Outliers in spectrum of sparse Wigner matrices
In this paper, we study the effect of sparsity on the appearance of outliers in the semi‐circular law. Let (Wn)n=1∞ be a sequence of random symmetric matrices such that each Wn is n × n with i.i.d. entries above and on the main diagonal equidistributed with the product bnξ, where ξ is a real centere...
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| Veröffentlicht in: | Random structures & algorithms Jg. 58; H. 3; S. 517 - 605 |
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John Wiley & Sons, Inc
01.05.2021
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| Abstract | In this paper, we study the effect of sparsity on the appearance of outliers in the semi‐circular law. Let (Wn)n=1∞ be a sequence of random symmetric matrices such that each Wn is n × n with i.i.d. entries above and on the main diagonal equidistributed with the product bnξ, where ξ is a real centered uniformly bounded random variable of unit variance and bn is an independent Bernoulli random variable with a probability of success pn. Assuming that limn→∞npn=∞, we show that for the random sequence (ρn)n=1∞
given by ρn:=θn+npnθn,θn:=max(maxi≤n‖rowi(Wn)‖22−npn,npn), the ratio ‖Wn‖ρn converges to one in probability. A noncentered counterpart of the theorem allows to obtain asymptotic expressions for eigenvalues of the Erdős–Renyi graphs, which were unknown in the regime npn=Θ(logn). In particular, denoting by An the adjacency matrix of the Erdős–Renyi graph 𝒢(n,pn) and by λ|k|(An) its kth largest (by the absolute value) eigenvalue, under the assumptions limn→∞npn=∞ and limn→∞pn=0 we have (1) (No non‐trivial outliers): if liminfnpnlogn≥1log(4/e)
then for any fixed k ≥ 2, |λ|k|(An)|2npn converges to 1 in probability; and (2) (Outliers): if limsupnpnlogn<1log(4/e) then there is ε > 0 such that for any k∈ℕ, we have limn→∞ℙ|λ|k|(An)|2npn>1+ε=1. On a conceptual level, our result reveals similarities in appearance of outliers in spectrum of sparse matrices and the so‐called BBP phase transition phenomenon in deformed Wigner matrices. |
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| AbstractList | In this paper, we study the effect of sparsity on the appearance of outliers in the semi‐circular law. Let (Wn)n=1∞ be a sequence of random symmetric matrices such that each Wn is n × n with i.i.d. entries above and on the main diagonal equidistributed with the product bnξ, where ξ is a real centered uniformly bounded random variable of unit variance and bn is an independent Bernoulli random variable with a probability of success pn. Assuming that limn→∞npn=∞, we show that for the random sequence (ρn)n=1∞
given by ρn:=θn+npnθn,θn:=max(maxi≤n‖rowi(Wn)‖22−npn,npn), the ratio ‖Wn‖ρn converges to one in probability. A noncentered counterpart of the theorem allows to obtain asymptotic expressions for eigenvalues of the Erdős–Renyi graphs, which were unknown in the regime npn=Θ(logn). In particular, denoting by An the adjacency matrix of the Erdős–Renyi graph 𝒢(n,pn) and by λ|k|(An) its kth largest (by the absolute value) eigenvalue, under the assumptions limn→∞npn=∞ and limn→∞pn=0 we have (1) (No non‐trivial outliers): if liminfnpnlogn≥1log(4/e)
then for any fixed k ≥ 2, |λ|k|(An)|2npn converges to 1 in probability; and (2) (Outliers): if limsupnpnlogn<1log(4/e) then there is ε > 0 such that for any k∈ℕ, we have limn→∞ℙ|λ|k|(An)|2npn>1+ε=1. On a conceptual level, our result reveals similarities in appearance of outliers in spectrum of sparse matrices and the so‐called BBP phase transition phenomenon in deformed Wigner matrices. In this paper, we study the effect of sparsity on the appearance of outliers in the semi‐circular law. Let (Wn)n=1∞ be a sequence of random symmetric matrices such that each Wn is n × n with i.i.d. entries above