The Marčenko-Pastur law for sparse random bipartite biregular graphs
We prove that the empirical spectral distribution of a (dL, dR)‐biregular, bipartite random graph, under certain conditions, converges to a symmetrization of the Marčenko‐Pastur distribution of random matrix theory. This convergence is not only global (on fixed‐length intervals) but also local (on i...
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| Published in | Random structures & algorithms Vol. 48; no. 2; pp. 313 - 340 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
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Blackwell Publishing Ltd
01.03.2016
Wiley Subscription Services, Inc |
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| ISSN | 1042-9832 1098-2418 |
| DOI | 10.1002/rsa.20581 |
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| Abstract | We prove that the empirical spectral distribution of a (dL, dR)‐biregular, bipartite random graph, under certain conditions, converges to a symmetrization of the Marčenko‐Pastur distribution of random matrix theory. This convergence is not only global (on fixed‐length intervals) but also local (on intervals of increasingly smaller length). Our method parallels the one used previously by Dumitriu and Pal (2012). © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 48, 313–340, 2016 |
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| AbstractList | We prove that the empirical spectral distribution of a (dL, dR)-biregular, bipartite random graph, under certain conditions, converges to a symmetrization of the Marcenko-Pastur distribution of random matrix theory. This convergence is not only global (on fixed-length intervals) but also local (on intervals of increasingly smaller length). Our method parallels the one used previously by Dumitriu and Pal (2012). © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 48, 313-340, 2016 We prove that the empirical spectral distribution of a (dL, dR)‐biregular, bipartite random graph, under certain conditions, converges to a symmetrization of the Marčenko‐Pastur distribution of random matrix theory. This convergence is not only global (on fixed‐length intervals) but also local (on intervals of increasingly smaller length). Our method parallels the one used previously by Dumitriu and Pal (2012). © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 48, 313–340, 2016 |
| Author | Johnson, Tobias Dumitriu, Ioana |
| Author_xml | – sequence: 1 givenname: Ioana surname: Dumitriu fullname: Dumitriu, Ioana email: dumitriu@math.washington.edu organization: Department of Mathematics, University of Washington, Box 354350, Washington, 98195, Seattle – sequence: 2 givenname: Tobias surname: Johnson fullname: Johnson, Tobias email: tobias.johnson@usc.edu organization: Department of Mathematics, University of Southern California, KAP 108, California, 90089, Los Angeles |
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| References | Z. Bai and J. W. Silverstein, Spectral analysis of large dimensional random matrices, Springer Series in Statistics, Second edition, Springer, New York, 2010. L. Erdős, B. Schlein, and H.-T. Yau, Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices, Ann Probab 37 (2009), 815-852. B. D. McKay, N. C. Wormald, and B. Wysocka, Short cycles in random regular graphs, Electron J Comb 11 (2004), Research Paper 66, 12. L. Erdős, J. Ramírez, B. Schlein, and H.-T. Yau, Universality of sine-kernel for Wigner matrices with a small Gaussian perturbation, Electron J Probab 15 (2010), 526-603. T. Tao and V. Vu, Random matrices: Universality of local eigenvalue statistics up to the edge, Comm Math Phys 298 (2010), 549-572. L. Erdős, B. Schlein, and H.