Large complex correlated Wishart matrices: the Pearcey kernel and expansion at the hard edge

Large complex correlated Wishart matrices: the Pearcey kernel and expansion at the hard edge

We study the eigenvalue behaviour of large complex correlated Wishart matrices near an interior point of the limiting spectrum where the density vanishes (cusp point), and refine the existing results at the hard edge as well. More precisely, under mild assumptions for the population covariance matri...

Full description

Saved in:
Bibliographic Details
Published in:Electronic journal of probability Vol. 21; no. none; pp. 1 - 36
Main Authors: Hachem, Walid, Hardy, Adrien, Najim, Jamal
Format: Journal Article
Language:English
Published: Institute of Mathematical Statistics (IMS) 01-01-2016
Series:https://projecteuclid.org/journals/electronic-journal-of-probability/volume-21/issue-none/Large-complex-correlated-Wishart-matrices--the-Pearcey-kernel-and/10.1214/15-EJP4441.full
Subjects:
Mathematics
Probability
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Abstract We study the eigenvalue behaviour of large complex correlated Wishart matrices near an interior point of the limiting spectrum where the density vanishes (cusp point), and refine the existing results at the hard edge as well. More precisely, under mild assumptions for the population covariance matrix, we show that the limiting density vanishes at generic cusp points like a cube root, and that the local eigenvalue behaviour is described by means of the Pearcey kernel if an extra decay assumption is satisfied. As for the hard edge, we show that the density blows up like an inverse square root at the origin. Moreover, we provide an explicit formula for the 1/N correction term for the fluctuation of the smallest random eigenvalue.
AbstractList We study the eigenvalue behaviour of large complex correlated Wishart matrices near an interior point of the limiting spectrum where the density vanishes (cusp point), and refine the existing results at the hard edge as well. More precisely, under mild assumptions for the population covariance matrix, we show that the limiting density vanishes at generic cusp points like a cube root, and that the local eigenvalue behaviour is described by means of the Pearcey kernel if an extra decay assumption is satisfied. As for the hard edge, we show that the density blows up like an inverse square root at the origin. Moreover, we provide an explicit formula for the 1/N correction term for the fluctuation of the smallest random eigenvalue.
Author Najim, Jamal
Hachem, Walid
Hardy, Adrien
Author_xml – sequence: 1
  givenname: Walid
  surname: Hachem
  fullname: Hachem, Walid
– sequence: 2
  givenname: Adrien
  surname: Hardy
  fullname: Hardy, Adrien
– sequence: 3
  givenname: Jamal
  surname: Najim
  fullname: Najim, Jamal
BackLink https://hal.science/hal-01834495$$DView record in HAL
BookMark eNpNkE9Lw0AUxBepYFu9-An2qhDdTXaTjbdSWqsE7EHxIiyvLy9NNH_KbpD229vaIp5mGH4zhxmxQdu1xNi1FHcylOpe6mD2vFRKyTM2lMJEQaxMOvjnL9jI-08hQqFiM2QfGbg1ceyaTU3bvTpHNfSU8_fKl-B63kDvKiT_wPuS-JLAIe34F7mWag5tzmm7gdZXXcuh_2X2tX2ar-mSnRdQe7o66Zi9zWev00WQvTw-TSdZgJE2fYA6zgsNiIXRQBAjCoxlkqrUCJFAaIwhVYRJIlCKWEssVgZMDmmaJKtIyGjMbo67JdR246oG3M52UNnFJLOHTEgTKZXq7wN7e2TRdd47Kv4KUtjDh1Zqe_ow-gGeKWVY
CitedBy_id crossref_primary_10_1214_15_AOP1022
crossref_primary_10_2140_pmp_2024_5_321
crossref_primary_10_1142_S2010326321500398
crossref_primary_10_1111_sapm_12279
crossref_primary_10_1002_cpa_22202
crossref_primary_10_1080_10652469_2021_1952200
crossref_primary_10_1007_s00023_023_01266_5
crossref_primary_10_11948_2018_965
crossref_primary_10_1214_22_AAP1890
crossref_primary_10_1007_s00220_019_03657_4
crossref_primary_10_1214_19_AIHP1040
crossref_primary_10_1016_j_jat_2021_105633
crossref_primary_10_1016_j_aim_2022_108291
crossref_primary_10_1016_j_physd_2022_133262
crossref_primary_10_1007_s00220_021_03986_3
Cites_doi 10.1214/009117906000000917
10.1103/PhysRevE.58.7176
