research33 min studyLesson 856 in Mathematics

Gaussian ensembles and joint eigenvalue densities

Derive eigenvalue repulsion while keeping variance, symmetry, ordering, and normalization explicit.

Before you begin

On this page

The question for today

What you’ll understand

Match GOE and GUE entry variances to their matrix densities.
Normalize a labelled Gaussian eigenvalue density and convert it to an ordered density.
Explain why a Vandermonde factor prevents independent Gaussian eigenvalues.

Gaussian matrix entries do not produce independent Gaussian eigenvalues. Diagonalization changes volume, introducing a factor that suppresses nearby eigenvalues. The factor depends on whether the matrix is real symmetric or complex Hermitian. To use the resulting formula, one must specify the entry variances, the integration variables, and whether the eigenvalues have been ordered. A missing permutation factor is a normalization error, even when the qualitative picture of repulsion is correct.

Choose a Gaussian convention before diagonalizing

Our normalized GOE is real symmetric: its independent upper off-diagonal entries have variance $1/N$ , and its independent diagonal entries have variance $2/N$ . Our normalized GUE is complex Hermitian: off-diagonal real and imaginary parts are independent centered Gaussians, each of variance $1/(2N)$ , while diagonal entries are real Gaussians of variance $1/N$ . The lower triangle is fixed by symmetry or conjugation. Both conventions give the limiting semicircle support from minus two to two.

With respect to Lebesgue measure on the independent real matrix coordinates, the density is proportional to the exponential in the display. The symmetry parameter is beta one for GOE and beta two for GUE; it does not count matrix size. Since an off-diagonal squared modulus appears twice in the trace, expanding the trace recovers the stated coordinate variances. Other books rescale Gaussian ensembles differently. Their eigenvalue formulas must be transformed with the corresponding change of variables before numerical edges can be compared.

d\Pr(H)\propto e^{-\beta N\operatorname{Tr}(H^2)/4}\,dH,\qquad \beta=1\ \text{(GOE)},\quad \beta=2\ \text{(GUE)}.

The matrix coordinate measure includes one real coordinate per GOE off-diagonal entry and two per GUE off-diagonal entry.↔ Scroll wide equations horizontally.

The Jacobian creates a dependent point process

Diagonalizing a matrix separates eigenvalue coordinates from angular coordinates. Invariant Gaussian laws make the angular part uniform after the eigenvector phase or sign ambiguity is treated appropriately. Integrating those coordinates leaves the absolute Vandermonde determinant to the symmetry power. For uniformly labelled, unordered eigenvalues, the displayed expression is a probability density on all of real N-dimensional space. Its normalizing constant is the integral of its numerator there. For GUE that integral has the explicit product shown, obtainable from Gaussian orthogonal polynomials.

If instead the eigenvalues are required to increase, the density is multiplied by $N!$ and restricted to that ordered chamber. Coincidences have probability zero, so the chambers partition the integral up to a null set. The Vandermonde power explains repulsion: holding the other coordinates distinct, bringing one pair together suppresses the density by their separation to power beta. This is a joint-density statement, not a claim that a marginal eigenvalue has zero density at every possible location.

p_{N,\beta}(\lambda)=\frac{e^{-\beta N\sum_j\lambda_j^2/4}\prod_{i<j}|\lambda_i-\lambda_j|^\beta}{Z_{N,\beta}},\qquad Z_{N,2}=(2\pi)^{N/2}N^{-N^2/2}\prod_{j=1}^N j!.

This density uses product Lebesgue measure and unordered labels. For beta one, Z is defined by the integral of the displayed numerator; the explicit product here is for beta two only.↔ Scroll wide equations horizontally.

Repulsion and confinement play different roles

Taking the logarithm of the density produces a quadratic confinement term and pairwise logarithms of eigenvalue separation. The former limits the overall spectral spread, while the latter discourages collisions. This energy representation is useful for equilibrium measures, but a finite ensemble is still a probability distribution rather than a set of points exactly minimizing its energy. Fluctuations and entropy remain present. Changing beta changes local repulsion even when the leading limiting empirical density stays the same semicircle under the chosen normalization.

Gaussian invariance supplies exact formulas unavailable for a generic Wigner entry law. A Bernoulli Wigner matrix, for instance, has no joint eigenvalue density with respect to full-dimensional Lebesgue measure at fixed size, even though global limiting behavior can agree with the Gaussian model. Universality compares specified limiting statistics under hypotheses; it does not turn every finite discrete model into the Gaussian joint density. Exact solvability and asymptotic comparison are complementary tools with different logical scopes.

Work it through

Normalize the two-eigenvalue GUE density

For normalized GUE of size two, verify that the unordered eigenvalue density is $\pi^{-1}e^{-x^2-y^2}(x-y)^2$ .

Substitute $N=2$ into the product formula: $Z_{2,2}=(2\pi)2^{-2}(1!)(2!)=\pi$ . The exponential coefficient is $N/2=1$ .
Independently expand $(x-y)^2=x^2+y^2-2xy$ . The cross term integrates to zero because each one-dimensional odd Gaussian integral vanishes.
Each square term integrates to $(\sqrt\pi/2)\sqrt\pi=\pi/2$ , using the Gaussian zeroth and second moments. Their sum is pi, proving normalization. On the ordered chamber $x<y$ , the density is twice the unordered density.

The result

Both the product formula and direct integration give normalizer pi; ordering changes the density by a factor two.

The living notebook

Explore the model

Interactive lab

Interactive lab · function

Isolate the quadratic repulsion factor

Pause/Reset, set coefficient a to 1 and x to 1, giving y equal to 1. For a fresh comparison Pause/Reset, set both a to 1 and x to 0.5, giving y equal to 0.25. Halving a separation quarters its beta-two repulsion factor.

y = ax²TangentSigned integral

y at chosen x: 0.608
Tangent slope · 2ax: 1.35
Signed area · ax³/3: 0.182

Coefficient a0.75Position x0.9

t = 0 s

This lab uses only y = ax². Playback traces the curve within the fixed integration interval from 0 to x; it does not estimate numerical error or change the chosen point. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forThe quadratic graph isolates one Vandermonde factor. It omits Gaussian confinement, other eigenvalues, and normalization, so its height is not an eigenvalue probability density.

Pause & reason

Check your understanding

Interactive checks need JavaScript. You can still read the explanation.

Read the checkpoint explanation

The unordered symmetric density allocates equal mass to each of the N factorial disjoint ordering chambers. Restricting to one chamber requires multiplication by N factorial to restore total mass one.

Your turn

Try it yourself

For the size-two GUE density in the worked example, compute the expected squared gap between the two labelled eigenvalues.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Set $u=(x+y)/2$ and $v=(x-y)/2$ . The Jacobian magnitude is two and the gap squared is $4v^2$ .

Compare with the solution

The transformed density is $(8/\pi)v^2e^{-2u^2-2v^2}$ . Consequently the marginal law of v is proportional to $v^2e^{-2v^2}$ . Its second moment is the ratio $\int v^4e^{-2v^2}dv/\int v^2e^{-2v^2}dv=3/4$ , by Gaussian integration. Multiplying by four gives expected squared gap three. The result concerns a finite normalized size-two ensemble, not the spacing scale in a large matrix. Sorting changes the sign of the gap convention but leaves its square unchanged.

Keep the idea

Take it with you

Entry variances and the trace exponent must describe the same Gaussian convention.
The Vandermonde factor introduces eigenvalue dependence and symmetry-specific repulsion.
Unordered and ordered densities differ by a permutation factor.

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