and on the main diagonal equidistributed with the product bnξ, where ξ is a real centered uniformly bounded random variable of unit variance and bn is an independent Bernoulli random variable with a probability of success pn. Assuming that limn→∞npn=∞, we show that for the random sequence (ρn)n=1∞given by ρn:=θn+npnθn,θn:=max(maxi≤n‖rowi(Wn)‖22−npn,npn), the ratio ‖Wn‖ρn converges to one in probability. A noncentered counterpart of the theorem allows to obtain asymptotic expressions for eigenvalues of the Erdős–Renyi graphs, which were unknown in the regime npn=Θ(logn). In particular, denoting by An the adjacency matrix of the Erdős–Renyi graph ð'¢(n,pn) and by λ|k|(An) its kth largest (by the absolute value) eigenvalue, under the assumptions limn→∞npn=∞ and limn→∞pn=0 we have (1) (No non‐trivial outliers): if liminfnpnlogn≥1log(4/e)then for any fixed k ≥ 2, |λ|k|(An)|2npn converges to 1 in probability; and (2) (Outliers): if limsupnpnlogn<1log(4/e) then there is ε > 0 such that for any k∈ℕ, we have limn→∞ℙ|λ|k|(An)|2npn>1+ε=1. On a conceptual level, our result reveals similarities in appearance of outliers in spectrum of sparse matrices and the so‐called BBP phase transition phenomenon in deformed Wigner matrices. In this paper, we study the effect of sparsity on the appearance of outliers in the semi‐circular law. Let be a sequence of random symmetric matrices such that each W n is n × n with i.i.d. entries above and on the main diagonal equidistributed with the product , where is a real centered uniformly bounded random variable of unit variance and b n is an independent Bernoulli random variable with a probability of success p n . Assuming that , we show that for the random sequence given by , the ratio converges to one in probability. A noncentered counterpart of the theorem allows to obtain asymptotic expressions for eigenvalues of the Erdős–Renyi graphs, which were unknown in the regime . In particular, denoting by A n the adjacency matrix of the Erdős–Renyi graph and by its k th largest (by the absolute value) eigenvalue, under the assumptions and we have (1) (No non‐trivial outliers): if then for any fixed k ≥ 2 , converges to 1 in probability; and (2) (Outliers): if then there is ε > 0 such that for any , we have . On a conceptual level, our result reveals similarities in appearance of outliers in spectrum of sparse matrices and the so‐called BBP phase transition phenomenon in deformed Wigner matrices. |
| Author | Tikhomirov, Konstantin Youssef, Pierre |
| Author_xml | – sequence: 1 givenname: Konstantin surname: Tikhomirov fullname: Tikhomirov, Konstantin organization: Georgia Tech – sequence: 2 givenname: Pierre surname: Youssef fullname: Youssef, Pierre email: youssef@lpsm.paris organization: Université Paris Diderot |
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| Cites_doi | 10.1214/009117905000000233 10.1016/j.aim.2017.11.001 10.1214/EJP.v16-934 10.1093/acprof:oso/9780199535255.001.0001 10.1007/s00440-011-0397-9 10.2307/1970008 10.1214/11-AIHP459 10.1214/EJP.v16-929 10.1002/rsa.20089 10.1002/cpa.21450 10.1007/BF02579329 10.1214/16-AIHP754 10.1214/11-AOP734 10.1007/s00493-016-3238-8 10.1142/S2010326312500153 10.1007/BF00324852 10.1017/CBO9780511814068 10.1007/s00220-012-1527-7 10.1007/BF00535717 10.1007/s00493-007-2190-z 10.1017/S0001867800010661 10.1007/s10959-016-0686-4 10.1017/9781108231596 10.1016/j.jmva.2005.08.003 10.1007/s00222-018-0817-x 10.1214/aop/1176992372 10.1002/cpa.21629 10.1017/S096354830000420X 10.1007/s00440-015-0632-x 10.1007/s00440-005-0466-z 10.1017/S0963548302005424 10.1007/s00220-007-0209-3 10.1016/j.aim.2011.02.007 10.1512/iumj.2014.63.5432 10.1214/08-AOP394 10.1214/13-AOP855 10.1007/BF02699376 10.1002/rsa.20713 10.1214/10-AIHP410 10.1007/s00440-017-0787-8 |
