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A refinement of Wigner’s semicircle law in a neighborhood of the spectrum edge for random symmetric matrices. (English. Russian original) Zbl 0930.15025
Funct. Anal. Appl. 32, No. 2, 114-131 (1998); translation from Funkts. Anal. Prilozh. 32, No. 2, 56-79 (1998).
The paper concerns the spectral properties of the Wigner ensemble of random symmetric \(N\times N\) matrices \(W_N\) whose elements \(\{w_{ ij}N^{-1/2}, i\leq j\}\) are jointly independent random variables with zero mean value and variance \(1/4N\). The main purpose is to study the asymptotic behaviour of the averaged traces \(M_{2p}(N) =\mathbb E \text{Tr} W_N^{2p}\) in the limit \(N,p\to \infty\). The principal result obtained is that if \(1\ll p\ll N^{2/3}\), then \[ M_{2p}(N)= {N\over \sqrt{\pi p^3}} \bigl(1+o(1) \bigr)\tag{1} \] provided \(V_k=\mathbb E [w_{ij}]^{2k} \leq(\text{const.} k)^k\). Expression (1) coincides with the leading term of the asymptotics of \(M_{2p} (N)\), \(N\to\infty\) derived for fixed but large enough \(p\) by E. Wigner [Ann. Math. (2) 62, 548–564 (1955; Zbl 0067.08403)]. This means that the eigenvalue distribution of \(W_N\) near the edge of the spectrum is governed by Wigner’s semicircle law not only on the “macroscopic” scale studied by E. Wigner, but on the “mesoscopic” one as well. It is also shown that on this scale the distribution of random variables \(\text{Tr } W_N^{2p} -M_{2p}(N)\) converges to the Gaussian one. The parameters of the latter do not depend on \(V_k\).
To prove (1), the authors develop deep modification of the approach originally suggested by E. Wigner. The principal idea here is to consider the averaged trace \[ \mathbb{E}\sum_i [W_N^{2p}]_{ii} ={1\over N} \sum_{i,\{j\}} \mathbb E w_{ij_1} w_{j_1j_2} \cdots w_{j_{2p-1}i} \tag{2} \] as the weighted sum over paths \(\eta= (i,j_1, \dots, j_{2p-1},i)\). The leading contribution to (2) gives the classes of paths \(\eta\) that correspond to trajectories of simple positive random \(2p\)-step walks. Consideration of the limit \(N,p\to \infty\) in (1) requires knowledge of the statistics self-intersections in the paths. This problem is studied in detail on the basis of combinatorial analysis.

15B52 Random matrices (algebraic aspects)
60F05 Central limit and other weak theorems
15A18 Eigenvalues, singular values, and eigenvectors
Full Text: DOI
[1] E. Wigner, ”Characteristic vectors of bordered matrices with infinite dimensions,” Ann. Math.,62, 548–564 (1955). · Zbl 0067.08403 · doi:10.2307/1970079
[2] E. Wigner, ”On the distribution of the roots of certain symmetric matrices,” Ann. Math.,67, 325–328 (1958). · Zbl 0085.13203 · doi:10.2307/1970008
[3] V. A. Marchenko and L. A. Pastur, ”The distribution of the eigenvalues in some ensembles of random matrices,” Mat. Sb.,72, 507–536 (1967). · Zbl 0152.16101
[4] L. A. Pastur, ”On the spectrum of random matrices,” Teor. Mat. Fiz.,10, 102–112 (1972).
[5] L. A. Pastur, ”The spectra of random self-adjoint operators,” Usp. Mat. Nauk,28, 3–64 (1973). · Zbl 0277.60049
[6] L. Arnold, ”On the asymptotic distribution of the eigenvalues of random matrices,” J. Math. Anal. Appl.,20, 262–268 (1967). · Zbl 0246.60029 · doi:10.1016/0022-247X(67)90089-3
[7] L. Arnold, ”On Wigner’s semicircle law for the eigenvalues of random matrices,” Z. Wahrscheinlichkeitstheorie Verw. Gebiete,19, 191–198 (1971). · Zbl 0212.51006 · doi:10.1007/BF00534107
[8] K. W. Wachter, ”The strong limits of random matrix spectra for sample matrices of independent elements,” Ann. Probab.,6, No. 1, 1–18 (1978). · Zbl 0374.60039 · doi:10.1214/aop/1176995607
[9] V. L. Girko, The Spectral Theory of Random Matrices [in Russian], Nauka, Moscow, 1988. · Zbl 0656.15012
[10] Z. D. Bai, ”Convergence rate of expected spectral distributions of large random matrices. I. Wigner Matrices,” Ann. Probab.,21, 625–648 (1993). · Zbl 0779.60024 · doi:10.1214/aop/1176989261
[11] A. M. Khorunzhy, B. A. Khoruzhenko, L. A. Pastur, and M. V. Shcherbina, ”The largen-limit in statistical mechanics and the spectral theory of disordered systems,” in: Phase Transitions and Critical Phenomena, Vol. 15 (C. Domb and J. L. Lebowitz, eds.). Academic Press, London, 1992.
[12] A. M. Khorunzhy, B. A. Khoruzhenko, and L. A. Pastur, ”Asymptotic properties of large random matrices with independent entries,” J. Math. Phys.,37, 5033–5059 (1996). · Zbl 0866.15014 · doi:10.1063/1.531589
[13] A. Khorunzhy, ”On smoothed density of states for Wigner random matrices,” Random Oper. Stochastic Equations,5, No. 2, 147–162 (1997). · Zbl 0929.60045 · doi:10.1515/rose.1997.5.2.147
[14] Ya. Sinai and A. Soshnikov, ”Central limit theorem for traces of large random symmetric matrices with independent matrix elements,” Bol. Soc. Brasil. Mat.,29, No. 1, 1–24 (1998). · Zbl 0912.15027 · doi:10.1007/BF01245866
[15] C. Tracy and H. Widom, ”On orthogonal and symplectic matrix ensembles,” Commun. Math. Phys.,177, 727–754 (1996). · Zbl 0851.60101 · doi:10.1007/BF02099545
[16] C. Tracy and H. Widom, ”Level-spacing distribution and the Airy kernel,” Commun. Math. Phys.,159, 151–174 (1994). · Zbl 0789.35152 · doi:10.1007/BF02100489
[17] P. J. Forrester, ”The spectrum edge of random matrix ensembles,” Nucl. Phys. B,402, 709–728 (1994). · Zbl 1043.82538 · doi:10.1016/0550-3213(93)90126-A
[18] Z. Füredi and J. Komlóz, ”The eigenvalues of random symmetric matrices,” Combinatorica,1, No. 3, 233–241 (1981). · Zbl 0494.15010 · doi:10.1007/BF02579329
[19] A. Boutet de Monvel and M. V. Shcherbina, ”On the norm of random matrices,” Mat. Zametki,57, No. 5, 688–698 (1995). · Zbl 0847.15010
[20] M. L. Mehta, Random Matrices, Academic Press, New York, 1991.
[21] O. Costin and J. L. Lebowitz, ”Gaussian fluctuations in random matrices,” Phys. Rev. Lett.,75, No. 1, 69–72 (1995). · doi:10.1103/PhysRevLett.75.69
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