| 1. |
- Bao, Zhigang, et al.
(författare)
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Central Limit Theorem for Mesoscopic Eigenvalue Statistics of the Free Sum of Matrices
- 2020
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Ingår i: International mathematics research notices. - : Oxford University Press (OUP). - 1073-7928 .- 1687-0247. ; 2022:7, s. 5320-5382
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Tidskriftsartikel (refereegranskat)abstract
- We consider random matrices of the form H-N = A(N) + UNBNUN*, where A(N) and B-N are two N by N deterministic Hermitian matrices and U-N is a Haar distributed random unitary matrix. We establish a universal central limit theorem for the linear eigenvalue statistics of H-N on all mesoscopic scales inside the regular bulk of the spectrum. The proof is based on studying the characteristic function of the linear eigenvalue statistics and consists of two main steps: (1) generating Ward identities using the left-translation invariance of the Haar measure, along with a local law for the resolvent of H-N and analytic subordination properties of the free additive convolution, allows us to derive an explicit formula for the derivative of the characteristic function; (2) a local law for two-point product functions of resolvents is derived using a partial randomness decomposition of the Haar measure. We also prove the corresponding results for orthogonal conjugations.
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| 2. |
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| 3. |
- Bao, Zhigang, et al.
(författare)
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Equipartition principle for Wigner matrices
- 2021
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Ingår i: Forum of Mathematics, Sigma. - : Cambridge University Press (CUP). - 2050-5094. ; 9
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Tidskriftsartikel (refereegranskat)abstract
- We prove that the energy of any eigenvector of a sum of several independent large Wigner matrices is equally distributed among these matrices with very high precision. This shows a particularly strong microcanonical form of the equipartition principle for quantum systems whose components are modelled by Wigner matrices.
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| 4. |
- Bao, Z., et al.
(författare)
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Local Law of Addition of Random Matrices on Optimal Scale
- 2017
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Ingår i: Communications in Mathematical Physics. - : Springer-Verlag New York. - 0010-3616 .- 1432-0916. ; 349:3, s. 947-990
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 5. |
- Bao, Zhigang, et al.
(författare)
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LOCAL SINGLE RING THEOREM ON OPTIMAL SCALE
- 2019
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Ingår i: Annals of Probability. - : Institute of Mathematical Statistics. - 0091-1798 .- 2168-894X. ; 47:3, s. 1270-1334
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Tidskriftsartikel (refereegranskat)abstract
- 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).
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| 6. |
- Bao, Zhigang, et al.
(författare)
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On the support of the free additive convolution
- 2020
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Ingår i: Journal d'Analyse Mathematique. - : Springer Nature. - 0021-7670 .- 1565-8538. ; 142:1, s. 323-348
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Tidskriftsartikel (refereegranskat)abstract
- We consider the free additive convolution of two probability measures mu and nu on the real line and show that mu boxed plus nu is supported on a single interval if mu and nu each has single interval support. Moreover, the density of mu boxed plus nu is proven to vanish as a square root near the edges of its support if both mu and nu have power law behavior with exponents between -1 and 1 near their edges. In particular, these results show the ubiquity of the conditions in our recent work on optimal local law at the spectral edges for addition of random matrices [5].
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| 7. |
- Bao, Z., et al.
(författare)
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Spectral rigidity for addition of random matrices at the regular edge
- 2020
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Ingår i: Journal of Functional Analysis. - : Academic Press. - 0022-1236 .- 1096-0783. ; 279:7
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Tidskriftsartikel (refereegranskat)abstract
- We consider the sum of two large Hermitian matrices A and B with a Haar unitary conjugation bringing them into a general relative position. We prove that the eigenvalue density on the scale slightly above the local eigenvalue spacing is asymptotically given by the free additive convolution of the laws of A and B as the dimension of the matrix increases. This implies optimal rigidity of the eigenvalues and optimal rate of convergence in Voiculescu's theorem. Our previous works [4,5] established these results in the bulk spectrum, the current paper completely settles the problem at the spectral edges provided they have the typical square-root behavior. The key element of our proof is to compensate the deterioration of the stability of the subordination equations by sharp error estimates that properly account for the local density near the edge. Our results also hold if the Haar unitary matrix is replaced by the Haar orthogonal matrix.
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| 8. |
- Hwang, Jong Yun, et al.
(författare)
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LOCAL LAW AND TRACY-WIDOM LIMIT FOR SPARSE SAMPLE COVARIANCE MATRICES
- 2019
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Ingår i: The Annals of Applied Probability. - : INST MATHEMATICAL STATISTICS. - 1050-5164 .- 2168-8737. ; 29:5, s. 3006-3036
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Tidskriftsartikel (refereegranskat)abstract
- We consider spectral properties of sparse sample covariance matrices, which includes biadjacency matrices of the bipartite Erdos-Renyi graph model. We prove a local law for the eigenvalue density up to the upper spectral edge. Under a suitable condition on the sparsity, we also prove that the limiting distribution of the rescaled, shifted extremal eigenvalues is given by the GOE Tracy-Widom law with an explicit formula on the deterministic shift of the spectral edge. For the biadjacency matrix of an Erdos-Renyi graph with two vertex sets of comparable sizes M and N, this establishes Tracy-Widom fluctuations of the second largest eigenvalue when the connection probability p is much larger than N-2/3 with a deterministic shift of order (Np)(-1).
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| 9. |
- Lee, Ji Oon, et al.
(författare)
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Local law and Tracy-Widom limit for sparse random matrices
- 2018
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Ingår i: Probability theory and related fields. - : Springer Berlin/Heidelberg. - 0178-8051 .- 1432-2064. ; 171:1-2, s. 543-616
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Tidskriftsartikel (refereegranskat)abstract
- 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)..
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| 10. |
- Li, Yiting, et al.
(författare)
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Central limit theorem for mesoscopic eigenvalue statistics of deformed Wigner matrices and sample covariance matrices
- 2021
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Ingår i: Annales de l'I.H.P. Probabilites et statistiques. - : Project Euclid. - 0246-0203 .- 1778-7017. ; 57:1, s. 506-546
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Tidskriftsartikel (refereegranskat)abstract
- We consider N by N deformed Wigner random matrices of the form X-N = H-N + A(N), where H-N is a real symmetric or complex Hermitian Wigner matrix and A(N) is a deterministic real bounded diagonal matrix. We prove a universal Central Limit Theorem for the linear eigenvalue statistics of X-N for all mesoscopic scales both in the spectral bulk and at regular edges where the global eigenvalue density vanishes as a square root. The method relies on studying the characteristic function of the linear statistics (Landon and Sosoe (2018)) by using the cumulant expansion method, along with local laws for the Green function of X-N (Ann. Probab. 48 (2020) 963-1001; Probab. Theory Related Fields 169 (2017) 257-352; J. Math. Phys. 54 (2013) 103504) and analytic subordination properties of the free additive convolution (Dallaporta and Fevrier (2019); Random Matrices Theory Appl. 9 (2020) 2050011). We also prove the analogous results for high-dimensional sample covariance matrices.
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