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| References | 2006; 97 1986; 73 2013; 49 2013; 66 1981; 1 2013; 02 2000; 9 1986; 14 2018; 324 2013; 41 2016; 165 2014; 63 2011; 16 2005; 27 2001; 89 2006; 134 2014; 42 2003; 12 2017; 51 2017; 53 2017; 30 2018; 171 1995; 81 2011; 227 1990 2007; 272 1982; 60 2018; 214 1958; 67 2014; III 2013; 155 2018 2017 2015 2014 2013 2012; 48 2001; 33 2012; 314 2005; 33 2001; 73 2016; 69 2018; 38 2009; 37 2007; 27 e_1_2_16_25_1 e_1_2_16_23_1 e_1_2_16_29_1 e_1_2_16_46_1 e_1_2_16_27_1 e_1_2_16_48_1 e_1_2_16_44_1 e_1_2_16_21_1 e_1_2_16_15_1 e_1_2_16_38_1 e_1_2_16_13_1 e_1_2_16_19_1 e_1_2_16_34_1 e_1_2_16_17_1 e_1_2_16_30_1 e_1_2_16_32_1 e_1_2_16_11_1 e_1_2_16_51_1 e_1_2_16_7_1 e_1_2_16_9_1 e_1_2_16_5_1 Ledoux M. (e_1_2_16_36_1) 2001 e_1_2_16_49_1 e_1_2_16_24_1 e_1_2_16_45_1 e_1_2_16_28_1 e_1_2_16_47_1 e_1_2_16_41_1 e_1_2_16_2_1 e_1_2_16_43_1 e_1_2_16_22_1 e_1_2_16_20_1 e_1_2_16_14_1 e_1_2_16_39_1 e_1_2_16_12_1 e_1_2_16_18_1 e_1_2_16_35_1 e_1_2_16_16_1 e_1_2_16_37_1 e_1_2_16_31_1 Rajagopalan A. B. (e_1_2_16_42_1) 2015 e_1_2_16_52_1 e_1_2_16_33_1 e_1_2_16_10_1 e_1_2_16_50_1 e_1_2_16_8_1 Ash R. (e_1_2_16_3_1) 1990 e_1_2_16_4_1 e_1_2_16_6_1 (e_1_2_16_26_1) 2014 Péché S. (e_1_2_16_40_1) 2014 |
| References_xml | – volume: 12 start-page: 61 year: 2003 end-page: 72 article-title: The largest eigenvalue of sparse random graphs publication-title: Combin. Probab. Comput. – volume: 214 start-page: 1031 year: 2018 end-page: 1080 article-title: The dimension‐free structure of nonhomogeneous random matrices publication-title: Invent. Math. – volume: 16 start-page: 1621 year: 2011 end-page: 1662 article-title: Fluctuations of the extreme eigenvalues of finite rank deformations of random matrices publication-title: Electron. J. Probab. – volume: 97 start-page: 1382 year: 2006 end-page: 1408 article-title: Eigenvalues of large sample covariance matrices of spiked population models publication-title: J. Multivariate Anal. – volume: 63 start-page: 1875 year: 2014 end-page: 1910 article-title: Exact separation phenomenon for the eigenvalues of large information‐plus‐noise type matrices, and an application to spiked models publication-title: Indiana Univ. Math. J. – volume: 314 start-page: 587 year: 2012 end-page: 640 article-title: Spectral statistics of Erdős‐‐Rényi graphs II: Eigenvalue spacing and the extreme eigenvalues publication-title: Commun. Math. Phys. – volume: 60 start-page: 291 year: 1982 end-page: 314 article-title: A chaos hypothesis for some large systems of random equations publication-title: Z. Wahrsch. Verw. Gebiete – volume: 171 start-page: 543 year: 2018 end-page: 616 article-title: Local law and Tracy‐Widom limit for sparse random matrices publication-title: Probab. Theory Related Fields – volume: 37 start-page: 1 year: 2009 end-page: 47 article-title: The largest eigenvalues of finite rank deformation of large Wigner matrices: convergence and non universality of the fluctuations publication-title: Ann. Probab. – volume: 67 start-page: 325 year: 1958 end-page: 327 article-title: On the distribution of the roots of certain symmetric matrices publication-title: Ann. Math. – volume: 66 start-page: 1663 year: 2013 end-page: 1750 article-title: The isotropic semicircle law and deformation of Wigner matrices publication-title: Commun. Pure Appl. Math. – volume: 27 start-page: 251 year: 2005 end-page: 275 article-title: Spectral techniques applied to sparse random graphs publication-title: Random Structures Algorithms – volume: 324 start-page: 40 year: 2018 end-page: 83 article-title: Norms of random matrices: Local and global problems publication-title: Adv. Math. – volume: 33 start-page: 1643 year: 2005 end-page: 1697 article-title: Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices publication-title: Ann. Probab. – year: 1990 – volume: 38 start-page: 665 