10.1016/S0924-8099(06)80038-7
10.1007/s002200050027
10.1142/S0129167X15500937
10.1016/j.jmva.2009.12.004
10.1214/009117905000000233
10.1214/07-AAP454
10.1006/jmva.1995.1058
10.1214/16-AAP1193
10.1093/imrn/rnr066
10.1214/15-AOP1022
10.1007/978-3-0348-8401-3
10.1142/S2010326316500015
10.1017/CBO9780511801334
10.1214/aos/1009210544
10.1214/15-AAP1121
10.1016/0550-3213(93)90126-A
10.1007/s00440-016-0730-4
10.1214/154957806000000078
10.1051/proc/201551009
10.1016/j.physd.2012.08.016
10.1007/s00220-006-0159-1
ContentType Journal Article
Copyright Distributed under a Creative Commons Attribution 4.0 International License
Copyright_xml – notice: Distributed under a Creative Commons Attribution 4.0 International License
DBID AAYXX
CITATION
1XC
DOI 10.1214/15-EJP4441
DatabaseName CrossRef
Hyper Article en Ligne (HAL)
DatabaseTitle CrossRef
DatabaseTitleList
DeliveryMethod fulltext_linktorsrc
Discipline Mathematics
EISSN 1083-6489
EndPage 36
ExternalDocumentID oai_HAL_hal_01834495v1
10_1214_15_EJP4441
GroupedDBID -~9
29G
2WC
5GY
5VS
AAYXX
ACGFO
ACIPV
AENEX
AFFOW
ALMA_UNASSIGNED_HOLDINGS
CITATION
E3Z
EJD
GR0
GROUPED_DOAJ
KWQ
M~E
OK1
P2P
RBV
REM
RNS
RPE
TR2
XSB
1XC
C1A
H13
ID FETCH-LOGICAL-c358t-c56df5accf85aea6cc0c6179498007a2888e4f2770c10651cfb8a8da9977b3013
ISSN 1083-6489
IngestDate Wed Sep 18 06:57:57 EDT 2024
Fri Aug 23 02:03:15 EDT 2024
IsDoiOpenAccess false
IsOpenAccess true
IsPeerReviewed true
IsScholarly true
Issue none
Language English
License Distributed under a Creative Commons Attribution 4.0 International License: http://creativecommons.org/licenses/by/4.0
LinkModel OpenURL
MergedId FETCHMERGED-LOGICAL-c358t-c56df5accf85aea6cc0c6179498007a2888e4f2770c10651cfb8a8da9977b3013
ORCID 0000-0001-7452-2838
0000-0001-8499-2761
OpenAccessLink https://doi.org/10.1214/15-ejp4441
PageCount 36
ParticipantIDs hal_primary_oai_HAL_hal_01834495v1
crossref_primary_10_1214_15_EJP4441
PublicationCentury 2000
PublicationDate 2016-1-1
2016
PublicationDateYYYYMMDD 2016-01-01
PublicationDate_xml – month: 01
  year: 2016
  text: 2016-1-1
  day: 01
PublicationDecade 2010
PublicationSeriesTitle https://projecteuclid.org/journals/electronic-journal-of-probability/volume-21/issue-none/Large-complex-correlated-Wishart-matrices--the-Pearcey-kernel-and/10.1214/15-EJP4441.full
PublicationTitle Electronic journal of probability
PublicationYear 2016
Publisher Institute of Mathematical Statistics (IMS)
Publisher_xml – name: Institute of Mathematical Statistics (IMS)
References 22
23
24
25
26
27
28
29
30
31
10
32
11
33
12
34
13
35
14
15
16
17
18
19
1
2
3
4
5
6
7
8
9
20
21
References_xml – ident: 13
  doi: 10.1214/009117906000000917
– ident: 7
  doi: 10.1103/PhysRevE.58.7176
– ident: 12
– ident: 35
– ident: 21
  doi: 10.1016/S0924-8099(06)80038-7
– ident: 33
– ident: 14
– ident: 20
  doi: 10.1007/s002200050027
– ident: 10
  doi: 10.1142/S0129167X15500937
– ident: 26
  doi: 10.1016/j.jmva.2009.12.004
– ident: 9
– ident: 3
  doi: 10.1214/009117905000000233
– ident: 28
  doi: 10.1214/07-AAP454
– ident: 31
  doi: 10.1006/jmva.1995.1058
– ident: 24
  doi: 10.1214/16-AAP1193
– ident: 4
  doi: 10.1093/imrn/rnr066
– ident: 17
  doi: 10.1214/15-AOP1022
– ident: 16
  doi: 10.1007/978-3-0348-8401-3
– ident: 29
  doi: 10.1142/S2010326316500015
– ident: 34
– ident: 32
– ident: 30
– ident: 2
  doi: 10.1017/CBO9780511801334
– ident: 22
  doi: 10.1214/aos/1009210544
– ident: 6
  doi: 10.1214/15-AAP1121
– ident: 15
  doi: 10.1016/0550-3213(93)90126-A
– ident: 23
  doi: 10.1007/s00440-016-0730-4
– ident: 19
  doi: 10.1214/154957806000000078
– ident: 18
  doi: 10.1051/proc/201551009
– ident: 1
  doi: 10.1016/j.physd.2012.08.016
– ident: 5
  doi: 10.1007/s00220-006-0159-1
– ident: 11
  doi: 10.1142/S0129167X15500937
– ident: 8
– ident: 27
– ident: 25
SSID ssj0020468
Score 2.2559423
Snippet We study the eigenvalue behaviour of large complex correlated Wishart matrices near an interior point of the limiting spectrum where the density vanishes (cusp...
