Geodesic
Partial linear eigenvalue statistics for non-hermitian random matrices
Teoriâ veroâtnostej i ee primeneniâ, Tome 67 (2022) no. 4, pp. 768-791 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

For an $n \times n$ independent-entry random matrix $X_n$ with eigenvalues $\lambda_1, \dots, \lambda_n$, the seminal work of Rider and Silverstein [Ann. Probab., 34 (2006), pp. 2118–2143] asserts that the fluctuations of the linear eigenvalue statistics $\sum_{i=1}^n f(\lambda_i)$ converge to a Gaussian distribution for sufficiently nice test functions $f$. We study the fluctuations of $\sum_{i=1}^{n-K} f(\lambda_i)$, where $K$ randomly chosen eigenvalues have been removed from the sum. In this case, we identify the limiting distribution and show that it need not be Gaussian. Our results hold for the case when $K$ is fixed as well as for the case when $K$ tends to infinity with $n$. The proof utilizes the predicted locations of the eigenvalues introduced by E. Meckes and M. Meckes, [Ann. Fac. Sci. Toulouse Math. (6), 24 (2015), pp. 93–117]. As a consequence of our methods, we obtain a rate of convergence for the empirical spectral distribution of $X_n$ to the circular law in Wasserstein distance, which may be of independent interest.
Mots-clés : random matrix
Keywords: independent and identically distributed matrices, spectral statistic, linear eigenvalue statistics, rate of convergence, circular law, Wasserstein distance. 
@article{TVP_2022_67_4_a6,
     author = {S. O'Rourke and N. Williams},
     title = {Partial linear eigenvalue statistics for non-hermitian random matrices},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {768--791},
     year = {2022},
     volume = {67},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2022_67_4_a6/}
}
TY  - JOUR
AU  - S. O'Rourke
AU  - N. Williams
TI  - Partial linear eigenvalue statistics for non-hermitian random matrices
JO  - Teoriâ veroâtnostej i ee primeneniâ
PY  - 2022
SP  - 768
EP  - 791
VL  - 67
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/TVP_2022_67_4_a6/
LA  - ru
ID  - TVP_2022_67_4_a6
ER  - 
%0 Journal Article
%A S. O'Rourke
%A N. Williams
%T Partial linear eigenvalue statistics for non-hermitian random matrices
%J Teoriâ veroâtnostej i ee primeneniâ
%D 2022
%P 768-791
%V 67
%N 4
%U http://geodesic.mathdoc.fr/item/TVP_2022_67_4_a6/
%G ru
%F TVP_2022_67_4_a6
S. O'Rourke; N. Williams. Partial linear eigenvalue statistics for non-hermitian random matrices. Teoriâ veroâtnostej i ee primeneniâ, Tome 67 (2022) no. 4, pp. 768-791. http://geodesic.mathdoc.fr/item/TVP_2022_67_4_a6/

[1] J. Alt, L. Erdős, T. Krüger, “Local inhomogeneous circular law”, Ann. Appl. Probab., 28:1 (2018), 148–203 | DOI | MR | Zbl

[2] J. Alt, L. Erdős, T. Krüger, “Spectral radius of random matrices with independent entries”, Probab. Math. Phys., 2:2 (2021), 221–280 ; (2019), 44 pp., arXiv: 1907.13631 | DOI | MR

[3] Z. D. Bai, “Circular law”, Ann. Probab., 25:1 (1997), 494–529 | DOI | MR | Zbl

[4] Zhigang Bao, Guangming Pan, Wang Zhou, “Central limit theorem for partial linear eigenvalue statistics of Wigner matrices”, J. Stat. Phys., 150:1 (2013), 88–129 | DOI | MR | Zbl

[5] T. Berggren, M. Duits, “Mesoscopic fluctuations for the thinned circular unitary ensemble”, Math. Phys. Anal. Geom., 20:3 (2017), 19, 40 pp. | DOI | MR | Zbl

[6] O. Bohigas, M. P. Pato, “Missing levels in correlated spectra”, Phys. Lett. B, 595:1-4 (2004), 171–176 | DOI

[7] O. Bohigas, M. P. Pato, “Randomly incomplete spectra and intermediate statistics”, Phys. Rev. E (3), 74:3 (2006), 036212, 6 pp. | DOI | MR

[8] C. Bordenave, D. Chafaï, “Around the circular law”, Probab. Surv., 9 (2012), 1–89 | DOI | MR | Zbl

[9] P. Bourgade, Horng-Tzer Yau, Jun Yin, “Local circular law for random matrices”, Probab. Theory Related Fields, 159:3-4 (2014), 545–595 | DOI | MR | Zbl

[10] P. Bourgade, Horng-Tzer Yau, Jun Yin, “The local circular law II: the edge case”, Probab. Theory Related Fields, 159:3-4 (2014), 619–660 | DOI | MR | Zbl

[11] D. Chafaï, A. Hardy, M. Maïda, “Concentration for Coulomb gases and Coulomb transport inequalities”, J. Funct. Anal., 275:6 (2018), 1447–1483 | DOI | MR | Zbl

[12] G. Cipolloni, L. Erdős, D. Schröder, Central limit theorem for linear eigenvalue statistics of non-Hermitian random matrices, 2021 (v1 – 2019), 65 pp., arXiv: 1912.04100

