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

by Daniil Kochergin, Ivan M. Khaymovich, Olga Valba, Alexander Gorsky

This Submission thread is now published as

Submission summary

Authors (as registered SciPost users): Ivan Khaymovich
Submission information
Preprint Link:  (pdf)
Date accepted: 2024-03-11
Date submitted: 2023-12-05 07:06
Submitted by: Khaymovich, Ivan
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
  • Mathematical Physics
  • Quantum Physics
Approaches: Theoretical, Computational


In this work we analytically explain the origin of the mobility edge in the ensemble of random regular graphs (RRG), with the connectivity $d$ and the fraction $\beta$ of disordered nodes, the location of which is under control. It is shown that the mobility edge in the spectrum survives in a certain range of parameters $(d,\beta)$ at infinitely large uniformly distributed disorder. The critical curve separating extended and localized states is derived analytically and confirmed numerically. The duality in the localization properties between the sparse and extremely dense RRG has been found and understood. The mobility edge physics has been analyzed numerically for the above partially disordered RRG, perturbed by the non-reciprocity parameter of node as well as by the enhanced number of short cycles, usually almost absent on RRG.

Author comments upon resubmission

Dear Editor,

We have addressed all Referees' questions and comments.
Please see the replies to both referees in the previous version, followed by the manuscript, where changes are highlighted with red font.

Thus, we resubmit our manuscript to SciPost Physics and believe that our work meets all the criteria of the journal.

Sincerely yours,
the authors.

List of changes

1 - the abstract has been reformulated to make it clearer.
2 - in the introduction, the references on MBL and typically protected modes there have been added with the corresponding discussion.
3 - the motivation of the generalisations of the model has been clarified in the end of the introduction.
4 - all the figure captions have been extended to make it clearer.
5 - an additional energy level, starting at $E=d$, has been described in the text.
6 - Fig. 2 clarifying the finite-size effects up to $N=3 \cdot 10^4$ with the corresponding discussion has been added to the text.
7 - some deviations from Poisson statistics in Fig. 3 have been discussed in the text.
8 - the contribution to the density of states (DOS) from the localized states, barely seen in Figs. 4 and 5(a), has been discussed in detail and shown in Fig. 3(b) at intermediate disorder values.
9 - the qualitative summary of the analytical results, based on the graph of clean nodes, has been added to Sec. 3.
10 - the self-averaging property of the Green's function at large effective connectivity has been discussed before Eq. (14).
11 - the mobility edge has been explicitly added as Eq. (18).
12 - some discussions of possible origins of DOS deviations from the analytical predictions have been added to the end of Sec. 3.
13 - the motivation of the generalisations of the model and the robustness of the mobility edge to perturbations has been added to the end of Sec. 3 and the beginning of Sec. 4.
14 - the references [65, 66] on the localisation in Erdos-Renyi graph due to the focusing of node degree have been added and their relations to our results have been discussed.
15 - Sec. 5 has been moved to the Appendix B and restructured, including some figures.
16 - discussions in the Conclusion have been slightly reformulated.
17 - Discussions in Appendix A have been extended and clarified.

Published as SciPost Phys. 16, 106 (2024)

Reports on this Submission

Anonymous Report 2 on 2024-1-9 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2309.05691v2, delivered 2024-01-09, doi: 10.21468/SciPost.Report.8386


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).

Requested changes

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.

  • validity: good
  • significance: good
  • originality: good
  • clarity: good
  • formatting: reasonable
  • grammar: acceptable

Author:  Ivan Khaymovich  on 2024-02-19  [id 4316]

(in reply to Report 2 on 2024-01-09)
answer to question

We thank the referee for thorough submitted referee report.

Please see our detailed reply with the updated manuscript version with the red-colored main changes.



Anonymous Report 1 on 2023-12-6 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2309.05691v2, delivered 2023-12-06, doi: 10.21468/SciPost.Report.8227


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"?

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

  • validity: ok
  • significance: good
  • originality: good
  • clarity: low
  • formatting: acceptable
  • grammar: below threshold

Author:  Ivan Khaymovich  on 2023-12-10  [id 4180]

(in reply to Report 1 on 2023-12-06)
answer to question

We thank the referee for thorough and quickly submitted referee report.

Please see our reply with the updated manuscript version with the red-colored main changes.



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