-T. Yau, Local semicircle law and complete delocalization for Wigner random matrices, Comm Math Phys 287 (2009), 641-655. T. Johnson and S. Pal, Cycles and eigenvalues of sequentially growing random regular graphs, Ann Probab 42 (2014), 1396-1437. B. D. McKay, Subgraphs of random graphs with specified degrees, Proceedings of the Twelfth Southeastern Conference on Combinatorics, Graph Theory and Computing, Vol. II (Baton Rouge, La., 1981), Congressus Numerantium, Vol. 33, 1981, pp. 213-223. L. Erdős, J. Ramírez, B. Schlein, T. Tao, V. Vu, and H.-T. Yau, Bulk universality for Wigner Hermitian matrices with subexponential decay, Math Res Lett 17 (2010), 667-674. L. V. Tran, V. H. Vu, and K. Wang, Sparse random graphs: Eigenvalues and eigenvectors, Random Structures Algorithms 42 (2013), 110-134. L. Erdős, S. Péché, J. A. Ramírez, B. Schlein, and H.-T. Yau, Bulk universality for Wigner matrices, Commun Pure Appl Math 63 (2010), 895-925. H. Mizuno and I. Sato, The semicircle law for semiregular bipartite graphs, J Combin Theory Ser A 101 (2003), 174-190. C. D. Godsil and B. Mohar, Walk generating functions and spectral measures of infinite graphs, Proceedings of the Victoria Conference on Combinatorial Matrix Analysis (Victoria, BC, 1987), Linear Algebra and Its Applications, Volume 107 (1988), pp. 191-206, DOI: 10.1016/0024-3795(88)90245-5, URL: http://dx.doi.org/10.1016/0024-3795(88)90245-5. I. Dumitriu, T. Johnson, S. Pal, and E. Paquette, Functional limit theorems for random regular graphs, Probab Theory Relat Fields 156 (2013), 921-975. T. Tao and V. Vu, Random matrices: Universality of local eigenvalue statistics, Acta Math 206 (2011), 127-204. I. Dumitriu and S. Pal, Sparse regular random graphs: Spectral density and eigenvectors, Ann Probab 40 (2012), 2197-2235. I. Dumitriu, Eigenvalue statistics for beta-ensembles, PhD thesis, Massachusetts Institute of Technology, Cambridge, MA, 2003. S. Brooks and E. Lindenstrauss, Non-localization of eigenfunctions on large regular graphs, Israel J Math 193 (2013) 1-14. 2004; 11 2010; 15 2011; 206 2010 2010; 17 2013; 42 2013; 156 2009; 287 2010; 298 2003 2014 1999; 2 2008; 1 2013; 193 2003; 101 2010; 63 1988; 107 2009; 37 2012; 40 2014; 42 1981; 33 10.1002/rsa.20581-BIB0016|rsa20581-cit-0016 Mizuno (10.1002/rsa.20581-BIB0017|rsa20581-cit-0017) 2003; 101 Tao (10.1002/rsa.20581-BIB0021|rsa20581-cit-0021) 2011; 206 Dumitriu (10.1002/rsa.20581-BIB0005|rsa20581-cit-0005) 2003 Erdős (10.1002/rsa.20581-BIB0007|rsa20581-cit-0007) 2010; 17 Brooks (10.1002/rsa.20581-BIB0001|rsa20581-cit-0001) 2013; 193 McKay (10.1002/rsa.20581-BIB0015|rsa20581-cit-0015) 1981; 33 Erdős (10.1002/rsa.20581-BIB0006|rsa20581-cit-0006) 2010; 63 Erdős (10.1002/rsa.20581-BIB0008|rsa20581-cit-0008) 2010; 15 Godsil (10.1002/rsa.20581-BIB0011|rsa20581-cit-0011) 1988; 107 Lorentzen (10.1002/rsa.20581-BIB0014|rsa20581-cit-0014) 2008; 1 Dumitriu (10.1002/rsa.20581-BIB0004|rsa20581-cit-0004) 2012; 40 McKay (10.1002/rsa.20581-BIB0018|rsa20581-cit-0018) 2004; 11 Erdős (10.1002/rsa.20581-BIB0010|rsa20581-cit-0010) 2009; 37 Johnson (10.1002/rsa.20581-BIB0013|rsa20581-cit-0013) 2014; 42 Dumitriu (10.1002/rsa.20581-BIB0003|rsa20581-cit-0003) 2013; 156 Tran (10.1002/rsa.20581-BIB0022|rsa20581-cit-0022) 2013; 42 10.1002/rsa.20581-BIB0012|rsa20581-cit-0012 Bai (10.1002/rsa.20581-BIB0002|rsa20581-cit-0002) 2010 Erdős (10.1002/rsa.20581-BIB0009|rsa20581-cit-0009) 2009; 287 Stanley (10.1002/rsa.20581-BIB0019|rsa20581-cit-0019) 1999; 2 Tao (10.1002/rsa.20581-BIB0020|rsa20581-cit-0020) 2010; 298 |
