# Statistics of Green's functions on a disordered Cayley tree and the validity of forward scattering approximation

### Submission summary

 As Contributors: Ivan Khaymovich · Pavel Nosov Arxiv Link: https://arxiv.org/abs/2108.10326v2 (pdf) Date submitted: 2021-11-04 20:12 Submitted by: Khaymovich, Ivan Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Mathematical Physics Quantum Physics Approaches: Theoretical, Computational

### Abstract

The accuracy of the forward scattering approximation for two-point Green's functions of the Anderson localization model on the Cayley tree is studied. A relationship between the moments of the Green's function and the largest eigenvalue of the linearized transfer-matrix equation is proved in the framework of the supersymmetric functional-integral method. The new large-disorder approximation for this eigenvalue is derived and its accuracy is established. Using this approximation the probability distribution of the two-point Green's function is found and compared with that in the forward scattering approximation (FSA). It is shown that FSA overestimates the role of resonances and thus the probability for the Green's function to be significantly larger than its typical value. The error of FSA increases with increasing the distance between points in a two-point Green's function.

###### Current status:
Editor-in-charge assigned

### Author comments upon resubmission

Dear Editor,

We are grateful to the referee for the high evaluation of our paper and for thorough careful reading of the manuscript. His/her critical remarks have allowed us to significantly improve the presentation of our work.

In the revised version of our manuscript, we address all the points mentioned by the referee.
The point-to-point reply to the referee is given below the report, while the list of changes is placed below.

Sincerely yours, the authors.

### List of changes

1. On page 6 before Eq. (15) we have clarified the control parameter of the Anderson transition in Eqs. (9-14).
2. We have developed and described a physical meaning of the parameter $\Omega_0(v)$ on page 5 around Eq. (11), on page 6 after Eq. (13), on page 6 after Eq. (15), and in the beginning of page 7.
3. We have added a paragraph at the end of Section 2 on page 7 explaining the physical meaning of the Lyapunov exponents in the problem and the difference of a single-orbital model on the Cayley tree with respect to the non-linear sigma model.
4. We have described the roadmap of calculations in the beginning of Sec. 3 on page 7.
5. We have clarified the non-commutativity of limits of $\eta\to 0$ and $N\to\infty$ on page 8 after Eq. (19).
6. The discussion on the boundary condition and the order of limits has been added on page 5 after Eq. (10).
7. The discussion on the deviations of the forward scattering approximation from the exact analytical results of our paper has been added on pages 19 and 20.
8. The new perspective of research opened by our paper has been discussed in the Conclusion.
9. We have added the clarification of the meaning of the arguments t and v of $\Omega_r$ on page 5 before Eq. (11).
10. In order to avoid repeating notations, we have changed them in Eqs. (80-81).
11. Several relevant references (including the ones mentioned by the Referee) have been added to the Introduction and the Conclusion sections.
12. Several clarifications, minor amendments, and corrections of typos have been done throughout the manuscript.

### Submission & Refereeing History

Resubmission 2108.10326v2 on 4 November 2021
Submission 2108.10326v1 on 25 August 2021

## Reports on this Submission

### Report

In the new version of their article, the authors have greatly clarified the points I had raised. I find the given physical interpretations very interesting, especially the new section 4.

I am not sure I agree with the statement in the conclusion that the agreement of numerical simulations (Refs. [6,7]) with the forward scattering approximation, in dimension two and in the strongly localized regime, is specific to this dimension. On the contrary, I believe that this agreement should persist in higher dimensions. My interpretation is that in the strongly localized regime, it is the competition between the paths that dominates (contrary to the weakly localized regime, where it is the interference between the paths which dominates). This competition is done by a global optimization, in a similar way to the physics of directed polymers. But I recognize that this remains an open problem which could be addressed in 3D or on random regular graphs for example. The question therefore arises as to whether the effects described by the authors are specific to 1D and to the Cayley tree where a single path connects two points of the network, or if these effects are important even in generic graphs where many paths contribute.

This discussion shows that the authors' interesting approach not only answers but also opens up interesting questions, in addition to being a technical  tour de force ''. So I can only recommend the publication of the manuscript in SciPost.

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### Author:  Ivan Khaymovich  on 2021-11-12  [id 1935]

(in reply to Report 1 on 2021-11-11)
Category:
remark

We thank the referee for his/her thorough reading of our revised manuscript and for high evaluation of changes.

We agree that the issue with FSA in higher dimensions in an open question, especially analytically.
We have only meant that in numerical simulations it seems that a specific Tracy-Widom form works only for 2D.