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These notes provide an introduction to the local semicircle law from random matrix theory, as well as some of its applications.
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L. Erdős, B. Schlein, H.T. Yau, Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices . Ann. Probab. 37 (2009), no. 3, 815–852
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L. Erdős, B. Schlein, H.T. Yau, Local semicircle law and complete delocalization for Wigner random matrices . Comm. Math. Phys. 287 (2009), 641–655
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Cited alongside, same era.
L. Erdős, B. Schlein, H.T. Yau, Wegner estimate and level repulsion for Wigner random matrices . Int. Math. Res. Not. 2010 (2009), 436–479
2009
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L. Erdős, J. Ramírez, B. Schlein, H.-T. Yau, Universality of sine-kernel for Wigner matrices with a small Gaussian perturbation . Electron. J. Probab. 15 (2010), no. 18, 526–603
2010
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M. Shcherbina, B. Tirozzi, Central limit theorem for fluctuations of linear eigenvalue statistics of large random graphs . J. Math. Phys. 51 (2010), no. 2, 20 pp
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C. Bordenave, A. Guionnet, Localization and delocalization of eigenvectors for heavy-tailed random matrices . Probab. Theory Related Fields 157 (2013), no. 3–4, 885–953
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C. Cacciapuoti , A. Maltsev, B. Schlein, Local Marchenko-Pastur law at the hard edge of sample covariance matrices . J. Math. Phys. 54 (2013)
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L. Erdős, A. Knowles, H.-T. Yau, J. Yin, Spectral Statistics of Erdős-Rényi Graphs I: Local Semicircle Law . Ann. Probab. 41 (2013), no. 3B, 2279–2375
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2010
Cited alongside, same era.
F. Benaych-Georges, A. Guionnet, M. Maïda, Fluctuations of the extreme eigenvalues of finite rank deformations of random matrices . Electron. J. Probab. Vol. 16 (2011), no. 60, 1621–1662
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Cited alongside, same era.
C. Bordenave, P. Caputo, D. Chafaï, Spectrum of non-Hermitian heavy tailed random matrices . Comm. Math. Phys. 307 (2011), no. 2, 513–560
2011
Cited alongside, same era.
L. Erdős, Universality of Wigner random matrices: a survey of recent results . Russian Mathematical Surveys 66 (2011), no. 3, 507–626
2011
Cited alongside, same era.
L. Erdős, A. Knowles, Quantum diffusion and eigenfunction delocalization in a random band matrix model . Comm. Math. Phys. 303 (2011), no. 2, 509–554
2011
Cited alongside, same era.
L. Erdős, A. Knowles, Quantum diffusion and delocalization for band matrices with general distribution . Ann. Henri Poincaré 12 (2011), no. 7, 1227–1319
2011
Cited alongside, same era.
L. Erdős, H.-T. Yau, J. Yin, Universality for generalized Wigner matrices with Bernoulli distribution . J. of Combinatorics 1 (2011), no. 2, 15–85
2011
Cited alongside, same era.
L.A. Pastur, M. Shcherbina, Eigenvalue distribution of large random matrices . Mathematical Surveys and Monographs, 171. American Mathematical Society, Providence, RI, 2011
2011
Cited alongside, same era.
L. Erdős, A. Knowles, H.-T. Yau, J. Yin, Delocalization and Diffusion Profile for Random Band Matrices . Comm. Math. Phys. 323 (2013), no. 1, 367–416
2013
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L. Erdős, A. Knowles, H.-T. Yau, J. Yin, The local semicircle law for a general class of random matrices . Electron. J. Probab 18 (2013), 1–58
2013
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V. Kargin, An inequality for the distance between densities of free convolutions. Ann. Probab. 41 (2013), no. 5, 3241–3260
2013
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A. Knowles, J. Yin, Eigenvector Distribution of Wigner Matrices . Probab. Theory Related Fields 155 (2013), 543–582
2013
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A. Knowles, J. Yin, The isotropic semicircle law and deformation of Wigner matrices . Comm. Pure Appl. Math. 66 (2013), no. 11, 1663–1750
2013
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F. Benaych-Georges, A. Guionnet, C. Male, Central limit theorems for linear statistics of heavy tailed random matrices . Comm. Math. Phys. 329 (2014), no. 2, 641–686
2014
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A. Bloemendal, L. Erdős, A. Knowles, H.-T. Yau, J. Yin, Isotropic local laws for sample covariance and generalized Wigner matrices . Electron. J. Probab. 19 (2014), no. 33, 1–53
2014
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P. Bourgade, L. Erdős, H.-T. Yau, Universality of general β \beta -ensembles . Duke Math. J. 163 (2014), no. 6, 1127–1190
2014
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P. Bourgade, L. Erdős, H.-T. Yau, Edge Universality of β \beta -ensembles . Comm. Math. Phys. 332 (2014), no. 1, 261–353
2014
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P. Bourgade, H.-T. Yau, J. Yin, Local circular law for random matrices , Probab. Theory Related Fields 159 (2014), no. 3, 545–595
2014
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P. Bourgade, H.-T. Yau, J. Yin, The local circular law II: the edge case . Probab. Theory Related Fields 159 (2014), no. 3–4, 619–660
2014
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C. Cacciapuoti , A. Maltsev, B. Schlein, Bounds for the Stieltjes transform and the density of states of Wigner matrices . Probab. Theory Related Fields 163 (2014), no. 1, 1–29
2014
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S. O’Rourke, V. Vu, Universality of local eigenvalue statistics in random matrices with external source . Random Matrices Theory Appl. 3 (2014), no. 2, 37 pp
2014
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N.S. Pillai, J. Yin, Universality of covariance matrices . Ann. Appl. Probab. 24 (2014), no. 3, 935–1001
2014
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2014
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V. Kargin, Subordination of the resolvent for a sum of random matrices . Ann. Probab. 43 (2015), no. 4, 2119–2150
2015
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A. Knowles, J. Yin, The outliers of a deformed Wigner matrix . Ann. Probab. 42 (2014), no. 5, 1980–2031
2031
Closest in time.
S. Chatterjee, A generalization of the Lindeberg principle . Ann. Probab. 34 (2006), no. 6, 2061–2076
2076
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