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On the statistical mechanics approach in the random matrix theory: Integrated density of states. (English) Zbl 1081.82569
Summary: We consider the ensemble of random symmetric \(n\times n\) matrices specified by an orthogonal invariant probability distribution. We treat this distribution as a Gibbs measure of a mean-field-type model. This allows us to show that the normalized eigenvalue counting function of this ensemble converges in probability to a nonrandom limit as \(n\to \infty\) and that this limiting distribution is the solution of a certain self-consistent equation.

82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
15B52 Random matrices (algebraic aspects)
Full Text: DOI
[1] E. Wigner, Statistical properties of real symmetric matrices with many dimensions,Can. Math. Proc. 1957:174–198.
[2] M. L. Mehta,Random Matrices (Academic Press, New York, 1967).
[3] V. L. Girko,Spectral Theory of Random Matrices (Kluwer, Dordrecht, 1990). · Zbl 0717.60047
[4] F. Haake,Quantum Signatures of Chaos (Springer-Verlag, Heidelberg, 1991). · Zbl 0741.58055
[5] R. Fernandez, J. Frohlich, and A. Sokal,Random Walks, Critical Phenomena and Triviality in the Quantum Field Theory (Springer-Verlag, Heidelberg, 1992).
[6] E. Brezin, C. Itzykson, G. Parisi, and J. Zuber, Planar diagrams,Commun. Math. Phys. 59:35–51 (1978). · Zbl 0997.81548 · doi:10.1007/BF01614153
[7] D. Bessis, C. Itzykson, and J. Zuber, Quantum field theory techniques in graphical enumeration.Adv. Appl. Math. 1:109–157 (1980). · Zbl 0453.05035 · doi:10.1016/0196-8858(80)90008-1
[8] D. Lechtenfeld, A. Ray, and R. Ray, Phase diagram and orthogonal polynomials in multiple-well matrix models,Int. J. Mod. Phys. 6:4491–4515 (1991). · doi:10.1142/S0217751X91002148
[9] L. Pastur, On the universality of the level spacing distribution for some ensemble of random matrices,Lett. Math. Phys. 25:259–265 (1992). · Zbl 0758.15017 · doi:10.1007/BF00398398
[10] F. J. Dyson, Statistical theory of the energy levels of the complex system. I–III,J. Math. Phys. 3:379–414 (1962). · Zbl 0105.41604
[11] M. Kac, G. Uhlenbeck, and P. Hemmer, On the van-der-Waals theory of vapor liquid equilibrium,J. Math. Phys. 4:216–247 (1963). · Zbl 0938.82518 · doi:10.1063/1.1703946
[12] P. Hemmer and J. Lebowitz, System with weak long-range potentials, inPhase Transitions and Critical Phenomena (Academic Press, New York, 1973).
[13] L. Pastur and M. Shcherbina, Long-range limit of correlation functions of lattice systems,Teor. Mat. Fiz. 61:3–16 (1984) [in Russian].
[14] M. Shcherbina, Classical Heisenberg model at zero temperature,Teor. Mat. Fiz. 81:134–144 (1989) [in Russian].
[15] K. Demetrefi, Two-dimensional quantum gravity, matrix models and string theory,Int. J. Mod. Phys. A 8:1185–1244 (1993). · doi:10.1142/S0217751X93000497
[16] E. Brezin and A. Zee, Universality of the correlations between eigenvalues of large random matrices,Nucl. Phys. B 402:613–627 (1993). · Zbl 1043.82534 · doi:10.1016/0550-3213(93)90121-5
[17] G. M. Cicuta, L. Molinari, and E. Montaldi, Largen phase transition in low dimensions,Mod. Phys. Lett. A 1:125–129 (1986). · doi:10.1142/S021773238600018X
[18] N. I. Muskhelishvili,Singular Integral Equations (Noordhoff, Groningen, 1953). · Zbl 0051.33203
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