| References_xml | – reference: I. Dumitriu and S. Pal, Sparse regular random graphs: Spectral density and eigenvectors, Ann Probab 40 (2012), 2197-2235. – reference: L. Erdős, S. Péché, J. A. Ramírez, B. Schlein, and H.-T. Yau, Bulk universality for Wigner matrices, Commun Pure Appl Math 63 (2010), 895-925. – reference: T. Tao and V. Vu, Random matrices: Universality of local eigenvalue statistics, Acta Math 206 (2011), 127-204. – reference: L. Erdős, B. Schlein, and H.-T. Yau, Local semicircle law and complete delocalization for Wigner random matrices, Comm Math Phys 287 (2009), 641-655. – reference: L. Erdős, B. Schlein, and H.-T. Yau, Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices, Ann Probab 37 (2009), 815-852. – reference: Z. Bai and J. W. Silverstein, Spectral analysis of large dimensional random matrices, Springer Series in Statistics, Second edition, Springer, New York, 2010. – reference: I. Dumitriu, T. Johnson, S. Pal, and E. Paquette, Functional limit theorems for random regular graphs, Probab Theory Relat Fields 156 (2013), 921-975. – reference: C. D. Godsil and B. Mohar, Walk generating functions and spectral measures of infinite graphs, Proceedings of the Victoria Conference on Combinatorial Matrix Analysis (Victoria, BC, 1987), Linear Algebra and Its Applications, Volume 107 (1988), pp. 191-206, DOI: 10.1016/0024-3795(88)90245-5, URL: http://dx.doi.org/10.1016/0024-3795(88)90245-5. – reference: B. D. McKay, N. C. Wormald, and B. Wysocka, Short cycles in random regular graphs, Electron J Comb 11 (2004), Research Paper 66, 12. – reference: S. Brooks and E. Lindenstrauss, Non-localization of eigenfunctions on large regular graphs, Israel J Math 193 (2013) 1-14. – reference: B. D. McKay, Subgraphs of random graphs with specified degrees, Proceedings of the Twelfth Southeastern Conference on Combinatorics, Graph Theory and Computing, Vol. II (Baton Rouge, La., 1981), Congressus Numerantium, Vol. 33, 1981, pp. 213-223. – reference: L. Erdős, J. Ramírez, B. Schlein, T. Tao, V. Vu, and H.-T. Yau, Bulk universality for Wigner Hermitian matrices with subexponential decay, Math Res Lett 17 (2010), 667-674. – reference: L. Erdős, J. Ramírez, B. Schlein, and H.-T. Yau, Universality of sine-kernel for Wigner matrices with a small Gaussian perturbation, Electron J Probab 15 (2010), 526-603. – reference: T. Johnson and S. Pal, Cycles and eigenvalues of sequentially growing random regular graphs, Ann Probab 42 (2014), 1396-1437. – reference: H. Mizuno and I. Sato, The semicircle law for semiregular bipartite graphs, J Combin Theory Ser A 101 (2003), 174-190. – reference: L. V. Tran, V. H. Vu, and K. Wang, Sparse random graphs: Eigenvalues and eigenvectors, Random Structures Algorithms 42 (2013), 110-134. – reference: I. Dumitriu, Eigenvalue statistics for beta-ensembles, PhD thesis, Massachusetts Institute of Technology, Cambridge, MA, 2003. – reference: T. Tao and V. Vu, Random matrices: Universality of local eigenvalue statistics up to the edge, Comm Math Phys 298 (2010), 549-572. – volume: 40 start-page: 2197 year: 2012 end-page: 2235 article-title: Sparse regular random graphs: Spectral density and eigenvectors publication-title: Ann Probab – volume: 101 start-page: 174 year: 2003 end-page: 190 article-title: The semicircle law for semiregular bipartite graphs publication-title: J Combin Theory Ser A – volume: 298 start-page: 549 year: 2010 end-page: 572 article-title: Random matrices: Universality of local eigenvalue statistics up to the edge publication-title: Comm Math Phys – year: 2014 article-title: Exchangeable pairs, switchings, and random regular graphs – volume: 287 start-page: 641 year: 2009 end-page: 655 article-title: Local semicircle law and complete delocalization for Wigner random matrices publication-title: Comm Math Phys – volume: 107 start-page: 191 year: 1988 end-page: 206 