Abstract
We study ensembles of random symmetric matrices whose entries exhibit certain correlations. Examples are distributions of Curie–Weiss type. We provide a criterion on the correlations ensuring the validity of Wigner’s semicircle law for the eigenvalue distribution measure. In case of Curie–Weiss distributions, this criterion applies above the critical temperature (i.e., \(\beta \,<\,1\)). We also investigate the largest eigenvalue of certain ensembles of Curie–Weiss type and find a transition in its behavior at the critical temperature.
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References
Aldous, D.: Exchangeability and related topics. In: Lecture Notes in Mathematics 117, (Springer, 1985), pp. 1–198
Anderson, G., Guionnet, A., Zeitouni, O.: An Introduction to Random Matrices. Cambridge University Press, Cambridge (2010)
Arnold, L.: On Wigner’s semicircle law for the eigenvalues of random matrices. Z. Wahrsch. Verw. Gebiete 19, 191–198 (1971)
Bai, Z., Silverstein, J.: Spectral Analysis of Large Dimensional Random Matrices. Springer, New York (2010)
Baik, J., Ben Arous, G., Péché, S.: Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices. Ann. Probab. 33, 1643–1697 (2005)
Bryc, W., Dembo, A., Jiang, T.: Spectral measure of large random Hankel, Markov and Toeplitz matrices. Ann. Probab. 34, 1–38 (2006)
Chatterjee, S.: A generalization of the Lindeberg principle. Ann. Probab. 34, 2061–2076 (2006)
Ellis, R.: Entropy, Large Deviations, and Statistical Mechanics. Springer, New York (1985)
de Finetti, B.: Funzione caratteristica di un fenomeno aleatorio, Atti della R. Academia Nazionale dei Lincei, Serie 6. Memorie, Classe di Scienze Fisiche, Mathematice e Naturale, 4, 251–299 (1931)
Friesen, O., Löwe, M.: The semicircle law for matrices with independent diagonals. J. Theor. Probab. 26, 1084–1096 (2013)
Friesen, O., Löwe, M.: A phase transition for the limiting spectral density of random matrices. Electron. J. Probab. 18, 1–17 (2013)
Götze, F., Naumov, A., Tikhomirov, A.: Semicircle Law for a Class of Random Matrices with Dependent Entries, Preprint arXiv:1211.0389v2
Götze, F., Tikhomirov, A.: Limit theorems for spectra of random matrices with martingale structure. Theory Probab. Appl. 51, 42–64 (2007)
Kirsch, W.: A Review of the Moment Method, in preparation
Latała, R.: Some estimates of norms of random matrices. Proc. Am. Math. Soc. 133, 1273–1282 (2005)
Olver, F.: Asymptotics and Special Functions. Academic Press, Waltham (1974)
Pastur, L.: Spectra of random self adjoint operators. Russian Math. Surv. 28, 1–67 (1973)
Pastur, L., Sherbina, M.: Eigenvalue Distribution of Large Random Matrices, Mathematical Surveys and Monographs 171. AMS, Providence (2011)
Stanley, R.: Enumerative Combinatorics, vol. 2. Cambridge University Press, Cambridge (1999)
Schenker, J., Schulz-Baldes, H.: Semicircle law and freeness for random matrices with symmetries or correlations. Math. Res. Lett. 12, 531–542 (2005)
Tao, T.: Topics in Random Matrix Theory. AMS, Providence (2012)
Thompson, C.: Mathematical Statistical Mechanics. Princeton University Press, Princeton (1979)
Wigner, E.: On the distribution of the roots of certain symmetric matrices. Ann. Math. 67, 325–328 (1958)
Acknowledgments
It is a pleasure to thank Matthias Löwe, Münster, and Wolfgang Spitzer, Hagen, for valuable discussion. Two of us (WK and SW) would like to thank the Institute for Advanced Study in Princeton, USA, where part of this work was done, for support and hospitality.
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Hochstättler, W., Kirsch, W. & Warzel, S. Semicircle Law for a Matrix Ensemble with Dependent Entries. J Theor Probab 29, 1047–1068 (2016). https://doi.org/10.1007/s10959-015-0602-3
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DOI: https://doi.org/10.1007/s10959-015-0602-3