# zbMATH — the first resource for mathematics

On fluctuations of eigenvalues of random Hermitian matrices. (English) Zbl 1039.82504
The author considers quite general random matrix ensembles of $$N \times N$$ Hermitian matrices. The density is given by $$c_{N, M} \exp (-M \roman{trace} V(H) )$$ where $$V$$ is a polynomial of even degree with positive leading coefficient and $$H$$ is thought of as a random matrix. Linear statistics problems involve the computation of $$(1/N)\sum_{i=1}^{N}f(x_{i})$$ where $$x_{i}, i=1, \cdots,N$$, are the random eigenvalues of the Hermitian matrices for suitably nice functions $$f$$. In particular, one can ask for the distribution of such sums. The main result of the paper is that if $$M/N \rightarrow 1$$ then the sums converge in distribution to a normal random variable. The mean, which is zero for suitably normalized $$f$$, and variance are directly related to the constants that appear in the asymptotic formula for the strong Szegő limit theorem. The above result is obtained for general even polynomials provided that certain limiting measures have support in a single interval.
The author shows the connection with the Szegő theorem, and also discusses the orthogonal and symplectic ensembles for general $$\beta$$ in the statistical mechanics interpretation and finally some results for orthonormal polynomials.

##### MSC:
 82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics 82B05 Classical equilibrium statistical mechanics (general)
Full Text:
##### References:
 [1] C. W. J. Beenakker, Universality of Brézin and Zee’s spectral correlator , Nuclear Phys. B 422 (1994), 515-520. [2] D. Bessis, C. Itzykson, and J. B. Zuber, Quantum field theory techniques in graphical enumeration , Adv. in Appl. Math. 1 (1980), no. 2, 109-157. · Zbl 0453.05035 [3] P. Billingsley, Convergence of probability measures , John Wiley & Sons Inc., New York, 1968. · Zbl 0172.21201 [4] A. Boutet de Monvel, L. Pastur, and M. Shcherbina, On the statistical mechanics approach in the random matrix theory: integrated density of states , J. Statist. Phys. 79 (1995), no. 3-4, 585-611. · Zbl 1081.82569 [5] E. Brézin, C. Itzykson, G. Parisi, and J. B. Zuber, Planar diagrams , Comm. Math. Phys. 59 (1978), no. 1, 35-51. · Zbl 0997.81548 [6] E. Brézin and A. Zee, Universality of the correlations between eigenvalues of large random matrices , Nuclear Phys. B 402 (1993), no. 3, 613-627. · Zbl 1043.82534 [7] T. Chan, The Wigner semi-circle law and eigenvalues of matrix-valued diffusions , Probab. Theory Related Fields 93 (1992), no. 2, 249-272. · Zbl 0767.60050 [8] P. Diaconis and M. Shahshahani, On the eigenvalues of random matrices , J. Appl. Probab. 31A (1994), 49-62. JSTOR: · Zbl 0807.15015 [9] P. Di Francesco, P. Ginsparg, and J. Zinn-Justin, $$2$$D gravity and random matrices , Phys. Rep. 254 (1995), no. 1-2, 133. [10] 1 F. Dyson, Statistical theory of the energy levels of complex systems. I , J. Mathematical Phys. 3 (1962), 140-156. · Zbl 0105.41604 [11] 2 F. Dyson, Statistical theory of the energy levels of complex systems. II , J. Mathematical Phys. 3 (1962), 157-165. · Zbl 0105.41604 [12] 3 F. Dyson, Statistical theory of the energy levels of complex systems. III , J. Mathematical Phys. 3 (1962), 166-175. · Zbl 0105.41604 [13] R. Fernandez, J. Fröhlich, and A. D. Sokal, Random walks, critical phenomena, and triviality in quantum field theory , Texts and Monographs in Physics, Springer-Verlag, Berlin, 1992. · Zbl 0761.60061 [14] P. J. Forrester, Global fluctuation formulas and universal correlations for random matrices and log-gas systems at infinite density , Nuclear Phys. B 435 (1995), no. 3, 421-429. · Zbl 1020.82585 [15] Giannoni, M.