article-title: Walk generating functions and spectral measures of infinite graphs, Proceedings of the Victoria Conference on Combinatorial Matrix Analysis (Victoria, BC, 1987) publication-title: Linear Algebra and Its Applications – volume: 42 start-page: 110 year: 2013 end-page: 134 article-title: Sparse random graphs: Eigenvalues and eigenvectors publication-title: Random Structures Algorithms – volume: 2 year: 1999 – volume: 17 start-page: 667 year: 2010 end-page: 674 article-title: Bulk universality for Wigner Hermitian matrices with subexponential decay publication-title: Math Res Lett – volume: 37 start-page: 815 year: 2009 end-page: 852 article-title: Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices publication-title: Ann Probab – volume: 1 year: 2008 – volume: 33 start-page: 213 year: 1981 end-page: 223 article-title: Subgraphs of random graphs with specified degrees, Proceedings of the Twelfth Southeastern Conference on Combinatorics, Graph Theory and Computing publication-title: Congressus Numerantium – year: 2003 – volume: 42 start-page: 1396 year: 2014 end-page: 1437 article-title: Cycles and eigenvalues of sequentially growing random regular graphs publication-title: Ann Probab – volume: 11 year: 2004 article-title: Short cycles in random regular graphs publication-title: Electron J Comb – volume: 193 start-page: 1 year: 2013 end-page: 14 article-title: Non‐localization of eigenfunctions on large regular graphs publication-title: Israel J Math – volume: 156 start-page: 921 year: 2013 end-page: 975 article-title: Functional limit theorems for random regular graphs publication-title: Probab Theory Relat Fields – volume: 206 start-page: 127 year: 2011 end-page: 204 article-title: Random matrices: Universality of local eigenvalue statistics publication-title: Acta Math – year: 2010 – volume: 15 start-page: 526 year: 2010 end-page: 603 article-title: Universality of sine‐kernel for Wigner matrices with a small Gaussian perturbation publication-title: Electron J Probab – volume: 63 start-page: 895 year: 2010 end-page: 925 article-title: Bulk universality for Wigner matrices publication-title: Commun Pure Appl Math – volume: 42 start-page: 1396 year: 2014 ident: 10.1002/rsa.20581-BIB0013|rsa20581-cit-0013 article-title: Cycles and eigenvalues of sequentially growing random regular graphs publication-title: Ann Probab doi: 10.1214/13-AOP864 – volume: 1 volume-title: volume 1 of atlantis studies in mathematics for engineering and science year: 2008 ident: 10.1002/rsa.20581-BIB0014|rsa20581-cit-0014 – volume: 101 start-page: 174 year: 2003 ident: 10.1002/rsa.20581-BIB0017|rsa20581-cit-0017 article-title: The semicircle law for semiregular bipartite graphs publication-title: J Combin Theory Ser A doi: 10.1016/S0097-3165(02)00010-9 – volume-title: Spectral analysis of large dimensional random matrices, Springer Series in Statistics year: 2010 ident: 10.1002/rsa.20581-BIB0002|rsa20581-cit-0002 – volume: 156 start-page: 921 year: 2013 ident: 10.1002/rsa.20581-BIB0003|rsa20581-cit-0003 article-title: Functional limit theorems for random regular graphs publication-title: Probab Theory Relat Fields doi: 10.1007/s00440-012-0447-y – volume: 107 start-page: 191 year: 1988 ident: 10.1002/rsa.20581-BIB0011|rsa20581-cit-0011 article-title: Walk generating functions and spectral measures of infinite graphs, Proceedings of the Victoria Conference on Combinatorial Matrix Analysis (Victoria, BC, 1987) publication-title: Linear Algebra and Its Applications doi: 10.1016/0024-3795(88)90245-5 – volume: 63 start-page: 895 year: 2010 ident: 10.1002/rsa.20581-BIB0006|rsa20581-cit-0006 article-title: Bulk universality for Wigner matrices publication-title: Commun