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Replica method for eigenvalues of real Wishart product matrices

by Jacob A. Zavatone-Veth, Cengiz Pehlevan

This Submission thread is now published as

Submission summary

Authors (as registered SciPost users): Jacob Zavatone-Veth
Submission information
Preprint Link: scipost_202212_00021v1  (pdf)
Date accepted: 2023-03-07
Date submitted: 2022-12-08 20:16
Submitted by: Zavatone-Veth, Jacob
Submitted to: SciPost Physics Lecture Notes
Ontological classification
Academic field: Physics
  • Mathematical Physics
  • Statistical and Soft Matter Physics
Approach: Theoretical


We show how the replica method can be used to compute the asymptotic eigenvalue spectrum of a real Wishart product matrix. For unstructured factors, this provides a compact, elementary derivation of a polynomial condition on the Stieltjes transform first proved by M\"uller [IEEE Trans. Inf. Theory. 48, 2086-2091 (2002)]. We then show how this computation can be extended to ensembles where the factors have correlated rows. Finally, we derive polynomial conditions on the average values of the minimum and maximum eigenvalues, which match the results obtained by Akemann, Ipsen, and Kieburg [Phys. Rev. E 88, 052118 (2013)] for the complex Wishart product ensemble.

Published as SciPost Phys. Core 6, 026 (2023)

Author comments upon resubmission

We thank the referees for their careful reading of our submission and for their helpful suggestions. We have revised our manuscript to address all of their specific concerns. We have uploaded a point-by-point response to the referees' specific concerns as a response to their comments.

We hope that the referees agree that these changes have improved the readability of the paper, and both expanded and clarified the relevance of its results. We are happy to divert this manuscript to SciPost Physics Lecture Notes if the referees and the editor believe that a venue more focused on pedagogy than SciPost Physics Core would be more suitable, and have accordingly resubmitted it to that journal.

List of changes

In brief, we have:
- Added a new analysis of the bulk spectrum for ensembles with correlated rows. This computation is easily performed using the replica approach, further illustrating its utility. To our knowledge, this result has not previously been reported outside the depth-1 case (Burda et al., PRE 2005).
- Reorganized the manuscript to better separate background information from results.
- Expanded our discussions of related work and of the possible utility of the replica approach in obtaining results not previously established using rigorous methods.
- Made the requested notational changes, and added a section where we define our conventions.

Please see our detailed response letter, uploaded as a reply to the referees' comments, for a more detailed list of changes.

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