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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.

MSC:
15B52 Random matrices (algebraic aspects)
60F05 Central limit and other weak theorems
15A18 Eigenvalues, singular values, and eigenvectors
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