Gravity without averaging

The paper studies matrix model ensembles which interpolate between the standard $U(N)$-invariant ensembles and trivial ensembles where the matrix is essentially fixed. Specifically, the authors are interested in how such deformation affects the Jackiw—Teitelboim(JT) gravity dual. They derive useful formulae both in the standard and the double-scaled cases.
The main result is the appearance of holes in the space-time which eventually tear it down when the random matrix is almost fixed. Interestingly, in Section 5 the authors point out that if not all the eigenvalues are fixed then the space-time picture continue to exists even if the fixing is complete(i.e. the eigenvalues are fixed with a delta-function rather than a smeared Gaussian). Also they argue that in such regime there is a local factorization in the path-integral.
The paper is definitely interesting and provides a useful set
of results regarding such matrix model approach to the factorization problem. While reading the manuscript I got two small questions:
1. In Section 5.1, the authors say “…when considering boundary energies close to $\kappa$… amplitude essentially factorizes”. I assume that the magnitude of non-factorization depends on the difference between the energy at hand and $\kappa$. I think it would be useful to have an estimate of how it scales with the energy difference.
2. Below eq. (6.3) the authors ask how to see the trivial identity (6.3) directly in JT. A naive proposal would be that the space-times ending on auxiliary holes are weighted with $H_0$ eigenvalues. Hence, upon the integration over $H_0$ they produce zero. I would be grateful to the authors if they could comment if this picture is correct.

Finally I noticed a few typos:
Figure 5 caption: “de” -> “the”
Is eq. (3.5) missing a sum over $m$ in the RHS?
Below eq. (3.23): “Tachyon” -> “tachyon”

1. Contains many different calculations approaching the problem from various points of view

2. Potentially interesting observation concerning the description of the theory of interest using a dual matrix model

1. The abstract is misleading

2. Many inconclusive sections, or sections whose conclusions do not address the stated motivation

3. Somewhat poorly written and overly colloquial English

4. The paper is rather technical and could contain a bit more background material for non-experts.

This article studies a matrix model that interpolates between a completely random matrix ensemble and one sharply peaked on a given Hamiltonian, $H_0$. This is achieved by introducing a Gaussian factor centered on $H_0$ and varying the width, $\sigma$, of the Gaussian. The completely random matrix model $(\sigma =\infty)$ is dual to JT gravity, and the authors attempt to understand the gravitational dual to the one at finite or zero $\sigma $.

I think it is fair to say that the authors do not succeed in finding the gravitational description they are looking for near $\sigma =0$. They do have some results in a perturbative expansion around infinite $\sigma $, both in a "closed string'' description where one obtains specific corrections to the dilaton potential, and an "open string'' one. Using the latter description, they argue that the specetime will be torn as $\sigma $ is made finite, and thus the gravitational description of the system breaks down as one tries to go towards $\sigma =0$.

They do make, however, two interesting observations concerning the $\sigma =0$ theory. The first observation is that by studying a certain dual matrix model, the $\sigma =0$ effects are captured by a very simple saddle point, which the authors also study perturbatively in $\sigma $ and argue it gives the expected physics. The second observation is that a different setup which involves fixing just a few (rather than all) eigenvalues of the target Hamiltonian does have a tractable gravitational interpretation near $\sigma =0$ in terms of spacetimes ending on "eigenbranes'' (corresponding to fixed energy boundary conditions). At small but finite $\sigma $ the boundary conditions become smeared, but going to finite $\sigma $ does not again appear possible due to the likelihood of spacetime-tearing transitions.

Despite this insuccess, I think this article could still be published as per the general acceptance criteria (all required) of Scipost, since it appears to contain several valuable calculations; however, it should be clear from the abstract and the main text what the paper achieves and what it does not. A major rewriting would thus be necessary.

1. Please change the abstract so as to make clear that the theory being studied is a matrix model, and its gravitational description is currently only understood near $\sigma=\infty$

2. Please rewrite the sections as needed in order to clarify what is being computed and achieved

3. Please check the equations; do e.g. (2.16) and (2.29) miss a summation sign?

4. The English can be considerably improved by using fewer colloquial words, e.g. get-> obtain, do -> perform and avoiding non-existing verbs such as "Hubbard-Stratonovich'ed"

5. Insert an explanatory/introductory sentence in needed places such that non-experts can also follow the calculations and understand why certain objects are being introduced.

1- Well-written paper with timely contents of current interest.
2- Thorough analysis with detailed computations.

None

In this paper the authors analyze a matrix model that interpolates between the standard randomness in the Gaussian matrix model and a model with fixed eigenvalues and hence no randomness.
They then generalize these results to the JT matrix model and interpret these gravitationally in terms of a deformed dilaton gravity model, and a tearing phase transition.

I have a couple of conceptual questions regarding the results.

* In section 3 the authors identify a deformed dilaton gravity model that corresponds to the ghost brane insertions they obtain as the deformation of the matrix model at large sigma.

The mapping from a genus zero spectral density to a dilaton gravity potential found in the references [65,66] is not sufficient to conclude that these models are entirely equal at the quantum level. For instance, in ref. [66] the authors employ a deformation theorem of Eynard and Orantin to fully show a match.
Is the identification of the authors in eq. (3.40) sufficient to conclude that this dilaton gravity model, when computing higher genus and wormhole contributions, fully matches with the matrix model the authors start with?
If so, then this should be emphasized and it would be interesting to see arguments. If not, then I believe the limitations of this identification would need to be properly emphasized.

The authors state that the main physical lesson here is that there exist dilaton gravity models that are less random.
However, this conclusion seems to only be obtained in the large L regime (where the genus zero string equation is used, and the mapping from this spectral density to the dilaton potential), and
at large sigma. Whereas the less random phase occurs at small sigma. Is it obvious why this interpretation must be correct, and in particular why the small sigma regime would have a dilaton gravity description?

* Have the authors attempted to understand the classical gravity features of the deformed dilaton gravity model eq (3.40)? For instance, what is the large \Phi asymptotics of eq (3.40)? This could be interpreted classically as the near-boundary asymptotics of the gravitational description.

* The authors utilize Kazakov's analysis of the tearing phase of gravity in section 4. Do the authors have intuition on what this tearing picture implies in Lorentzian signature? For instance, it looks quite dramatic for bulk probes.

* On p44, the authors mention a relation with the fuzzball program, as Lorentzian signature microstructure that is close to the horizon. However, their spacetime brane-like boundaries seem to be qualitatively very different from the worldsheet brane boundaries one uses in string theory.
Is there a direct relation or more concrete identification the authors have in mind here?

If the authors can briefly comment on these questions, the paper would be suitable for publication.

Clarifications in the text based on the above comments where deemed applicable.