year: 2018 end-page: 708 article-title: A proof of the block model threshold conjecture publication-title: Combinatorica – year: 2018 – volume: 48 start-page: 107 year: 2012 end-page: 133 article-title: Central limit theorems for eigenvalues of deformations of Wigner matrices publication-title: Ann. Inst. Henri Poincaré Probab. Stat. – volume: 272 start-page: 185 year: 2007 end-page: 228 article-title: The largest eigenvalue of rank one deformation of large Wigner matrices publication-title: Commun. Math. Phys. – year: 2014 – volume: 73 year: 2001 – volume: 9 start-page: 149 year: 2000 end-page: 166 article-title: The expected norm of random matrices publication-title: Combin. Probab. Comput. – volume: 30 start-page: 1624 year: 2017 end-page: 1654 article-title: Complex outliers of Hermitian random matrices publication-title: J. Theoret. Probab. – volume: 155 start-page: 231 year: 2013 end-page: 263 article-title: Outliers in the spectrum of iid matrices with bounded rank perturbations publication-title: Probab. Theory Related Fields – volume: III start-page: 1159 year: 2014 end-page: 1174 article-title: Deformed ensembles of random matrices publication-title: Proc. ICM‐Seoul – volume: 1 start-page: 233 year: 1981 end-page: 241 article-title: The eigenvalues of random symmetric matrices publication-title: Combinatorica – volume: 27 start-page: 721 year: 2007 end-page: 736 article-title: Spectral norm of random matrices publication-title: Combinatorica – volume: 16 start-page: 1750 year: 2011 end-page: 1792 article-title: Free convolution with a semicircular distribution and eigenvalues of spiked deformations of Wigner matrices publication-title: Electron. J. Probab. – volume: 49 start-page: 64 year: 2013 end-page: 94 article-title: On finite rank deformations of Wigner matrices publication-title: Ann. Inst. Henri Poincaré Probab. Stat. – volume: 02 year: 2013 article-title: On finite rank deformations of Wigner matrices II. Delocalized perturbations publication-title: Random Matrices Theory Appl. – volume: 227 start-page: 494 year: 2011 end-page: 521 article-title: The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices publication-title: Adv. Math. – volume: 51 start-page: 538 year: 2017 end-page: 561 article-title: Concentration and regularization of random graphs publication-title: Random Structures Algorithms – volume: 33 start-page: 124 year: 2001 end-page: 140 article-title: Sparse random matrices: Spectral edge and statistics of rooted trees publication-title: Adv. Appl. Probab. – volume: 41 start-page: 2279 year: 2013 end-page: 2375 article-title: Spectral statistics of Erdős‐‐Rényi graphs I: Local semicircle law publication-title: Ann. Probab. – volume: 165 start-page: 313 year: 2016 end-page: 363 article-title: Outliers in the single ring theorem publication-title: Probab. Theory Relat. Fields – volume: 81 start-page: 73 year: 1995 end-page: 205 article-title: Concentration of measure and isoperimetric inequalities in product spaces publication-title: Inst. Hautes Études Sci. Publ. Math. – volume: 14 start-page: 1318 year: 1986 end-page: 1328 article-title: The spectral radius of large random matrices publication-title: Ann. Probab. – volume: 42 start-page: 1980 year: 2014 end-page: 2031 article-title: The outliers of a deformed Wigner matrix publication-title: Ann. Probab. – volume: 89 year: 2001 – volume: 53 start-page: 1241 year: 2017 end-page: 1279 article-title: Spectra of nearly Hermitian random matrices publication-title: Ann. Inst. Henri Poincaré Probab. Stat. – year: 2017 – volume: 73 start-page: 555 year: 1986 end-page: 569 article-title: Limiting behavior of the norm of products of random matrices and two problems of Geman–Hwang publication-title: Probab. Theory Related Fields – volume: 134 start-page: 127 year: 2006 end-page: 