Pure Appl Math doi: 10.1002/cpa.20317 – volume-title: Eigenvalue statistics for beta-ensembles, PhD thesis year: 2003 ident: 10.1002/rsa.20581-BIB0005|rsa20581-cit-0005 – volume: 37 start-page: 815 year: 2009 ident: 10.1002/rsa.20581-BIB0010|rsa20581-cit-0010 article-title: Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices publication-title: Ann Probab doi: 10.1214/08-AOP421 – volume: 40 start-page: 2197 year: 2012 ident: 10.1002/rsa.20581-BIB0004|rsa20581-cit-0004 article-title: Sparse regular random graphs: Spectral density and eigenvectors publication-title: Ann Probab doi: 10.1214/11-AOP673 – volume: 17 start-page: 667 year: 2010 ident: 10.1002/rsa.20581-BIB0007|rsa20581-cit-0007 article-title: Bulk universality for Wigner Hermitian matrices with subexponential decay publication-title: Math Res Lett doi: 10.4310/MRL.2010.v17.n4.a7 – ident: 10.1002/rsa.20581-BIB0016|rsa20581-cit-0016 doi: 10.1201/9781420036114 – volume: 287 start-page: 641 year: 2009 ident: 10.1002/rsa.20581-BIB0009|rsa20581-cit-0009 article-title: Local semicircle law and complete delocalization for Wigner random matrices publication-title: Comm Math Phys doi: 10.1007/s00220-008-0636-9 – volume: 2 volume-title: Volume 62 of Cambridge studies in advanced mathematics year: 1999 ident: 10.1002/rsa.20581-BIB0019|rsa20581-cit-0019 – volume: 298 start-page: 549 year: 2010 ident: 10.1002/rsa.20581-BIB0020|rsa20581-cit-0020 article-title: Random matrices: Universality of local eigenvalue statistics up to the edge publication-title: Comm Math Phys doi: 10.1007/s00220-010-1044-5 – volume: 206 start-page: 127 year: 2011 ident: 10.1002/rsa.20581-BIB0021|rsa20581-cit-0021 article-title: Random matrices: Universality of local eigenvalue statistics publication-title: Acta Math doi: 10.1007/s11511-011-0061-3 – volume: 193 start-page: 1 year: 2013 ident: 10.1002/rsa.20581-BIB0001|rsa20581-cit-0001 article-title: Non-localization of eigenfunctions on large regular graphs publication-title: Israel J Math doi: 10.1007/s11856-012-0096-y – ident: 10.1002/rsa.20581-BIB0012|rsa20581-cit-0012 doi: 10.37236/4659 – volume: 11 year: 2004 ident: 10.1002/rsa.20581-BIB0018|rsa20581-cit-0018 article-title: Short cycles in random regular graphs publication-title: Electron J Comb – volume: 33 start-page: 213 year: 1981 ident: 10.1002/rsa.20581-BIB0015|rsa20581-cit-0015 article-title: Subgraphs of random graphs with specified degrees, Proceedings of the Twelfth Southeastern Conference on Combinatorics, Graph Theory and Computing publication-title: Congressus Numerantium – volume: 15 start-page: 526 year: 2010 ident: 10.1002/rsa.20581-BIB0008|rsa20581-cit-0008 article-title: Universality of sine-kernel for Wigner matrices with a small Gaussian perturbation publication-title: Electron J Probab doi: 10.1214/EJP.v15-768 – volume: 42 start-page: 110 year: 2013 ident: 10.1002/rsa.20581-BIB0022|rsa20581-cit-0022 article-title: Sparse random graphs: Eigenvalues and eigenvectors publication-title: Random Structures Algorithms doi: 10.1002/rsa.20406 |
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| Snippet | We prove that the empirical spectral distribution of a (dL, dR)‐biregular, bipartite random graph, under certain conditions, converges to a symmetrization of... We prove that the empirical spectral distribution of a (dL, dR)-biregular, bipartite random graph, under certain conditions, converges to a symmetrization of... |
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| SubjectTerms | Marčenko-Pastur law random bipartite graphs spectral distribution Wishart matrix |
| Title | The Marčenko-Pastur law for sparse random bipartite biregular graphs |
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