Singularity of discrete random matrices
Let ξ be a non-constant real-valued random variable with finite support and let M n ( ξ ) denote an n × n random matrix with entries that are independent copies of ξ . For ξ which is not uniform on its support, we show that P [ M n ( ξ ) is singular ] = P [ zero row or column ] + ( 1 + o n ( 1 ) ) P...
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| Veröffentlicht in: | Geometric and functional analysis 2021-10, Vol.31 (5), p.1160-1218 |
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| creator | Jain, Vishesh Sah, Ashwin Sawhney, Mehtaab |
| description | Let
ξ
be a non-constant real-valued random variable with finite support and let
M
n
(
ξ
)
denote an
n
×
n
random matrix with entries that are independent copies of
ξ
. For
ξ
which is not uniform on its support, we show that
P
[
M
n
(
ξ
)
is singular
]
=
P
[
zero row or column
]
+
(
1
+
o
n
(
1
)
)
P
[
two equal (up to sign) rows or columns
]
,
thereby confirming a folklore conjecture. As special cases, we obtain:
For
ξ
=
Bernoulli
(
p
)
with fixed
p
∈
(
0
,
1
/
2
)
,
P
[
M
n
(
ξ
)
is singular
]
=
2
n
(
1
-
p
)
n
+
(
1
+
o
n
(
1
)
)
n
(
n
-
1
)
(
p
2
+
(
1
-
p
)
2
)
n
,
which determines the singularity probability to
two
asymptotic terms. Previously, no result of such precision was available in the study of the singularity of random matrices. The first asymptotic term confirms a conjecture of Litvak and Tikhomirov.
For
ξ
=
Bernoulli
(
p
)
with fixed
p
∈
(
1
/
2
,
1
)
,
P
[
M
n
(
ξ
)
is singular
]
=
(
1
+
o
n
(
1
)
)
n
(
n
-
1
)
(
p
2
+
(
1
-
p
)
2
)
n
.
Previously, only the much weaker upper bound of
(
p
+
o
n
(
1
)
)
n
was known due to the work of Bourgain–Vu–Wood.
For
ξ
which is uniform on its support:
We show that
P
[
M
n
(
ξ
)
is singular
]
=
(
1
+
o
n
(
1
)
)
n
P
[
two rows or columns are equal
]
.
Perhaps more importantly, we provide a sharp analysis of the contribution of the ‘compressible’ part of the unit sphere to the lower tail of the smallest singular value of
M
n
(
ξ
)
. |
| doi_str_mv | 10.1007/s00039-021-00580-6 |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_2614796884</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2614796884</sourcerecordid><originalsourceid>FETCH-LOGICAL-c319t-584d8d871fe17b342d621bbf90fcd85c45543d04f4015247106140b67dd0485b3</originalsourceid><addsrcrecordid>eNp9kE1LxDAQhoMouK7-AU8FD56iM_nuURa_YMGDCt5C2yRLl227Ju1h_73RCt48zTA87zvwEHKJcIMA-jYBAC8pMKQA0gBVR2SBggE1pYbjvAMqKgT_OCVnKW0zLqWQC3L92vabaVfFdjwUQyhcm5roR1_EqndDV3TVGNvGp3NyEqpd8he_c0neH-7fVk90_fL4vLpb04ZjOVJphDPOaAwedc0Fc4phXYcSQuOMbET-yh2IIAAlExpBoYBaaZePRtZ8Sa7m3n0cPiefRrsdptjnl5ZlVJfKGJEpNlNNHFKKPth9bLsqHiyC_RZiZyE2C7E_QqzKIT6HUob7jY9_1f-kvgBWY2Dw</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2614796884</pqid></control><display><type>article</type><title>Singularity of discrete random matrices</title><source>SpringerNature Journals</source><creator>Jain, Vishesh ; Sah, Ashwin ; Sawhney, Mehtaab</creator><creatorcontrib>Jain, Vishesh ; Sah, Ashwin ; Sawhney, Mehtaab</creatorcontrib><description>Let
ξ
be a non-constant real-valued random variable with finite support and let
M
n
(
ξ
)
denote an
n
×
n
random matrix with entries that are independent copies of
ξ
. For
ξ
which is not uniform on its support, we show that
P
[
M
n
(
ξ
)
is singular
]
=
P
[
zero row or column
]
+
(
1
+
o
n
(
1
)
)
P
[
two equal (up to sign) rows or columns
]
,
thereby confirming a folklore conjecture. As special cases, we obtain:
For
ξ
=
Bernoulli
(
p
)
with fixed
p
∈
(
0
,
1
/
2
)
,
P
[
M
n
(
ξ
)
is singular
]
=
2
n
(
1
-
p
)
n
+
(
1
+
o
n
(
1
)
)
n
(
n
-
1
)
(
p
2
+
(
1
-
p
)
2
)
n
,
which determines the singularity probability to
two
asymptotic terms. Previously, no result of such precision was available in the study of the singularity of random matrices. The first asymptotic term confirms a conjecture of Litvak and Tikhomirov.
