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Invertibility of symmetric random matrices
dc.contributor.authorVershynin, Romanen_US
dc.date.accessioned2014-02-11T17:57:20Z
dc.date.available2015-04-16T14:24:20Zen_US
dc.date.issued2014-03en_US
dc.identifier.citationVershynin, Roman (2014). "Invertibility of symmetric random matrices ." Random Structures & Algorithms 44(2): 135-182.en_US
dc.identifier.issn1042-9832en_US
dc.identifier.issn1098-2418en_US
dc.identifier.urihttps://hdl.handle.net/2027.42/102712
dc.description.abstractWe study n × n symmetric random matrices H , possibly discrete, with iid above‐diagonal entries. We show that H is singular with probability at most exp ( − n c ) , and | | H − 1 | | = O ( n ) . Furthermore, the spectrum of H is delocalized on the optimal scale o ( n − 1 / 2 ) . These results improve upon a polynomial singularity bound due to Costello, Tao and Vu, and they generalize, up to constant factors, results of Tao and Vu, and Erdös, Schlein and Yau.Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 135‐182, 2014en_US
dc.publisherWiley‐Interscienceen_US
dc.subject.otherInvertibility Problemen_US
dc.subject.otherSymmetric Random Matricesen_US
dc.subject.otherSingularity Probabilityen_US
dc.titleInvertibility of symmetric random matricesen_US
dc.typeArticleen_US
dc.rights.robotsIndexNoFollowen_US
dc.subject.hlbsecondlevelMathematicsen_US
dc.subject.hlbtoplevelScienceen_US
dc.description.peerreviewedPeer Revieweden_US
dc.description.bitstreamurlhttp://deepblue.lib.umich.edu/bitstream/2027.42/102712/1/rsa20429.pdf
dc.identifier.doi10.1002/rsa.20429en_US
dc.identifier.sourceRandom Structures & Algorithmsen_US
dc.identifier.citedreferenceT. Tao and, V. Vu, From the Littlewood‐Offord problem to the Circular Law: Universality of the spectral distribution of random matrices, Bull Am Math Soc 46 2009, 377 – 396.en_US
dc.identifier.citedreferenceJ. Kahn, J. Komlós, and E. Szemerédi, On the probability that a random ± 1 ‐matrix is singular, J Am Math Soc, 8, ( 1995 ), 223 – 240.en_US
dc.identifier.citedreferenceR. Latala, Some estimates of norms of random matrices, Proc Am Math Soc 133 ( 2005 ), 1273 – 1282.en_US
dc.identifier.citedreferenceH. Nguyen, Inverse Littlewood‐Offord problems and the singularity of random symmetric matrices, Duke Math J (in press).en_US
dc.identifier.citedreferenceH. Nguyen, On the least singular value of random symmetric matrices, Submitted for Publication.en_US
dc.identifier.citedreferenceM. Rudelson, Invertibility of random matrices: Norm of the inverse, Ann Math 168 ( 2008 ), 575 – 600.en_US
dc.identifier.citedreferenceM. Rudelson and, R. Vershynin, The Littlewood‐Offord Problem and invertibility of random matrices, Adv Math 218 ( 2008 ), 600 – 633.en_US
dc.identifier.citedreferenceM. Rudelson and, R. Vershynin, Smallest singular value of a random rectangular matrix, Commun Pure Appl Math 62 ( 2009 ), 1707 – 1739.en_US
dc.identifier.citedreferenceM. Rudelson and, R. Vershynin, Non‐asymptotic theory of random matrices: Extreme singular values, In Proceedings of the International Congress of Mathematicians, Hyderabad, India, 2010.en_US
dc.identifier.citedreferenceA. Sidorenko, A correlation inequality for bipartite graphs, Graphs Combin 9 1993, 201 – 204.en_US
dc.identifier.citedreferenceT. Tao and, V. Vu, Additive combinatorics, In Cambridge Studies in Advanced Mathematics, 105, Cambridge University Press, Cambridge, 2006.en_US
dc.identifier.citedreferenceT. Tao and, V. Vu, Inverse Littlewood‐Offord theorems and the condition number of random discrete matrices, Ann Math 169 2009, 595 – 632.en_US
dc.identifier.citedreferenceT. Tao and, V. Vu, Random matrices: The distribution of the smallest singular values, Geom Funct Anal 20 2010, 260 – 297.en_US
dc.identifier.citedreferenceT. Tao and, V. Vu, Random matrices: Universality of local eigenvalue statistics up to the edge, Comm Math Phys, 298 ( 2010 ), 549 – 572.en_US
dc.identifier.citedreferenceT. Tao and, V. Vu, Random matrices: Universality of local eigenvalue statistics, Acta Math 206 ( 2011 ), 127 – 204.en_US
dc.identifier.citedreferenceT. Tao and V. Vu, Random matrices: Localization of the eigenvalues and the necessity of four moments, Acta Math Vietnam 36 ( 2011 ), 431 – 449.en_US
dc.identifier.citedreferenceR. Vershynin, Introduction to the non‐asymptotic analysis of random matrices, In Y. Eldar and G. Kutyniok (Editors), Compressed sensing: Theory and Applications, Cambridge University Press, in press.en_US
dc.identifier.citedreferenceN. Alon and J. Spencer, The probabilistic method, 2nd edition, Wiley‐Interscience, New York, 2000.en_US
dc.identifier.citedreferenceJ. Bourgain, V. Vu, and P. Wood, On the singularity probability of discrete random matrices, J Funct Anal, 258, 2010, 559 – 603.en_US
dc.identifier.citedreferenceK. Costello, Bilinear and quadratic variants on the Littlewood‐Offord problem, 2009, Submitted for Publication ( 2009 ).en_US
dc.identifier.citedreferenceK. Costello, T. Tao and, V. Vu, Random symmetric matrices are almost surely non‐singular, Duke Math J, 135, 2006, 395 – 413.en_US
dc.identifier.citedreferenceL. Erdös, B. Schlein, and H.‐T. Yau, Wegner estimate and level repulsion for Wigner random matrices, Int Math Res Not, 3, 2010, 436 – 479.en_US
dc.identifier.citedreferenceF. Götze, Asymptotic expansions for bivariate von Mises functionals, Z Wahrsch Verw Gebiete 50 1979, 333 – 355.en_US
dc.owningcollnameInterdisciplinary and Peer-Reviewed


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