173 article-title: The largest eigenvalue of small rank perturbations of Hermitian random matrices publication-title: Probab. Theory Related Fields – volume: 69 start-page: 2131 year: 2016 end-page: 2194 article-title: Outlier eigenvalues for deformed i.i.d. random matrices publication-title: Commun. Pure Appl. Math. – year: 2015 – year: 2013 – ident: e_1_2_16_43_1 – ident: e_1_2_16_5_1 doi: 10.1214/009117905000000233 – ident: e_1_2_16_44_1 doi: 10.1016/j.aim.2017.11.001 – ident: e_1_2_16_18_1 doi: 10.1214/EJP.v16-934 – volume-title: Mathematical Surveys and Monographs year: 2001 ident: e_1_2_16_36_1 – ident: e_1_2_16_14_1 doi: 10.1093/acprof:oso/9780199535255.001.0001 – ident: e_1_2_16_48_1 doi: 10.1007/s00440-011-0397-9 – start-page: 1159 year: 2014 ident: e_1_2_16_40_1 article-title: Deformed ensembles of random matrices publication-title: Proc. ICM‐Seoul – ident: e_1_2_16_52_1 doi: 10.2307/1970008 – ident: e_1_2_16_41_1 doi: 10.1214/11-AIHP459 – ident: e_1_2_16_9_1 doi: 10.1214/EJP.v16-929 – ident: e_1_2_16_21_1 doi: 10.1002/rsa.20089 – ident: e_1_2_16_30_1 doi: 10.1002/cpa.21450 – ident: e_1_2_16_23_1 doi: 10.1007/BF02579329 – volume-title: Information theory year: 1990 ident: e_1_2_16_3_1 – ident: e_1_2_16_37_1 doi: 10.1214/16-AIHP754 – ident: e_1_2_16_20_1 doi: 10.1214/11-AOP734 – ident: e_1_2_16_38_1 doi: 10.1007/s00493-016-3238-8 – ident: e_1_2_16_45_1 doi: 10.1142/S2010326312500153 – ident: e_1_2_16_4_1 doi: 10.1007/BF00324852 – ident: e_1_2_16_12_1 doi: 10.1017/CBO9780511814068 – ident: e_1_2_16_19_1 doi: 10.1007/s00220-012-1527-7 – ident: e_1_2_16_25_1 doi: 10.1007/BF00535717 – ident: e_1_2_16_50_1 doi: 10.1007/s00493-007-2190-z – ident: e_1_2_16_27_1 – ident: e_1_2_16_29_1 doi: 10.1017/S0001867800010661 – volume-title: Outlier eigenvalue fluctuations of perturbed iid matrices year: 2015 ident: e_1_2_16_42_1 – ident: e_1_2_16_46_1 doi: 10.1007/s10959-016-0686-4 – ident: e_1_2_16_51_1 doi: 10.1017/9781108231596 – ident: e_1_2_16_6_1 doi: 10.1016/j.jmva.2005.08.003 – ident: e_1_2_16_33_1 doi: 10.1007/s00222-018-0817-x – ident: e_1_2_16_7_1 – volume-title: Discrete Mathematics and its Applications (Boca Raton) year: 2014 ident: e_1_2_16_26_1 – ident: e_1_2_16_2_1 – ident: e_1_2_16_24_1 doi: 10.1214/aop/1176992372 – ident: e_1_2_16_13_1 doi: 10.1002/cpa.21629 – ident: e_1_2_16_47_1 doi: 10.1017/S096354830000420X – ident: e_1_2_16_11_1 doi: 10.1007/s00440-015-0632-x – ident: e_1_2_16_8_1 – ident: e_1_2_16_39_1 doi: 10.1007/s00440-005-0466-z – ident: e_1_2_16_32_1 doi: 10.1017/S0963548302005424 – ident: e_1_2_16_22_1 doi: 10.1007/s00220-007-0209-3 – ident: e_1_2_16_28_1 – ident: e_1_2_16_10_1 doi: 10.1016/j.aim.2011.02.007 – ident: e_1_2_16_15_1 doi: 10.1512/iumj.2014.63.5432 – ident: e_1_2_16_16_1 doi: 10.1214/08-AOP394 – ident: e_1_2_16_31_1 doi: 10.1214/13-AOP855 – ident: e_1_2_16_49_1 doi: 10.1007/BF02699376 – ident: e_1_2_16_34_1 doi: 10.1002/rsa.20713 – ident: e_1_2_16_17_1 doi: 10.1214/10-AIHP410 – ident: e_1_2_16_35_1 doi: 10.1007/s00440-017-0787-8 |
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| Snippet | In this paper, we study the effect of sparsity on the appearance of outliers in the semi‐circular law. Let (Wn)n=1∞ be a sequence of random symmetric matrices... In this paper, we study the effect of sparsity on the appearance of outliers in the semi‐circular law. Let be a sequence of random symmetric matrices such that... |
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| SubjectTerms | BBP phase transition Convergence Eigenvalues Erdos‐Renyi graph Mathematical analysis Mathematics Matrix methods moment method outliers Phase transitions Probability Random variables Sparse matrices Sparsity spectral gap |
| Title | Outliers in spectrum of sparse Wigner matrices |
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