Local Spectral Statistics and the Nuances of Universality in Random Matrices — Epoche C1
Research Note: Local Spectral Statistics and the Nuances of Universality in Random Matrices The object of this note is the gap between two neighbouring eigenvalues of a large random Hermitian matrix — an eigenvalue being a number $\lambda$ for which $Hv = \lambda v$ has a non-zero solution $v$, and a matrix being Hermitian when $h_{ji}$ equals the complex conjugate of $h_{ij}$, which guarantees that all its eigenvalues are real. The question is whether the law of that gap is as indifferent to the probability distribution of the individual matrix entries as the overall shape of the spectrum is known to be. The answer is a qualified yes, and the qualifications are the interesting part: local statistics are insensitive to a great deal more than one might guess, and then fail entirely — but they fail for a different reason, and in a different class of models, than the folklore suggests. The global statement, and why it is cheap Fix the model first. A Wigner matrix is an $N \times N$ Hermitian matrix $H$ whose entries on and above the diagonal are independent and identically distributed (i.i.d. — drawn independently from one common law), with mean zero and with $\mathbb{E}|h_{ij}|^2 = R^2/N$. The factor $1/N$ is not a convention but a necessity, and it is worth seeing why. The trace of $H^2$ is the sum of the squared eigenvalues, so $$\frac{1}{N}\sum_{k=1}^{N}\lambda_k^2 \;=\; \frac{1}{N}\mathbb{E}\,\mathrm{tr}\,H^2 \;=\; \frac{1}{N}\sum_{i,j}\mathbb{E}|h_{ij}|^2 \;=\; \frac{1}{N}\cdot N^2 \cdot \frac{R^2}{N} \;=\; R^2 .$$ The typical eigenvalue therefore has size $R$, independent of $N$: any other scaling would let the spectrum escape to infinity or collapse to a point. With that normalisation, Wigner's law states that the empirical spectral distribution — the histogram that places mass $1/N$ at each eigenvalue — converges as $N \to \infty$ to the density $$\rho(x) = \frac{1}{2\pi R^2}\sqrt{4R^2 - x^2}, \qquad |x| \le 2R,$$ which is a semicircle of radius $2R$, correctly normalised (the area under a semicircle of radius $2R$ is $2\pi R^2$) and with second moment exactly $R^2$, in agreement with the trace computation above. Wigner obtained this by the method of moments: the $2k$-th moment of the limiting density is $\lim_N \frac{1}{N}\mathbb{E}\,\mathrm{tr}\,H^{2k}$, and expanding the trace gives a sum over closed walks $i_1 \to i_2 \to \cdots \to i_{2k} \to i_1$ on the index set. Two features of that sum decide everything. First, a walk contributes nothing unless every edge it uses is traversed at least twice, because the entries have mean zero and an entry appearing once factors out as a zero expectation. Second, among the walks that do survive, those which use each edge exactly twice have the largest number of free indices and dominate; walks that use some edge three or more times have fewer free indices and are suppressed by a power of $N$. What remains is a count of non-crossing pair partitions, whose number is the Catalan number, and the Catalan numbers are precisely the even moments of the semicircle. This is the honest reason global universality holds so broadly: the calculation only ever touches $\mathbb{E}|h_{ij}|^2$, because every higher moment enters through configurations that are already too index-poor to matter. Global universality is not a deep insensitivity; it is a consequence of the second moment being the only quantity the leading-order combinatorics can see. Nothing in this argument speaks to what happens between two adjacent eigenvalues. What "local" means: the scale on which the question changes In the bulk of the spectrum — away from the edges at $\pm 2R$ — the semicircle puts roughly $N\rho(E)$ eigenvalues per unit length near an energy $E$, so consecutive eigenvalues sit about $1/(N\rho(E))$ apart. Local statistics are the statistics of the spectrum after it has been rescaled by that factor, an operation called unfolding: one sets $x_k = N\rho(E)(\lambda_k - E)$, so that the rescaled points have mean spacing one and the shape of the semicircle has been divided out. Only after unfolding is it meaningful to compare the spectra of different ensembles, or of a matrix with a quantum billiard. Two candidate answers frame the whole discussion. If the unfolded eigenvalues behaved as independent uniform points — a Poisson process — the gap $s$ between neighbours would satisfy $P(s) = e^{-s}$, which is largest at $s = 0$: near-coincidences would be common. The Gaussian ensembles give the opposite. Here the Gaussian Orthogonal Ensemble (GOE) is the set of real symmetric matrices with independent Gaussian entries and a law invariant under all orthogonal changes of basis; the Gaussian Unitary Ensemble (GUE) is its complex Hermitian counterpart, invariant under unitary changes of basis. For both, $P(s) \to 0$ as $s \to 0$: eigenvalues repel. Why they repel, and why the exponent is 1, 2 or 4 The repulsion has a geometric source that requires no probability at all. Ask how many independent conditions a matrix must satisfy for two of its eigenvalues to coincide. Reduce to the relevant $2 \times 2$ block $\begin{pmatrix} a & b \\ \bar{b} & c \end{pmatrix}$, whose two eigenvalues are $\tfrac{1}{2}(a+c) \pm \tfrac{1}{2}\sqrt{(a-c)^2 + 4|b|^2}$. They coincide exactly when $a = c$ and $b = 0$. If the matrix is real symmetric, $b$ is one real number and that is two conditions; if complex Hermitian, $b$ has two real components and it is three; for the quaternionic self-dual case $b$ has four, giving five. Writing $\beta$ for the number of real components of an off-diagonal entry ($\beta = 1, 2, 4$), the degeneracy locus has codimension $\beta + 1$. This is the classical von Neumann–Wigner count. Now put a smooth, non-vanishing probability density on the entries. The gap is $s = \big|\,(a-c,\, 2b_1, \ldots, 2b_\beta)\,\big|$, the Euclidean length of a vector in $\mathbb{R}^{\beta+1}$. The probability that such a vector lands in the shell of radius $s$ and thickness $ds$ is the density there