Cavity and replica methods for the spectral density of sparse symmetric random matrices

1- very pedagogical and clearly written notes.

1- lack of a broad overview on the motivations to study the problem and its applications.
2- lack of a perspective on the open problems and interesting new directions.

The authors have written a pedagogical introduction to the replica and cavity methods for computing the spectrum of sparse random matrices.
The notes start from the Edward-Jones formula, and build progressively these tools. They focus on the Bray-Rodgers computation (section 4) and on the successive developments (section 5).
I found the notes very pedagogical and clearly written. I have only two main suggestions.
The problem, as well as the tools explained in the notes, have found applications in many different contexts. Despite this fact the motivation section (essentially the first six lines of page 3) is very short and not very useful for a young researcher starting to look at this problem. Correspondingly the bibliography appears to be rather short. I believe that it could be very useful to add a bit more of motivations/applications/bibliography.
The second suggestion concerns the conclusions. I think that it would be useful to mention the relevant open problems and to give some perspectives on the field.
Apart from this, I think that the notes meet the acceptance criteria and therefore I recommend them for publication in the “SciPost lecture notes” series.

1) The paper clearly discusses the two main techniques in the field.

2) All calculations are very detailed.

3) There is a growing interest in the applications of random matrix theory to networks.

1) The authors do not present a list of open problems or discuss promising research lines in the field.

2) Some sections present various technical details that are irrelevant for a person entering the field.

In this paper the authors review the cavity method and the replica method to
compute the spectral density of sparse (symmetric) random matrices. In my opinion, this
work complies with its main purpose, i.e., to present a detailed account of the
two main techniques to study the eigenvalue distribution of sparse random
matrices. The present paper is a useful source of detailed information for
students and young researchers that are entering the field of sparse random matrices. Thus, I recommend the present paper for publication in SciPost.

However, I have a couple of comments and suggestions that the authors should take
into consideration before the paper is published.

1) I'd like to suggest a couple of references that would enrich the present manuscript. The
cavity method for random matrices has been originally discussed in the following works:

P. Cizeau and J. P. Bouchaud, Phys. Rev. E 50, 1810 (1994)

A. Cavagna, I. Giardina and G. Parisi, Phys. Rev. Lett. 83 (1999)

Although the formulation of the cavity approach in the above papers is slightly different than in reference [31], the central ideas
are already there. Concerning localization and the study of the IPR for undirected graphs, besides reference [41], the authors
could also mention

G. Biroli, G. Semerjian and M. Tarzia, Prog. Theor. Phys. Suppl. 184, 187 (2010)

whose central ideas are similar to [41]. As far as I am aware, the first time that the replica-symmetric Gaussian ansatz appears in the
context of sparse random matrices is the following work

D. S. Dean, J. Phys. A: Math. Gen. 35, L153 (2002)

Finally, the authors could be more precise when mentioning the difficulties with the Bray-Rodgers equation. I agree
that a full analytic solution is not available, but this equation has been solved numerically in the above paper of Cavagna et al.
In this context, there is also the recent work

https://arxiv.org/abs/2102.09629

which solves numerically a pair of equations that is analogous to eqs. (60) and (61) that yield the Bray-Rodgers equation.

2) Equation (2) has the merit to show that the spectral density follows from
the averaged free energy of an analogous interacting system. However, the spectral density
can be also computed from the imaginary part of the resolvent, and the cavity method can be directly applied
to the resolvent matrix, which is a central object in random matrix theory. In my opinion, the cavity approach on the resolvent is more general, since the
resolvent elements allow to compute other spectral observables and study, for instance, eigenvector localization.
Given that the present paper is addressed to students and young researchers, I think it'd be interesting to briefly discuss the connection between the spectral density and the resolvent, to mention
that the cavity method can be applied to the resolvent, and to discuss the connection between the complex variances of [31] and the diagonal part of the resolvent matrix.

3) I got a bit frustrated with the main outcome of section 4.4. The authors
perform an involved perturbative expansion in powers of $1/c$, but in the end they present results only
for the spectral density when $c \rightarrow \infty$. I think it'd be nice to show the $1/c$ correction to the Wigner
law in eq. (117). If this is not possible, what is the point of expanding everything up to $O(1/c)$?
I'd also suggest the authors make an effort to shorten section 4.4 (some equations, like eq. (105), can be written in a single line).

4) Since the paper is mostly addressed to those entering the field, it'd be useful to include
in the final section a list of open problems or research lines that the authors find interesting to be further explored.

Minor comments:

1) One should mention that $c_{ii}=0$ around eq. (17).

2) The sentence "In this very sparse regime...thermodynamic limit is taken." on page 8 is a bit ambiguous. I suggest the authors
reformulate it in a clearer way.

3) Please, specify that eq. (24) is exact only for $N \rightarrow \infty$.

4) Maybe it is a good idea to highlight in section 3.2 that the cavity equations have been rigorously proven in [32].

5) The authors introduce the spectral density of a single instance in eq. (32). I wonder whether this quantity
has any interest, given that the spectral density of a finite system is a collection of $\delta$ peaks. In
practice, one always ends up taking the limit $N \rightarrow \infty$. In addition, the rhs of eq. (32) is valid strictly for $N \rightarrow \infty$.
Thus, in case the authors decide to keep eq. (32), it is better to highlight that this equation is an approximation for finite $N$.

6) It seems that a factor $N$ is missing in eq. (63).

7) I do not understand "which diverges for any finite $z$" just below eq. (100).

8) I found the explanation around eqs. (113) and (114) quite confusing. Maybe it is better to discuss
how the spectral density transforms as soon as you introduce the change of variables, eqs. (87-89).

9) The authors write "These effects are much...degree $c$ increases" on page 31. It is not possible
to draw this conclusion from figure 4, which presents results only for $\epsilon=10^{-3}$. It'd be interesting to complete figure 4 with results
for $\epsilon=10^{-300}$. It'd be even better to see a graph of the point where the continuous spectrum vanishes as a function of $c$.

10) Section 6 discusses too many implementation details that, in my
opinion, are not very helpful. The introduction of so many
quantities to describe precisely how to implement the algorithm
(in particular, the discussion about measurement sweeps) distracts
the reader. Are $\mathcal{M}$ and $N_P$ the same quantity?