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Schnelli, Kevin

Open this publication in new window or tab >>LOCAL SINGLE RING THEOREM ON OPTIMAL SCALE### Bao, Zhigang

### Erdos, Laszlo

### Schnelli, Kevin

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_0_j_idt188_some",{id:"formSmash:j_idt184:0:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_0_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_0_j_idt188_otherAuthors",{id:"formSmash:j_idt184:0:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_0_j_idt188_otherAuthors",multiple:true}); 2019 (English)In: Annals of Probability, ISSN 0091-1798, E-ISSN 2168-894X, Vol. 47, no 3, p. 1270-1334Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Institute of Mathematical Statistics, 2019
##### Keywords

Non-Hermitian random matrices, local eigenvalue density, single ring theorem, free convolution
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:kth:diva-251704 (URN)10.1214/18-AOP1284 (DOI)000466616100003 ()
#####

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##### Note

Hong Kong Univ Sci & Technol, Dept Math, Kowloon, Clear Water Bay, Hong Kong, Peoples R China..

IST Austria, Campus 1, A-3400 Klosterneuburg, Austria..

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.

Let U and V be two independent N by N random matrices that are distributed according to Haar measure on U(N). Let Sigma be a nonnegative deterministic N by N matrix. The single ring theorem [Ann. of Math. (2) 174 (2011) 1189-1217] asserts that the empirical eigenvalue distribution of the matrix X : = U Sigma V* converges weakly, in the limit of large N, to a deterministic measure which is supported on a single ring centered at the origin in C. Within the bulk regime, that is, in the interior of the single ring, we establish the convergence of the empirical eigenvalue distribution on the optimal local scale of order N-1/2+epsilon and establish the optimal convergence rate. The same results hold true when U and V are Haar distributed on O(N).

QC 20190521

Available from: 2019-05-21 Created: 2019-05-21 Last updated: 2019-05-21Bibliographically approvedOpen this publication in new window or tab >>Local law and Tracy-Widom limit for sparse random matrices### Lee, Ji Oon

### Schnelli, Kevin

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_some",{id:"formSmash:j_idt184:1:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_otherAuthors",{id:"formSmash:j_idt184:1:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_otherAuthors",multiple:true}); 2018 (English)In: Probability theory and related fields, ISSN 0178-8051, E-ISSN 1432-2064, Vol. 171, no 1-2, p. 543-616Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Springer Berlin/Heidelberg, 2018
##### Keywords

Local law, Sparse random matrices, Erdos-Renyi graph
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:kth:diva-229009 (URN)10.1007/s00440-017-0787-8 (DOI)000432129600012 ()
#####

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##### Funder

EU, European Research Council, 338804
##### Note

Korea Adv Inst Sci & Technol, Daejeon, South Korea..

KTH. IST Austria, Klosterneuburg, Austria..

We consider spectral properties and the edge universality of sparse random matrices, the class of random matrices that includes the adjacency matrices of the ErdAs-R,nyi graph model G(N, p). We prove a local law for the eigenvalue density up to the spectral edges. Under a suitable condition on the sparsity, we also prove that the rescaled extremal eigenvalues exhibit GOE Tracy-Widom fluctuations if a deterministic shift of the spectral edge due to the sparsity is included. For the adjacency matrix of the ErdAs-R,nyi graph this establishes the Tracy-Widom fluctuations of the second largest eigenvalue when p is much larger than wth a deterministic shift of order (Np)(-1)..

QC 20180531

Available from: 2018-05-31 Created: 2018-05-31 Last updated: 2018-05-31Bibliographically approvedOpen this publication in new window or tab >>Convergence rate for spectral distribution of addition of random matrices### Bao, Zhigang

### Erdos, Laszlo

### Schnelli, Kevin

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_some",{id:"formSmash:j_idt184:2:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_otherAuthors",{id:"formSmash:j_idt184:2:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_otherAuthors",multiple:true}); 2017 (English)In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 319, p. 251-291Article in journal (Refereed) Published
##### Place, publisher, year, edition, pages

Academic Press, 2017
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:kth:diva-216603 (URN)10.1016/j.aim.2017.08.028 (DOI)000412150400010 ()2-s2.0-85028323616 (Scopus ID)
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##### Note

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.

QC 20171115

Available from: 2017-11-15 Created: 2017-11-15 Last updated: 2017-11-15Bibliographically approvedOpen this publication in new window or tab >>Local Law of Addition of Random Matrices on Optimal Scale### Bao, Z.

### Erdős, L.

### Schnelli, Kevin

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_some",{id:"formSmash:j_idt184:3:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_otherAuthors",{id:"formSmash:j_idt184:3:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_otherAuthors",multiple:true}); 2017 (English)In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 349, no 3, p. 947-990Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Springer-Verlag New York, 2017
##### National Category

Probability Theory and Statistics
##### Identifiers

urn:nbn:se:kth:diva-202111 (URN)10.1007/s00220-016-2805-6 (DOI)000393696700005 ()2-s2.0-84995751210 (Scopus ID)
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##### Note

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.

The eigenvalue distribution of the sum of two large Hermitian matrices, when one of them is conjugated by a Haar distributed unitary matrix, is asymptotically given by the free convolution of their spectral distributions. We prove that this convergence also holds locally in the bulk of the spectrum, down to the optimal scales larger than the eigenvalue spacing. The corresponding eigenvectors are fully delocalized. Similar results hold for the sum of two real symmetric matrices, when one is conjugated by Haar orthogonal matrix.

QC 20170314

Available from: 2017-03-14 Created: 2017-03-14 Last updated: 2017-11-29Bibliographically approved