---
_id: '17281'
abstract:
- lang: eng
text: We extend the free convolution of Brown measures of R-diagonal elements introduced
by Kösters and Tikhomirov [ 28] to fractional powers. We then show how this fractional
free convolution arises naturally when studying the roots of random polynomials
with independent coefficients under repeated differentiation. When the proportion
of derivatives to the degree approaches one, we establish central limit theorem-type
behavior and discuss stable distributions.
acknowledgement: This work was supported by the National Science Foundation [Grant
No. DMS-2143142 to S.O.]; and the European Research Council [Grant No. 101020331].The
third author acknowledges the support of the University of Colorado Boulder, where
a portion of this work was completed. The authors thank Martin Auer, Vadim Gorin,
Brian Hall, and Noah Williams for comments, corrections, and references. The authors
also wish to thank the anonymous referees for useful feedback and corrections.
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Andrew J
full_name: Campbell, Andrew J
id: 582b06a9-1f1c-11ee-b076-82ffce00dde4
last_name: Campbell
- first_name: Sean
full_name: O'Rourke, Sean
last_name: O'Rourke
- first_name: David T
full_name: Renfrew, David T
id: 4845BF6A-F248-11E8-B48F-1D18A9856A87
last_name: Renfrew
orcid: 0000-0003-3493-121X
citation:
ama: Campbell AJ, O’Rourke S, Renfrew DT. The fractional free convolution of R-diagonal
elements and random polynomials under repeated differentiation. International
Mathematics Research Notices. 2024;2024(13):10189-10218. doi:10.1093/imrn/rnae062
apa: Campbell, A. J., O’Rourke, S., & Renfrew, D. T. (2024). The fractional
free convolution of R-diagonal elements and random polynomials under repeated
differentiation. International Mathematics Research Notices. Oxford University
Press. https://doi.org/10.1093/imrn/rnae062
chicago: Campbell, Andrew J, Sean O’Rourke, and David T Renfrew. “The Fractional
Free Convolution of R-Diagonal Elements and Random Polynomials under Repeated
Differentiation.” International Mathematics Research Notices. Oxford University
Press, 2024. https://doi.org/10.1093/imrn/rnae062.
ieee: A. J. Campbell, S. O’Rourke, and D. T. Renfrew, “The fractional free convolution
of R-diagonal elements and random polynomials under repeated differentiation,”
International Mathematics Research Notices, vol. 2024, no. 13. Oxford University
Press, pp. 10189–10218, 2024.
ista: Campbell AJ, O’Rourke S, Renfrew DT. 2024. The fractional free convolution
of R-diagonal elements and random polynomials under repeated differentiation.
International Mathematics Research Notices. 2024(13), 10189–10218.
mla: Campbell, Andrew J., et al. “The Fractional Free Convolution of R-Diagonal
Elements and Random Polynomials under Repeated Differentiation.” International
Mathematics Research Notices, vol. 2024, no. 13, Oxford University Press,
2024, pp. 10189–218, doi:10.1093/imrn/rnae062.
short: A.J. Campbell, S. O’Rourke, D.T. Renfrew, International Mathematics Research
Notices 2024 (2024) 10189–10218.
corr_author: '1'
date_created: 2024-07-21T22:01:01Z
date_published: 2024-07-01T00:00:00Z
date_updated: 2025-09-08T08:16:32Z
day: '01'
ddc:
- '510'
department:
- _id: LaEr
doi: 10.1093/imrn/rnae062
external_id:
isi:
- '001198019500001'
file:
- access_level: open_access
checksum: f36a7dbf53f23d5833db711052e69b49
content_type: application/pdf
creator: dernst
date_created: 2024-07-22T06:40:19Z
date_updated: 2024-07-22T06:40:19Z
file_id: '17288'
file_name: 2024_IMRN_Campbell.pdf
file_size: 1233508
relation: main_file
success: 1
file_date_updated: 2024-07-22T06:40:19Z
has_accepted_license: '1'
intvolume: ' 2024'
isi: 1
issue: '13'
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
month: '07'
oa: 1
oa_version: Published Version
page: 10189-10218
publication: International Mathematics Research Notices
publication_identifier:
eissn:
- 1687-0247
issn:
- 1073-7928
publication_status: published
publisher: Oxford University Press
quality_controlled: '1'
scopus_import: '1'
status: public
title: The fractional free convolution of R-diagonal elements and random polynomials
under repeated differentiation
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 2024
year: '2024'
...