For
ξ
=
Bernoulli
(
p
)
with fixed
p
∈
(
1
/
2
,
1
)
,
P
[
M
n
(
ξ
)
is singular
]
=
(
1
+
o
n
(
1
)
)
n
(
n
-
1
)
(
p
2
+
(
1
-
p
)
2
)
n
.
Previously, only the much weaker upper bound of
(
p
+
o
n
(
1
)
)
n
was known due to the work of Bourgain–Vu–Wood.
For
ξ
which is uniform on its support:
We show that
P
[
M
n
(
ξ
)
is singular
]
=
(
1
+
o
n
(
1
)
)
n
P
[
two rows or columns are equal
]
.
Perhaps more importantly, we provide a sharp analysis of the contribution of the ‘compressible’ part of the unit sphere to the lower tail of the smallest singular value of
M
n
(
ξ
)
.</description><identifier>ISSN: 1016-443X</identifier><identifier>EISSN: 1420-8970</identifier><identifier>DOI: 10.1007/s00039-021-00580-6</identifier><language>eng</language><publisher>Cham: Springer International Publishing</publisher><subject>Analysis ; Asymptotic properties ; Compressibility ; Mathematics ; Mathematics and Statistics ; Random variables ; Singularities ; Upper bounds</subject><ispartof>Geometric and functional analysis, 2021-10, Vol.31 (5), p.1160-1218</ispartof><rights>The Author(s), under exclusive licence to Springer Nature Switzerland AG 2021</rights><rights>The Author(s), under exclusive licence to Springer Nature Switzerland AG 2021.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c319t-584d8d871fe17b342d621bbf90fcd85c45543d04f4015247106140b67dd0485b3</citedby><cites>FETCH-LOGICAL-c319t-584d8d871fe17b342d621bbf90fcd85c45543d04f4015247106140b67dd0485b3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s00039-021-00580-6$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00039-021-00580-6$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27924,27925,41488,42557,51319</link.rule.ids></links><search><creatorcontrib>Jain, Vishesh</creatorcontrib><creatorcontrib>Sah, Ashwin</creatorcontrib><creatorcontrib>Sawhney, Mehtaab</creatorcontrib><title>Singularity of discrete random matrices</title><title>Geometric and functional analysis</title><addtitle>Geom. Funct. Anal</addtitle><description>Let
ξ
be a non-constant real-valued random variable with finite support and let
M
n
(
ξ
)
denote an
n
×
n
random matrix with entries that are independent copies of
ξ
. For
ξ
which is not uniform on its support, we show that
P
[
M
n
(
ξ
)
is singular
]
=
P
[
zero row or column
]
+
(
1
+
o
n
(
1
)
)
P
[
two equal (up to sign) rows or columns
]
,
thereby confirming a folklore conjecture. As special cases, we obtain:
For
ξ
=
Bernoulli
(
p
)
with fixed
p
∈
(
0
,
1
/
2
)
,
P
[
M
n
(
ξ
)
is singular
]
=
2
n
(
1
-
p
)
n
+
(
1
+
o
n
(
1
)
)
n
(
n
-
1
)
(
p
2
+
(
1
-
p
)
2
)
n
,
which determines the singularity probability to
two
asymptotic terms. Previously, no result of such precision was available in the study of the singularity of random matrices. The first asymptotic term confirms a conjecture of Litvak and Tikhomirov.
For
ξ
=
Bernoulli
(
p
)
with fixed
p
∈
(
1
/
2
,
1
)
,
P
[
M
n
(
ξ
)
is singular
]
=
(
1
+
o
n
(
1
)
)
n
(
n
-
1
)
(
p
2
+
(
1
-
p
)
2
)
n
.
Previously, only the much weaker upper bound of
(
p
+
o
n
(
1
)
)
n
was known due to the work of Bourgain–Vu–Wood.
For
ξ
which is uniform on its support:
We show that
P
[
M
n
(
ξ
)
is singular
]
=
(
1
+
o
n
(
1
)
)
n
P
[
two rows or columns are equal
]
.