[13] G. Cipolloni, L. Erdős, D. Schröder, “Fluctuation around the circular law for random matrices with real entries”, Electron. J. Probab., 26 (2021), 24, 61 pp. ; (2020), 50 pp., arXiv: 2002.02438 | DOI | MR | Zbl

[14] N. Coston, S. O'Rourke, “Gaussian fluctuations for linear eigenvalue statistics of products of independent iid random matrices”, J. Theoret. Probab., 33:3 (2020), 1541–1612 ; (2019 (v1 – 2018)), 64 pp., arXiv: 1809.08367 | DOI | MR | Zbl

[15] R. Durrett, Probability. Theory and examples, Camb. Ser. Stat. Probab. Math., 31, 4th ed., Cambridge Univ. Press, Cambridge, 2010, x+428 pp. | DOI | MR | Zbl

[16] A. Edelman, “The probability that a random real Gaussian matrix has $k$ real eigenvalues, related distributions, and the circular law”, J. Multivariate Anal., 60:2 (1997), 203–232 | DOI | MR | Zbl

[17] G. B. Folland, Real analysis. Modern techniques and their applications, Pure Appl. Math. (N. Y.), 2nd ed., Wiley-Interscience Publ. John Wiley Sons, Inc., New York, 1999, xvi+386 pp. | MR | Zbl

[18] J. Ginibre, “Statistical ensembles of complex, quaternion, and real matrices”, J. Math. Phys., 6 (1965), 440–449 | DOI | MR | Zbl

[19] V. L. Girko, “The circular law”, Theory Probab. Appl., 29:4 (1985), 694–706 | DOI | MR | Zbl

[20] F. Götze, A. Tikhomirov, “The circular law for random matrices”, Ann. Probab., 38:4 (2010), 1444–1491 | DOI | MR | Zbl

[21] I. Jana, “CLT for non-Hermitian random band matrices with variance profiles”, J. Stat. Phys., 187:2 (2022), 13, 25 pp. ; (2020 (v1 – 2019)), 21 pp., arXiv: 1904.11098 | DOI | MR | Zbl

[22] K. Johansson, “On Szegő's asymptotic formula for Toeplitz determinants and generalizations”, Bull. Sci. Math. (2), 112:3 (1988), 257–304 | MR | Zbl

[23] P. Kopel, Linear statistics of non-Hermitian matrices matching the real or complex Ginibre ensemble to four moments, 2015, 52 pp., arXiv: 1510.02987

[24] P. Kopel, S. O'Rourke, Van Vu, “Random matrix products: universality and least singular values”, Ann. Probab., 48:3 (2020), 1372–1410 ; (2019 (v1 – 2018)), 60 pp., arXiv: 1802.03004 | DOI | MR | Zbl

[25] G. Lambert, “Incomplete determinantal processes: from random matrix to Poisson statistics”, J. Stat. Phys., 176:6 (2019), 1343–1374 | DOI | MR | Zbl

[26] E. S. Meckes, M. W. Meckes, “A rate of convergence for the circular law for the complex Ginibre ensemble”, Ann. Fac. Sci. Toulouse Math. (6), 24:1 (2015), 93–117 | DOI | MR | Zbl

[27] M. L. Mehta, Random matrices and the statistical theory of energy levels, Academic Press, New York–London, 1967, x+259 pp. | MR | Zbl

[28] S. O'Rourke, D. Renfrew, “Central limit theorem for linear eigenvalue statistics of elliptic random matrices”, J. Theoret. Probab., 29:3 (2016), 1121–1191 | DOI | MR | Zbl

[29] S. O'Rourke, A. Soshnikov, “Partial linear eigenvalue statistics for Wigner and sample covariance random matrices”, J. Theoret. Probab., 28:2 (2015), 726–744 | DOI | MR | Zbl

[30] V. V. Petrov, Limit theorems of probability theory. Sequences of independent random variables, Oxford Stud. Probab., 4, The Clarendon Press, Oxford Univ. Press, New York, 1995, xii+292 pp. | MR | Zbl

[31] B. Rider, J. W. Silverstein, “Gaussian fluctuations for non-Hermitian random matrix ensembles”, Ann. Probab., 34:6 (2006), 2118–2143 | DOI | MR | Zbl

[32] B. Rider, B. Virág, “The noise in the circular law and the Gaussian free field”, Int. Math. Res. Not. IMRN, 2007:2 (2007), rnm006, 33 pp. | DOI | MR | Zbl

[33] T. Tao, “Outliers in the spectrum of iid matrices with bounded rank perturbations”, Probab. Theory Related Fields, 155:1-2 (2013), 231–263 | DOI | MR | Zbl

[34] T. Tao, Van Vu, “Random matrices: the circular law”, Commun. Contemp. Math., 10:2 (2008), 261–307 | DOI | MR | Zbl

[35] T. Tao, Van Vu, “Random matrices: universality of ESDs and the circular law”, Ann. Probab., 38:5 (2010), 2023–2065 | DOI | MR | Zbl

[36] T. Tao, Van Vu, “Random matrices: universality of local spectral statistics of non-Hermitian matrices”, Ann. Probab., 43:2 (2015), 782–874 | DOI | MR | Zbl

[37] Jun Yin, “The local circular law III: general case”, Probab. Theory Related Fields, 160:3-4 (2014), 679–732 | DOI | MR | Zbl