Semicircle law and Stieltjes transforms

For the real Wigner ensemble specified in the preceding lesson, with a fixed centered unit-variance off-diagonal law having all moments finite and an independent centered diagonal law with finite moments, the empirical spectral measures converge weakly in probability to the displayed semicircle measure. These are sufficient assumptions rather than a claim of minimal moment hypotheses. Weak convergence tests bounded continuous functions and describes a fraction of the eigenvalues. It does not alone control the largest eigenvalue or the spacing between neighbors.

The limiting density integrates to one, has support from minus two to two, and has the Catalan even moments found by the walk expansion. A finite number of outliers can lie far away while changing the empirical measure by only a vanishing amount. Likewise, a narrow interval whose width shrinks with dimension is outside the direct scope of a fixed-test-function weak limit. Global density, extremal eigenvalues, and local spacing laws are different statements even when they arise from the same ensemble.

For a probability measure on the real line, the chosen transform has positive imaginary part when the argument does, and behaves as minus the reciprocal of the argument at infinity away from the real axis. For the semicircle law it satisfies a quadratic equation. The algebraic equation has two roots; the half-plane and infinity conditions select exactly one. A numerical quadratic solver returning the other root has not found another possible spectral law for the same normalized matrix model.

Write the square root of $z^2-4$ as the analytic branch on the complement of the interval from minus two to two that behaves like $z$ at infinity. In particular it is positive for real arguments greater than two and negative for real arguments less than minus two. This is a statement about the composite analytic square root, not a prescription to take the principal scalar square root separately at every point. That distinction prevents a sign discontinuity when moving across the upper half-plane.

The imaginary part of the transform is an integral of $\eta/((x-E)^2+\eta^2)$ when $z=E+i\eta$. Dividing by pi gives convolution with a Cauchy probability density. At continuity points of a limiting density, sending the positive smoothing parameter to zero recovers that density. Integrating over an interval recovers its mass, with half weights at boundary atoms in the symmetric inversion convention. Atoms themselves are recovered by $\lim_{\eta\downarrow0}\eta\operatorname{Im}m(E+i\eta)$.

At finite dimension the empirical measure is atomic, so a plotted smoothed density depends on the chosen positive smoothing width. Taking that width to zero before a matrix-size limit is different from first obtaining a continuous limiting law. The transform bound $|m(z)|\leq1/\operatorname{Im}z$ also deteriorates as smoothing vanishes. Local laws control this shrinking-scale regime with extra estimates; the global transform convergence theorem does not automatically allow arbitrarily small widths or certify every numerical histogram.