Perhaps more importantly, we provide a sharp analysis of the contribution of the ‘compressible’ part of the unit sphere to the lower tail of the smallest singular value of
M
n
(
ξ
)
.</description><subject>Analysis</subject><subject>Asymptotic properties</subject><subject>Compressibility</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Random variables</subject><subject>Singularities</subject><subject>Upper bounds</subject><issn>1016-443X</issn><issn>1420-8970</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2021</creationdate><recordtype>article</recordtype><recordid>eNp9kE1LxDAQhoMouK7-AU8FD56iM_nuURa_YMGDCt5C2yRLl227Ju1h_73RCt48zTA87zvwEHKJcIMA-jYBAC8pMKQA0gBVR2SBggE1pYbjvAMqKgT_OCVnKW0zLqWQC3L92vabaVfFdjwUQyhcm5roR1_EqndDV3TVGNvGp3NyEqpd8he_c0neH-7fVk90_fL4vLpb04ZjOVJphDPOaAwedc0Fc4phXYcSQuOMbET-yh2IIAAlExpBoYBaaZePRtZ8Sa7m3n0cPiefRrsdptjnl5ZlVJfKGJEpNlNNHFKKPth9bLsqHiyC_RZiZyE2C7E_QqzKIT6HUob7jY9_1f-kvgBWY2Dw</recordid><startdate>20211001</startdate><enddate>20211001</enddate><creator>Jain, Vishesh</creator><creator>Sah, Ashwin</creator><creator>Sawhney, Mehtaab</creator><general>Springer International Publishing</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20211001</creationdate><title>Singularity of discrete random matrices</title><author>Jain, Vishesh ; Sah, Ashwin ; Sawhney, Mehtaab</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c319t-584d8d871fe17b342d621bbf90fcd85c45543d04f4015247106140b67dd0485b3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2021</creationdate><topic>Analysis</topic><topic>Asymptotic properties</topic><topic>Compressibility</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Random variables</topic><topic>Singularities</topic><topic>Upper bounds</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Jain, Vishesh</creatorcontrib><creatorcontrib>Sah, Ashwin</creatorcontrib><creatorcontrib>Sawhney, Mehtaab</creatorcontrib><collection>CrossRef</collection><jtitle>Geometric and functional analysis</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Jain, Vishesh</au><au>Sah, Ashwin</au><au>Sawhney, Mehtaab</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Singularity of discrete random matrices</atitle><jtitle>Geometric and functional analysis</jtitle><stitle>Geom. Funct. Anal</stitle><date>2021-10-01</date><risdate>2021</risdate><volume>31</volume><issue>5</issue><spage>1160</spage><epage>1218</epage><pages>1160-1218</pages><issn>1016-443X</issn><eissn>1420-8970</eissn><abstract>Let
ξ
be a non-constant real-valued random variable with finite support and let
M
n
(
ξ
)
denote an
n
×
n
random matrix with entries that are independent copies of
ξ
. For
ξ
which is not uniform on its support, we show that
P
[
M
n
(
ξ
)
is singular
]
=
P
[
zero row or column
]
+
(
1
+
o
n
(
1
)
)
P
[
two equal (up to sign) rows or columns
]
,
thereby confirming a folklore conjecture. As special cases, we obtain:
For
ξ
=
Bernoulli
(
p
)
with fixed
p
∈
(
0
,
1
/
2
)
,
P
[
M
n
(
ξ
)
is singular
]
=
2
n
(
1
-
p
)
n
+
(
1
+
o
n
(
1
)
)
n
(
n
-
1
)
(
p
2
+
(
1
-
p
)
2
)
n
,
which determines the singularity probability to
two
asymptotic terms. Previously, no result of such precision was available in the study of the singularity of random matrices. The first asymptotic term confirms a conjecture of Litvak and Tikhomirov.
For
ξ
=
Bernoulli
(
p
)
with fixed
p
∈
(
1
/
2
,
1
)
,
P
[
M
n
(
ξ
)
is singular
]
=
(
1
+
o
n
(
1
)
)
n
(
n
-
1
)
(
p
2
+
(
1
-
p
)
2
)
n
.
Previously, only the much weaker upper bound of
(
p
+
o
n
(
1
)
)
n
was known due to the work of Bourgain–Vu–Wood.
For
ξ
which is uniform on its support:
We show that
P
[
M
n
(
ξ
)
is singular
]
=
(
1
+
o
n
(
1
)
)
n
P
[
two rows or columns are equal
]
.
Perhaps more importantly, we provide a sharp analysis of the contribution of the ‘compressible’ part of the unit sphere to the lower tail of the smallest singular value of
M
n
(
ξ
)
.</abstract><cop>Cham</cop><pub>Springer International Publishing</pub><doi>10.1007/s00039-021-00580-6</doi><tpages>59</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 1016-443X |
| ispartof | Geometric and functional analysis, 2021-10, Vol.31 (5), p.1160-1218 |
| issn | 1016-443X 1420-8970 |
| language | eng |
| recordid | cdi_proquest_journals_2614796884 |
| source | SpringerNature Journals |
| subjects | Analysis Asymptotic properties Compressibility Mathematics Mathematics and Statistics Random variables Singularities Upper bounds |
| title | Singularity of discrete random matrices |
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