-J. and Voros, A. and Zinn-Justin, J., eds., Chaos et physique quantique , North-Holland Publishing Co., Amsterdam, 1991. [16] J. Harer and D. Zagier, The Euler characteristic of the moduli space of curves , Invent. Math. 85 (1986), no. 3, 457-485. · Zbl 0616.14017 [17] Helminck, G., ed., Geometric and quantum aspects of integrable systems (The Netherlands, 1992) , Lecture Notes in Physics, vol. 424, Springer-Verlag, Berlin, 1993. · Zbl 0782.00071 [18] D. M. Jackson, On an integral representation for the genus series for $$2$$-cell embeddings , Trans. Amer. Math. Soc. 344 (1994), no. 2, 755-772. JSTOR: · Zbl 0810.05004 [19] K. Johansson, On Szegő’s asymptotic formula for Toeplitz determinants and generalizations , Bull. Sci. Math. (2) 112 (1988), no. 3, 257-304. · Zbl 0661.30001 [20] K. Johansson, On random matrices from the compact classical groups , Ann. of Math. (2) 145 (1997), no. 3, 519-545. JSTOR: · Zbl 0883.60010 [21] D. S. Lubinsky, H. N. Mhaskar, and E. B. Saff, A proof of Freud’s conjecture for exponential weights , Constr. Approx. 4 (1988), no. 1, 65-83. · Zbl 0653.42024 [22] A. P. Magnus, On Freud’s equations for exponential weights , J. Approx. Theory 46 (1986), no. 1, 65-99. · Zbl 0619.42015 [23] M. L. Mehta, Random matrices , Academic Press Inc., Boston, MA, 1991. · Zbl 0780.60014 [24] H. N. Mhaskar and E. B. Saff, Where does the sup norm of a weighted polynomial live? (A generalization of incomplete polynomials) , Constr. Approx. 1 (1985), no. 1, 71-91. · Zbl 0582.41009 [25] H. N. Mhaskar and E. B. Saff, Weighted analogues of capacity, transfinite diameter, and Chebyshev constant , Constr. Approx. 8 (1992), no. 1, 105-124. · Zbl 0747.31001 [26] N. I. Muskhelishvili, Singular integral equations. Boundary problems of function theory and their application to mathematical physics , P. Noordhoff N. V., Groningen, 1953. · Zbl 0051.33203 [27] K. Okikiolu, An analogue of the strong Szegö limit theorem for a Sturm-Liouville operator on the interval , · Zbl 0866.47017 [28] H. D. Politzer, Random-matrix description of the distribution of mesoscopic conductance , Phys. Rev. B 40 (1989), 11917-11919. [29] C. E. Porter, Statistical Theories of Spectra: Fluctuations , Academic Press, New York, 1965. · Zbl 0144.22603 [30] G. Szegö, Orthogonal Polynomials , American Mathematical Society Colloquium Publications, vol. 23, American Mathematical Society, New York, 1939. · Zbl 0023.21505 [31] G. Szegö, On certain Hermitian forms associated with the Fourier series of a positive function , Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (1952), no. Tome Supplementaire, 228-238. · Zbl 0048.04203 [32] C. A. Tracy and H. Widom, Fredholm determinants, differential equations and matrix models , Comm. Math. Phys. 163 (1994), no. 1, 33-72. · Zbl 0813.35110 [33] F. G. Tricomi, Integral equations , Pure and Applied Mathematics. Vol. V, Interscience Publishers, Inc., New York, 1957. · Zbl 0078.09404 [34] H. Weyl, Theorie der Darstellungen kontinuerlichen halbeinfacher Gruppen durch lineare Transformationen, I , Math. Zeit. 23 (1925), 271-309. · JFM 51.0319.01 [35] H. Weyl, The Classical Groups. Their Invariants and Representations , Princeton University Press, Princeton, N.J., 1939. · Zbl 0020.20601
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.