In the same way as above, we obtain Using the second property of f (τ ) in eq. ( 4.34), one can see that the plateau is correctly reproduced lim One can consider other cases as well and evaluate the integral using the prescription in the last subsection.While there is no technical difficulty, one needs to handle many cases separately in order to fix τ (max) and τ (min) for each constituent integral, which is rather laborious.

Summary and discussion
In this paper, we have analyzed n-point SFFs in WK topological gravity, and as its special cases, in Airy and JT gravities in detail.While 2-point SFF of JT gravity was studied recently [33,44,34,35], our focus has been on the late time behavior of the n-point SFFs, especially on the two typical timescales, t ∼ −2/3 and t ∼ −1 .Moreover, for Airy gravity we have done full analysis at all time-scale using the exact result by Okounkov [38].
Regarding one point SFF (or equivalently disconnected two point SFF), we have found that it decays by power law at t ∼ −2/3 ( → 0).However at much later time t ∼ −1 , it decays exponentially.For connected two-point SFF in Airy gravity, we have found that it first shows the t-linear ramp behavior, which changes into plateau at t = t higher genus ∼ −1 .The dip time, where the disconnected and connected twopoint SFFs become the same order, is at t ∼ −1/2 .These are the same behavior seen in JT gravity [32,33,34,35].Not only one-and two-point SFFs, we have discussed general n(≥ 2)-point SFF at t ∼ τ −1 with τ fixed.The crucial point in this late time is that n(≥ 2)-point SFF is characterized by a single function F (τ ) or its normalized form f (τ ) given by eq.(4.28) or (4.33), respectively.F (τ ) is characterized by E τ given in (3.23), which is implicitly determined by the classical eigenvalue distribution ρ 0 (E) through eq.(4.27).f (τ ) essentially represents the connected two-point SFF.Therefore, general n ≥ 2-point SFFs at late time t ∼ −1 are characterized by its connected two-point SFF and are specified by the classical eigenvalue density.These suggest that the qualitative behaviors of general n(≥ 2)point SFFs in Airy and JT gravities are very similar.From the viewpoint of general topological gravity, these theories correspond to different values of parameters t k yielding different classical eigenvalue densities.Therefore the qualitative behavior mentioned above seems universal, independent of the particular choices of t k for topological gravity.
The essential point is that the connected two point SFF determines the multiple n-point SFFs.This is due to the fact that the n-point SFFs are determined by the Christoffel-Darboux (CD) kernel.Especially at the late time t ∼ −1 , CD kernel is approximated by the sine-kernel which represents the eigenvalue repulsion.This sine-kernel is essential for the ramp and plateau.The fact that general n-point SFFs are characterized by the sine-kernel is due to the equivalence of WK topological gravity with the one-matrix model [30,31].In one-matrix model, one can always diagonalize it with Vandermonde determinant, which is the origin of the eigenvalue repulsion.In this way, general n-point SFF inherits the characteristics of two point SFF.
These imply that topological gravity which is dual to the one-matrix model is too simple in some sense since two-point SFF is all the non-trivial information we need.To go beyond topological gravity and to study more realistic gravity as is higher dimensional one, one might need to study multi-matrix models.See for the recent approach along these lines [47].However, these multi-matrices are difficult to solve analytically in general.On the gravity side, JT gravity can acquire bulk degrees of freedom if one includes matter fields.However, the long thin tube limit of the moduli space show divergence due to the negative Casimir energy [21,48,49].It is very interesting to study how the correspondence between JT gravity with matters and multi-matrix models works in such cases and figure out how to cure these divergences.

Figure 3 :
Figure3: Plot of the same functions as in Figure2, with more focus on the ramp region t 100.One observes the t-linear growth, which is due to the eigenvalue repulsion.

Figure 5 :
Figure 5: Plot of the three-point function for i τ i = 0 as a function of τ 1 with fixed τ 2 .In this figure we set = 0.01/ √ 2,β i = 2 (i = 1, 2, 3) and τ 2 = 1.The blue dots represent the real part of the exact three-point function given by eq.(2.35), while the red solid curve is the late time τ -scaling limit given by eq.(2.40).

Figure 6 :
Figure 6: Log-plot of the absolute value of the three-point function (blue) in eq.(2.43) as a function of τ , where i τ i = 0.For comparison, we also log-plot the absolute value of the two-point function (red) in eq.(2.22) with β 1 = β 2 = β + iτ −1 .Both show exponential decay.In this figure we set = 0.01/ √ 2 and β = 2.