Wigner ensembles and moment expansions

Begin with a real symmetric matrix $H_N=N^{-1/2}(\xi_{ij})$. The entries strictly above the diagonal are independent copies of one fixed centered real variable of variance one, and the lower entries are obtained by symmetry. Assume all absolute moments of that variable are finite for the moment arguments here. Let the diagonal variables be independent of those entries and of one another, centered, with a fixed distribution having finite moments. They may have a different variance. Independence is required only among the genuinely free coordinates, not between reflected entries.

The empirical spectral measure assigns mass one over dimension to every eigenvalue, counted with multiplicity. Its moments equal normalized traces of matrix powers by the spectral theorem. The second moment is order one under the stated entry variance, explaining the square-root normalization. A diagonal variance of two, as in the normalized Gaussian orthogonal ensemble, changes finite corrections without changing the leading global limit. Complex Hermitian Wigner matrices require conjugate symmetry and variance specified as an absolute square; their real and imaginary coordinate variances must not both be mistaken for the full complex variance.

Represent an index tuple by a closed walk whose vertices are the distinct indices and whose edges are the unordered entry locations it uses. If an edge occurs exactly once, independence and centering force that term’s expectation to vanish. A surviving term uses every edge at least twice. For a fixed walk length, its number of distinct vertices determines how many index assignments it can have, while the normalization penalizes excessive repetitions. Finite moments bound the weights of the remaining patterns.

For an even length, the leading patterns have a tree of distinct vertices and use every tree edge exactly twice. Their ordered traversal patterns are counted by Catalan numbers. Odd lengths have no comparable leading contribution and their normalized expected moments vanish. To turn expected moment convergence into convergence of a random spectral measure, one also controls fluctuations and verifies that the limiting moments determine a unique measure. Counting a few expectations alone does not establish an almost-sure spectral theorem or a local eigenvalue statistic.

Leading moments may agree for Gaussian, bounded, or other suitably integrable entries, but exact finite expectations need not. A four-step walk that repeatedly traverses the same edge depends on the fourth entry moment. Its contribution is lower order than the leading tree terms but can be substantial for a small matrix. Diagonal loops create additional corrections. Universality of a limiting global measure therefore does not mean that all finite ensembles have identical eigenvalue distributions or identical low-dimensional moments.

Independence is equally important. A matrix with all entries equal can have the same normalized second trace moment as a Wigner matrix while concentrating its spectrum into one exceptional eigenvalue. Its fourth moment reveals the difference. Likewise, finite variance is enough for some global-law theorems, but the elementary high-order moment proof presented here deliberately assumes more moments. A proof’s sufficient assumptions should not be confused with the weakest possible assumptions known for a different method.