Random matrices and spectral laws

Consider one infinite real symmetric random array. Above the diagonal, entries are independent identically distributed bounded variables with mean zero and variance one. Let diagonal entries be independent identically distributed bounded mean-zero variables, independent of the off-diagonal entries, and mirror entries below the diagonal to enforce symmetry. Its upper-left n-by-n block M_n is a concrete Wigner-type model. Symmetry creates dependence between reflected entries, so it is incorrect to call all n² entries independent. The eigenvalues are real because each block is symmetric, not simply because its entries are random.

Normalize by H_n=M_n/√n. The unscaled row contains n order-one contributions, and the average squared eigenvalue is order n; division by √n keeps a nontrivial spectral scale. The empirical spectral measure places mass 1/n at every eigenvalue of H_n, counting multiplicity. It is itself random but always has total mass one. Using a matrix whose entries already have variance 1/n is another consistent convention; dividing again by √n would incorrectly shrink the spectrum toward zero. Centering also matters because a persistent nonzero mean can create a large structured component.

For the bounded Wigner model just specified, the empirical measures converge almost surely weakly to the semicircle distribution with density √(4−x²)/(2π) on [−2,2]. Weak convergence means averages of bounded continuous test functions converge to their integrals against that density. It does not mean each random eigenvalue converges to a common value, nor that a finite histogram exactly equals a smooth curve. Almost surely describes the probability of the limiting event across ensemble realizations; it is not an assertion that finite random fluctuations disappear at every dimension.

The law concerns the bulk distribution. Control of extreme eigenvalues, microscopic spacings, or eigenvector coordinates involves additional results and assumptions. A measure can have vanishing total mass far away while its largest point keeps escaping, so weak convergence alone does not bound the spectral norm. Other ensembles have other laws: sample covariance matrices lead to different densities and scalings, while nonsymmetric matrices can have complex spectra. Similar-looking histograms do not justify transferring a theorem between these settings without checking the ensemble definition.

Expanding tr(M_n^k) sums products of entries along closed index walks. Independence and centering force many expected products to vanish because an entry appearing only once contributes mean zero. The surviving patterns, after the √n normalization, explain why certain pairings dominate fixed even moments as n grows. Odd limiting moments vanish; even moments are Catalan numbers in the semicircle law. Establishing all convergence and concentration details is a theorem, not merely the observation that a few low-order expected moments have plausible values.

For a first numerical check, compute the second moment from the squared entries rather than diagonalizing the matrix. This is efficient because tr(M²) sums diagonal squares and twice the off-diagonal squares in a symmetric matrix. Its scaling immediately tests whether normalization is sensible. The spectral lab is deliberately a different object: a finite sine sum whose mode count and decay exponent change a deterministic profile. It illustrates how spectra encode scale, but it generates no random matrices, eigenvalue samples, empirical measures, or semicircle limit. A research claim needs the ensemble and probability argument above.