Abstract
Hochstättler et al. (J Theor Probab 29:1047–1068, 2016) showed that the semicircle law holds for generalized Curie–Weiss matrix ensembles at or above the critical temperature. We extend their result to the case of subcritical temperatures for which the correlations between the matrix entries are stronger. Nevertheless, one may use the concept of approximately uncorrelated ensembles that was first introduced in Hochstättler et al. (2016). In order to do so, one needs to remove the average magnetization of the entries by an appropriate modification of the ensemble that turns out to be of rank 1, thus not changing the limiting spectral measure.
Similar content being viewed by others
References
Anderson, G., Guionnet, A., Zeitouni, O.: An Introduction to Random Matrices. Cambridge University Press, Cambridge (2010)
Arnold, L.: On the asymptotic distribution of the eigenvalues of random matrices. J. Math. Anal. Appl. 20, 262–268 (1967)
Arnold, L.: On Wigner’s semicircle law for the eigenvalues of random matrices. Z. Wahrscheinlichkeitstheorie verw. Geb. 19, 191–198 (1971)
Bai, Z., Silverstein, J.: Spectral Analysis of Large Dimensional Random Matrices. Springer, New York (2010)
Bryc, W., Dembo, A., Jiang, T.: Spectral measure of large random Hankel, Markov and Toeplitz matrices. Ann. Prob. 34, 1–38 (2006)
Ellis, R.: Entropy, Large Deviations, and Statistical Mechanics. Springer, New York (2006)
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)
Füredi, Z., Komlós, J.: The eigenvalues of random symmetric matrices. Combinatorica 1(3), 233–241 (1981)
Götze, F., Naumov, A., Tikhomirov, A.: Limit theorems for two classes of random matrices with dependent entries. Theory Probab. Appl. 59, 23–39 (2015)
Götze, F., Tikhomirov, A.: Limit theorems for spectra of random matrices with martingale structure. Theory Probab. Appl. 51, 42–64 (2007)
Grenander, U.: Probabilities on Algebraic Structures. Wiley (1968)
Hochstättler, W., Kirsch, W., Warzel, S.: Semicircle law for a matrix ensemble with dependent entries. J. Theor. Probab. 29, 1047–1068 (2016)
Hofmann-Credner, K., Stolz, M.: Wigner theorems for random matrices with dependent entries: ensembles associated to symmetric spaces and sample covariance matrices. Electron. Commun. Probab. 13, 401–414 (2008)
Kirsch, W.: Moments in Probability, book in preparation, to appear at DeGruyter
Kirsch, W., Kriecherbauer, T.: Sixty years of moments for random matrices, Preprint arXiv:1612.06725 to appear in: F. Gesztesy et al (eds.) Partial Differential Equations, Mathematical Physics, and Stochastic Analysis. A Volume in Honor of Helge Holden’s 60th Birthday, EMS Congress Reports
Kirsch, W., Kriecherbauer, T.: Largest and second largest singular values of de Finetti random matrices; in preparation
Marchenko, V., Pastur, L.: Distribution of eigenvalues in certain sets of random matrices. Math. USSR-Sbornik 1, 457–483 (1967)
Löwe, M., Schubert, K.: On the limiting spectral density of random matrices filled with stochastic processes, to appear in: Random Oper. Stoch. Equ. arXiv:1512.02498
Olver, F.W.J.: Asymptotics and Special Functions. Academic Press, New York (1974)
Pastur, L.: On the spectrum of random matrices. Theor. Math. Phys. 10(1), 67–74 (1972)
Pastur, L.: Spectra of random selfadjoint operators. Russian Math. Surv. 28, 1–67 (1973)
Pastur, L., Sherbina M.: Eigenvalue distribution of large random matrices. Math. Surv. Monogr. 171, AMS (2011)
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.: Characteristic vectors of bordered matrices with infinite dimension. Ann. Math. 62, 548–564 (1955)
Wigner, E.: On the distribution of the roots of certain symmetric matrices. Ann. Math. 67, 325–328 (1958)
Acknowledgements
The second author would like to thank the Lehrgebiet Stochastics at the FernUniversität in Hagen, where most of the work was accomplished, for support and great hospitality. The authors are grateful to Michael Fleermann for valuable suggestions.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kirsch, W., Kriecherbauer, T. Semicircle Law for Generalized Curie–Weiss Matrix Ensembles at Subcritical Temperature. J Theor Probab 31, 2446–2458 (2018). https://doi.org/10.1007/s10959-017-0768-y
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10959-017-0768-y