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Replica approach to the generalized Rosenzweig-Porter model
by Davide Venturelli, Leticia F. Cugliandolo, Grégory Schehr, Marco Tarzia
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Gregory Schehr · Davide Venturelli |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202301_00042v1 (pdf) |
| Date accepted: | 2023-03-09 |
| Date submitted: | 2023-01-31 11:50 |
| Submitted by: | Venturelli, Davide |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
The generalized Rosenzweig-Porter model with real (GOE) off-diagonal entries arguably constitutes the simplest random matrix ensemble displaying a phase with fractal eigenstates, which we characterize here by using replica methods. We first derive analytical expressions for the average spectral density in the limit in which the size $N$ of the matrix is large but finite. We then focus on the number of eigenvalues in a finite interval and compute its cumulant generating function as well as the level compressibility, i.e., the ratio of the first two cumulants: these are useful tools to describe the local level statistics. In particular, the level compressibility is shown to be described by a universal scaling function, which we compute explicitly, when the system is probed over scales of the order of the Thouless energy. Interestingly, the same scaling function is found to describe the level compressibility of the complex (GUE) Rosenzweig-Porter model in this regime. We confirm our results with numerical tests.
Author comments upon resubmission
we thank you for handling our paper which we would like to resubmit to SciPost Physics. Our paper was reviewed by three referees, whom we want to thank for their careful reading of our manuscript and for their helpful suggestions, which have undoubtedly improved our work.
We were glad to read that the three referees acknowledged the scientific quality of our results. Unfortunately, we cannot agree with the judgment of Referee 1 who suggested to publish our paper in SciPost Core, since he considered that “none of the ``expectations'' for SciPost Physics is clearly met''. In fact, this opinion is in strong contrast with the report of Referee 2, who wrote explicitly that our paper ``meets all the criteria for acceptance, and meets (if not exceeds) the general expectation of clarity, correctness, and broad interest''. Similarly, Referee 3 also started his report by writing that ``[Our] calculation opens the possibility to understand the results for the level compressibility conjectured in [23] and checked numerically in [31]''.
After the extensive changes detailed below, and having provided a point-to-point response to all the issues raised by the three Referees, we hope that our manuscript will be accepted for publication in SciPost Physics.
Yours sincerely, the authors.
List of changes
We have attached a comprehensive file (Reply_to_referee_reports.pdf) in our reply to the Referee reports. In that document we give a detailed response to all the issues raised by the three Referees, indicating the location of the corresponding changes in the revised manuscript.
We have also highlighted in blue color, for convenience, all the changes in the manuscript.
These edits appear in the Abstract, Introduction and Conclusions, as well as in Sections 1.1-2, 2.0-1, 2.4, 3.0, 3.2, 3.5-6, B. A whole new section (Appendix F) has been included in response to comment 1 of the third Referee.
Published as SciPost Phys. 14, 110 (2023)