Abstract
Given an N × N random matrix ensemble, we often want to know, in addition to its limiting eigenvalue distribution, how the eigenvalues fluctuate around the limit. This is important in random matrix theory because in many ensembles, the eigenvalues exhibit repulsion, and this feature is often important in applications (see, e.g. [112]). If we take a diagonal random matrix ensemble with independent entries, then the eigenvalues are just the diagonal entries of the matrix and by independence do not exhibit any repulsion. If we take a self-adjoint ensemble with independent entries, i.e. the Wigner ensemble, the eigenvalues are not independent and appear to spread evenly, i.e. there are few bald spots and there is much less clumping; see Fig. 5.1. For some simple ensembles, one can obtain exact formulas measuring this repulsion, i.e. the two-point correlation functions; unfortunately these exact expressions are usually rather complicated. However, just as in the case of the eigenvalue distributions themselves, the large N limit of these distributions is much simpler and can be analysed.
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References
G.W. Anderson, O. Zeitouni, A CLT for a band matrix model. Probab. Theory Relat. Fields 134(2), 283–338 (2006)
Z.D. Bai, J.W. Silverstein, CLT for linear spectral statistics of large-dimensional sample covariance matrices. Ann. Probab. 32(1A), 553–605 (2004)
Z. Bai, J.W. Silverstein, Spectral Analysis of Large Dimensional Random Matrices. Springer Series in Statistics, 2nd edn. (Springer, New York, 2010)
P. Biane, Some properties of crossings and partitions. Discrete Math. 175(1–3), 41–53 (1997)
T. Cabanal-Duvillard, Fluctuations de la loi empirique de grandes matrices aléatoires. Ann. Inst. H. Poincaré Probab. Statist. 37(3), 373–402 (2001)
B. Collins, J.A. Mingo, P. Śniady, R. Speicher, Second order freeness and fluctuations of random matrices. III. Higher order freeness and free cumulants. Doc. Math. 12, 1–70 (2007) (electronic)
R. Cori, Un code pour les graphes planaires et ses applications (Société Mathématique de France, Paris, 1975). With an English abstract, Astérisque, No. 27
A. Jacques, Sur le genre d’une paire de substitutions. C. R. Acad. Sci. Paris Sér. A-B 267, A625–A627 (1968)
K. Johansson, On fluctuations of eigenvalues of random Hermitian matrices. Duke Math. J. 91(1), 151–204 (1998)
M. Krbalek, P. Šeba, P. Wagner, Headways in traffic flow: remarks from a physical perspective. Phys. Rev. E 64(6), 066119 (2001)
T. Kusalik, J.A. Mingo, R. Speicher, Orthogonal polynomials and fluctuations of random matrices. J. Reine Angew. Math. 604, 1–46 (2007)
J.A. Mingo, A. Nica, Annular noncrossing permutations and partitions, and second-order asymptotics for random matrices. Int. Math. Res. Not. 2004(28), 1413–1460 (2004)
J.A. Mingo, M. Popa, Real second order freeness and Haar orthogonal matrices. J. Math. Phys. 54(5), 051701, 35 (2013)
J.A. Mingo, R. Speicher, Second order freeness and fluctuations of random matrices. I. Gaussian and Wishart matrices and cyclic Fock spaces. J. Funct. Anal. 235(1), 226–270 (2006)
J.A. Mingo, P. Śniady, R. Speicher, Second order freeness and fluctuations of random matrices. II. Unitary random matrices. Adv. Math. 209(1), 212–240 (2007)
J.A. Mingo, R. Speicher, E. Tan, Second order cumulants of products. Trans. Am. Math. Soc. 361(9), 4751–4781 (2009)
A. Nica, R. Speicher, Lectures on the Combinatorics of Free Probability. London Mathematical Society Lecture Note Series, vol. 335 (Cambridge University Press, Cambridge, 2006)
C.E.I. Redelmeier, Real second-order freeness and the asymptotic real second-order freeness of several real matrix models. Int. Math. Res. Not. 2014(12), 3353–3395 (2014)
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Mingo, J.A., Speicher, R. (2017). Fluctuations and Second Order Freeness. In: Free Probability and Random Matrices. Fields Institute Monographs, vol 35. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-6942-5_5
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