Robust extended states in Anderson model on partially disordered random regular graphs

see my previous report

see my report

The main scientific question I raised in my previous report concerned the localization effects induced by fluctuating connectivity. I must express my dissatisfaction with the authors' response on this matter. It is well-established that fluctuating connectivity can induce localization of eigenstates in sparse adjacency matrices, as evidenced in Ref. G. Biroli and R. Monasson, J. Phys. A: Math. Gen. 32, L255 (1999), where it is stated that "localized eigenvectors are centered on geometrical defects, that is on sites whose number of neighbors is much smaller or much larger than the average connectivity." The condition for such an effect is not too stringent, given approximately by |q-c| > √q, where q is the anomalous connectivity of the site and c is the average connectivity.

It appears to me that for small enough beta, this effect should play a role. Hence, the rigorous arguments presented in Refs. [65,66] should be relevant here, appropriately adapted to the present context. The recent work by M. Tarzia, PHYSICAL REVIEW B 105, 174201 (2022), is also pertinent, in my opinion. Essentially, the authors make two approximations for d << N: infinite disorder and fixed connectivity d*. A precise numerical study of the effects of finite disorder and fluctuating connectivity is necessary to explain the deviations observed between the numerical data and the analytic predictions, especially at small beta.

The case of a dense graph with d ≈ N is less clear. The mapping of such a Random Regular Graph (RRG) to a Cayley tree should not be valid in this limit, so it is not surprising that a direct use of the previous cavity approach does not work. I still believe the authors could provide more details about their approach for this part to be clearer.

In my previous report, as well as that of the other referee, I highlighted formulations that were difficult to understand or grammatically incorrect. The corrections made by the authors do not entirely satisfy me. The abstract, as well as various passages in the articles, are still not well-written in my opinion. I provide below some examples, which are not exhaustive:

-The abstract states that the authors deal with the "ensemble of random regular graphs (RRG), with the connectivity d and the fraction β of disordered nodes." I would suggest stating that the authors consider partially disordered random regular graphs, i.e., with a fraction β of the sites being disordered, while the rest remain clean. The authors study the influence of the connectivity d and fraction β on the localization/delocalization properties of the states.
- The last sentence about the perturbations does not seem necessary to include in the abstract, as the material has been placed in the appendix. In any case, I think the authors should change the "non-reciprocity parameter for edges" to non-Hermitian directed hoppings (non-reciprocity is not clear to me).
- In the caption of Fig. 5, "Black solid (dashed) line denotes the mobility edge, |E_{M E} |^2 = 4(1 − β)(d − 1), Eq. (19), (|E_{M E} + 1|^2 =4(1 − β)(d − 1))." can be confusing: Firstly, the equation is not (19) but (18), and this refers to the case described in (a) while it should also be stated that the parenthesis refers to the case (b).
- Still in this caption of Fig. 5, the panel (c) corresponds to "the central band |E| < E_{ME}": this information is not sufficient to understand what is represented. Is what is plotted D_2 at E=0 or an average of D_2 over the central band? Eq. (19) gives 1-beta_c=1/(d-1) but for RRG it should be 1/d instead, so what was used in Fig. 5 (a)?
-The end of the first paragraph in section 4 is still confusing: "but for any values of the total vertex degree d of the graph." d* < d and d* >> 1, therefore d >> 1.
-I don't think the expression "complimentary graph" is the right one; rather, it should be "complementary graph."
-d_eff just before the conclusion should be written as d_\text{eff}.
-In the color plots, I still believe that a Perceptually Uniform Color Map should be used instead of the color map chosen by the authors. As I mentioned earlier, the chosen color map emphasizes D_2=0.5, which is not particularly distinctive in their data.
- In Fig. 4, the authors should indicate that the black line corresponds to the generalized Kesten-McKay formula.
- Below Eq. (13), the relative fluctuations sigma_d/d* = sqrt(beta)/sqrt((1-beta)d) are not small for small beta, even at d=10 or 20.
-In Eq. (19), either write the left equation or the one on the right side of the arrow. For RRG it should be d in the denominator, not d-1.

Once again, this is not an exhaustive list. Moreover, there is a significant amount of formulations that are not typical English. This can be polished very easily using, for example, ChatGPT or another such tool (which I have used to polish my report).

Citations for Fluctuating Connectivity and Localization Effects:
Kindly include appropriate references to prior works discussing the localization effects induced by fluctuating connectivity. A relevant study is provided by G. Biroli and R. Monasson in J. Phys. A: Math. Gen. 32, L255 (1999). Additionally, cite studies that have numerically investigated these effects in conjunction with finite disorder, such as the work by M. Tarzia in PHYSICAL REVIEW B 105, 174201 (2022).

Editing the Manuscript:
Requesting a thorough review and editing of the manuscript in accordance with the feedback provided in my report. Key areas of focus include refining the abstract to accurately describe the study as focusing on 'partially disordered random regular graphs.' Ensure that figure captions are clear, equations are correctly referenced, and overall grammatical issues are addressed for enhanced readability.

I thank the authors for addressing my scientific questions and agree that the
results merit publication in SciPost Physics. However, the presentation of
those results still can and should be improved.

In particular:

New Figure 2:
The authors added another Figure to the manuscript which illustrates the
fractal dimension (panel a)) and the density of states (panel b)) at moderate
disorder strength W=30. I believe this addition is useful, as it connects the
article to finite disorder strengths, complementing the previous focus on the
W→∞ limit. Do I understand correctly that the authors assume that the localized
states around E=0 in Fig. 5, central peak in the density of states in Fig. 2b)
and 4c)d) as well as the dip in the fractal dimension in Fig. 2a) are due to
isolated clean nodes and pairs of clean nodes? Something to this effect is
stated at the bottom of page 9, but it should probably be stated right were
Fig. 2 is discussed in the text. Instead the text is currently very vague about
these deviations, even though the black (theoretical) curve strongly differs
from the red (numerical) curve in panel b). The text describing Fig. 4 is
similarly vague. Furthermore, there is no explanation of the N→∞ extrapolation
in Panel a).

Section 4:
I can now follow the motivation of including the discussion on the duality
between sparse and dense RRG. However, the first couple sentences of this
sections are very difficult to understand. What do you mean by "bare degree d"?

Abstract:
I find the abstract now easier to understand. In the first sentence, I
recommend omitting the subclause ", the location of which is under control."
since it sounds awkward and seems unnecessary. The last sentence is still
hard to understand and contains grammatical errors. Maybe splitting it up and
giving more context would help.

Grammar and style:
The manuscript still contains frequent grammatical mistakes and awkward
phrasings. For example, in the first two sentences of the main text, there are
two missing definite articles. These mistakes distract the reader from the
scientific content. As the other referee pointed out, nowadays there are plenty
of tools widely available that can point out grammatical and stylistic
problems.