SourceID hal
crossref
SourceType Open Access Repository
Aggregation Database
StartPage 1
SubjectTerms Mathematics
Probability
Title Large complex correlated Wishart matrices: the Pearcey kernel and expansion at the hard edge
URI https://hal.science/hal-01834495
Volume 21
hasFullText 1
inHoldings 1
isFullTextHit
isPrint
link http://sdu.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwtV1Lb9NAEF415QIHxFOUl1bALZoSO97Y5hbRRKGkVaQUlQOStVmvVRU1RXGC2n_PzM76kdJDOXBJrM3GWn_fPmbG8xDiQ6pVZBOjoAiUQQUlV6BT1HnSRaCKtK_sIieD22QeH39PDkbRaKdTFUlt2v4r09iGXFPk7D-wXd8UG_AaOcdPZB0_78T7lFy72VPcXuH3ykWroFh5SnWWV-vuhUvKz65wJHXOXLjjdfenXS0tJw6wV7hHlM5R2YU6dik0q3vTb2jUVNBp5Z-gCjWc-_u62d1wZrh5d4pSf940e6eBYU7JlmuztD7nCs-H-sI_LAFnS-dZRcNGcKhmMNuQ7MbgPckbg7ZkHkdJnNSjA98KlwW0Rod9eGuGkHZzNwFheUkW3rEDETyI0IAIHkSoQARAeMBDCAwhOF_RMefUiOhKwehwFqFIuE9vO9qGFo4AZbW77bZxVGfTpWwt5C7B-bRJHziaV-YbPkxQvIVBxCWS9u0tbf4E4hjxrZXGx0nQkks4T8xfJ97NB2nO9cqXYTKcZ7ODcTb9cvx1-9c6v_hkOM3OcKr3Aiq7kqrfQUfcC3HrDisjhTdh9CIfXOqfwif8xUF8bIawJeJ1zqo3FE5iO3kkHnpVSw55jTwWO3b5RDxokC2fih-OaOmJlg3R0hMtK6I_Sfyb9ERLJloi0bJeK1KvXR9aK5LWyjPxbTw6-TwBX28ETF8lazBqkBdKG1MkSls9MKZnBnRgpahVxTpMksRGRRjHPROg5B6YYpHoJNcp6lALPCj7z8UuTdMXQpp-jopLYdIkRkBRDNdmUOC-19O5iYrY7on3FUTZL04rk5E6jkBmgco8kHviHbFSdbidqZd36fRK3KcpzbbD12J3vdrYN6JT5pu3juA_94jRtg
link.rule.ids 230,315,782,786,866,887,4028,27932,27933,27934
linkProvider Directory of Open Access Journals
openUrl ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Large+complex+correlated+Wishart+matrices%3A+the+Pearcey+kernel+and+expansion+at+the+hard+edge&rft.jtitle=Electronic+journal+of+probability&rft.au=Hachem%2C+Walid&rft.au=Hardy%2C+Adrien&rft.au=Najim%2C+Jamal&rft.series=https%3A%2F%2Fprojecteuclid.org%2Fjournals%2Felectronic-journal-of-probability%2Fvolume-21%2Fissue-none%2FLarge-complex-correlated-Wishart-matrices--the-Pearcey-kernel-and%2F10.1214%2F15-EJP4441.full&rft.date=2016&rft.pub=Institute+of+Mathematical+Statistics+%28IMS%29&rft.issn=1083-6489&rft.eissn=1083-6489&rft.volume=21&rft.issue=1&rft.spage=1&rft.epage=36&rft_id=info:doi/10.1214%2F15-EJP4441&rft.externalDBID=HAS_PDF_LINK&rft.externalDocID=oai_HAL_hal_01834495v1
thumbnail_l http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=1083-6489&client=summon
thumbnail_m http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=1083-6489&client=summon
thumbnail_s http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=1083-6489&client=summon