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Schnelli, Kevin
Publications (4 of 4) Show all publications
Bao, Z., Erdos, L. & Schnelli, K. (2019). LOCAL SINGLE RING THEOREM ON OPTIMAL SCALE. Annals of Probability, 47(3), 1270-1334
Open this publication in new window or tab >>LOCAL SINGLE RING THEOREM ON OPTIMAL SCALE
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]

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).

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 ()
Note

QC 20190521

Available from: 2019-05-21 Created: 2019-05-21 Last updated: 2019-05-21Bibliographically approved
Lee, J. O. & Schnelli, K. (2018). Local law and Tracy-Widom limit for sparse random matrices. Probability theory and related fields, 171(1-2), 543-616
Open this publication in new window or tab >>Local law and Tracy-Widom limit for sparse random matrices
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]

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)..

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 ()
Funder
EU, European Research Council, 338804
Note

QC 20180531

Available from: 2018-05-31 Created: 2018-05-31 Last updated: 2018-05-31Bibliographically approved
Bao, Z., Erdos, L. & Schnelli, K. (2017). Convergence rate for spectral distribution of addition of random matrices. Advances in Mathematics, 319, 251-291
Open this publication in new window or tab >>Convergence rate for spectral distribution of addition of random matrices
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)
Note

QC 20171115

Available from: 2017-11-15 Created: 2017-11-15 Last updated: 2017-11-15Bibliographically approved
Bao, Z., Erdős, L. & Schnelli, K. (2017). Local Law of Addition of Random Matrices on Optimal Scale. Communications in Mathematical Physics, 349(3), 947-990
Open this publication in new window or tab >>Local Law of Addition of Random Matrices on Optimal Scale
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]

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.

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)
Note

QC 20170314

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