Resolvents and self-consistent equations

For a Hermitian matrix $H$ and $z=E+i\eta$ with $\eta>0$, define $G_H(z)=(H-zI)^{-1}$. Real eigenvalues ensure the inverse exists and its operator norm is at most the reciprocal of eta. The normalized trace is precisely the empirical Stieltjes transform with the denominator convention used throughout this module. The imaginary part of the matrix resolvent is positive semidefinite and equals eta times its adjoint product, giving a useful entrywise summation identity as well as a sign check.

For two Hermitian matrices, multiplying the inverse difference on both sides yields the displayed perturbation identity. At the same spectral parameter, its operator norm is at most the matrix perturbation norm divided by eta squared. This can be a coarse but deterministic estimate; it does not need any randomness. As the imaginary part shrinks, the bound worsens, reflecting the greater sensitivity of a finely resolved spectral question. A small perturbation at a fixed smoothing scale need not remain negligible at every microscopic scale.

Delete one index and denote the complementary minor by $H^{(i)}$, with the removed column off that diagonal denoted by $a_i$. Block inversion gives the diagonal resolvent entry as the reciprocal of its diagonal entry minus the spectral parameter minus the quadratic form involving the minor resolvent. For a Wigner matrix, the removed free column is independent of the minor. Conditional on the minor, its covariance converts the quadratic form’s conditional mean into the minor trace divided by matrix dimension.

The quadratic form is not equal to its conditional mean on each realization. Bounding their difference requires assumptions on entry tails or moments and estimates on the minor resolvent. Comparing the minor trace with the full trace introduces another controlled error, usually via interlacing or a resolvent relation. At fixed positive imaginary part these ingredients can lead to the scalar closure $m\approx-1/(z+m)$. The approximation sign records genuine probabilistic work that must be supplied rather than erased by formal manipulation.

Once an approximate equation is obtained, one must show its solution remains near the intended semicircle branch. The derivative of the scalar quadratic at that branch becomes small near a spectral edge, so a small residual need not yield the same solution error everywhere. The positive imaginary part and normalization at infinity help select the correct root, while stability estimates quantify the remaining comparison. More general variance profiles can lead to coupled vector self-consistency equations rather than a single scalar equation.

A local law permits the imaginary part to decrease with matrix dimension, under specific tail and geometry assumptions. Its scale and error must be stated before using it to count eigenvalues in shrinking intervals or compare neighboring levels. The exact resolvent identities remain true at every positive imaginary part, but estimates derived with constants depending on a fixed lower bound for that part do not become local laws automatically. This separation of algebra, concentration, and stability is central to avoiding circular spectral arguments.