---
_id: '181'
abstract:
- lang: eng
text: We consider large random matrices X with centered, independent entries but
possibly di erent variances. We compute the normalized trace of f(X)g(X∗) for
f, g functions analytic on the spectrum of X. We use these results to compute
the long time asymptotics for systems of coupled di erential equations with random
coe cients. We show that when the coupling is critical, the norm squared of the
solution decays like t−1/2.
acknowledgement: The work of the second author was also partially supported by the
Hausdorff Center of Mathematics.
article_processing_charge: No
arxiv: 1
author:
- first_name: László
full_name: Erdös, László
id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
last_name: Erdös
orcid: 0000-0001-5366-9603
- first_name: Torben H
full_name: Krüger, Torben H
id: 3020C786-F248-11E8-B48F-1D18A9856A87
last_name: Krüger
orcid: 0000-0002-4821-3297
- first_name: David T
full_name: Renfrew, David T
id: 4845BF6A-F248-11E8-B48F-1D18A9856A87
last_name: Renfrew
orcid: 0000-0003-3493-121X
citation:
ama: Erdös L, Krüger TH, Renfrew DT. Power law decay for systems of randomly coupled
differential equations. SIAM Journal on Mathematical Analysis. 2018;50(3):3271-3290.
doi:10.1137/17M1143125
apa: Erdös, L., Krüger, T. H., & Renfrew, D. T. (2018). Power law decay for
systems of randomly coupled differential equations. SIAM Journal on Mathematical
Analysis. Society for Industrial and Applied Mathematics . https://doi.org/10.1137/17M1143125
chicago: Erdös, László, Torben H Krüger, and David T Renfrew. “Power Law Decay for
Systems of Randomly Coupled Differential Equations.” SIAM Journal on Mathematical
Analysis. Society for Industrial and Applied Mathematics , 2018. https://doi.org/10.1137/17M1143125.
ieee: L. Erdös, T. H. Krüger, and D. T. Renfrew, “Power law decay for systems of
randomly coupled differential equations,” SIAM Journal on Mathematical Analysis,
vol. 50, no. 3. Society for Industrial and Applied Mathematics , pp. 3271–3290,
2018.
ista: Erdös L, Krüger TH, Renfrew DT. 2018. Power law decay for systems of randomly
coupled differential equations. SIAM Journal on Mathematical Analysis. 50(3),
3271–3290.
mla: Erdös, László, et al. “Power Law Decay for Systems of Randomly Coupled Differential
Equations.” SIAM Journal on Mathematical Analysis, vol. 50, no. 3, Society
for Industrial and Applied Mathematics , 2018, pp. 3271–90, doi:10.1137/17M1143125.
short: L. Erdös, T.H. Krüger, D.T. Renfrew, SIAM Journal on Mathematical Analysis
50 (2018) 3271–3290.
date_created: 2018-12-11T11:45:03Z
date_published: 2018-01-01T00:00:00Z
date_updated: 2025-04-15T08:05:02Z
day: '01'
department:
- _id: LaEr
doi: 10.1137/17M1143125
ec_funded: 1
external_id:
arxiv:
- '1708.01546'
isi:
- '000437018500032'
intvolume: ' 50'
isi: 1
issue: '3'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1708.01546
month: '01'
oa: 1
oa_version: Published Version
page: 3271 - 3290
project:
- _id: 258DCDE6-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '338804'
name: Random matrices, universality and disordered quantum systems
- _id: 258F40A4-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: M02080
name: Structured Non-Hermitian Random Matrices
publication: SIAM Journal on Mathematical Analysis
publication_status: published
publisher: 'Society for Industrial and Applied Mathematics '
publist_id: '7740'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Power law decay for systems of randomly coupled differential equations
type: journal_article
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
volume: 50
year: '2018'
...