The largest eigenvalue has its own universal law, and it is not a bell curve
The eigenvalues of a large random matrix fill a semicircle on [−2, 2], and the largest one sits at the right-hand end. Where exactly? Near 2, obviously. The interesting question is how near, and how much it moves from one matrix to the next.
There is an obvious guess. A matrix of size n is built from about n²/2 independent numbers, and quantities built out of many independent numbers usually wobble by an amount of order n−1/2, with a Gaussian shape. Both halves of that guess are wrong. The fluctuation is of order n−2/3, which is smaller, and its shape is skewed, not symmetric.
The reason is that the largest eigenvalue is not free to wander. Every eigenvalue below it is pushing up against it, and the whole stack has to move for the top one to move far. The distribution that comes out of that squeeze was identified by Tracy and Widom in the 1990s, and the rest of this module is largely about how far it travels beyond matrices.
Take the largest eigenvalue, subtract 2, multiply by n2/3, and do it a few thousand times. The histogram that builds up is the Tracy–Widom distribution — and this demo compares it to the published law in the only way that can be done honestly in a browser.
There is no elementary formula for the density. Writing it down needs a solution of the Painlevé II differential equation, and drawing a smooth skewed hump that merely resembled one would be a picture of the answer masquerading as evidence for it. So no theoretical curve is drawn here. What is compared instead is three published numbers — the mean, the variance and the skewness — against the same three numbers measured from the sample. If a histogram matches all three, that means something; if it matches a curve that was drawn to match it, that means nothing.
| Statistic | Measured | Tracy–Widom | Difference |
|---|---|---|---|
| Mean | — | -1.7711 | — |
| Variance | — | 0.8132 | — |
| Skewness | — | 0.2241 | — |
Each sample is one tridiagonal β-ensemble matrix of size 200, diagonalised exactly, with its largest eigenvalue centred at 2 and multiplied by n2/3. No theoretical density is drawn: there is no elementary formula for one. What is being compared is the measured mean, variance and skewness against the published Tracy–Widom values — three numbers that can be checked without a curve.
Try it: Collect a few thousand samples at β = 2 and watch the measured column settle onto the published one. Then drop the matrix size to 50 and collect again — the shape survives, but the mean drifts, because n2/3 is an asymptotic statement and 50 is not asymptotic.
What was measured for this page: at n = 400 with 10,000 samples, β = 2 gave mean −1.7688, variance 0.8137 and skewness 0.2358, against the published −1.7711, 0.8132 and 0.2241. The β = 1 run at the same size gave −1.2787, 1.5805 and 0.2913 against −1.2065, 1.6078 and 0.2935: the variance and skewness land, and the mean is still about 0.07 high, shrinking as n grows. That residual is a real finite-size effect and not a disagreement with the theory, but it is visible, so it is worth stating rather than rounding away.
The exponent is the part of this that can be checked directly, without any special functions at all. Measure how much the largest eigenvalue varies at several matrix sizes, plot the results on logarithmic axes, and read off the slope of the line.
400 tridiagonal β-ensemble spectra at each size from 25 to 800, sampled one size per tick so the page keeps responding. The dashed lines are the two candidate slopes drawn through the leftmost measured point; the solid line is the least-squares fit to the data.
Try it: Run the measurement and compare the fitted slope to the two dashed reference lines. The last column of the table is the cleaner test: multiplying the spread by n2/3 gives a column of numbers that stops changing, which is what an exponent of exactly 2/3 means.
Running this offline over n = 25 to 800 with 2,500 samples at each size, the fitted exponent came out −0.6637 for β = 1 and −0.6671 for β = 2, against a predicted −2/3 = −0.6667 and a central-limit guess of −0.5. The measurement distinguishes the two candidates without any ambiguity at all.
Skewness is a single number, and single numbers are easy to nod at without seeing. So here is the asymmetry directly: the sample, and the best Gaussian that can be fitted to it — same mean, same variance, every advantage given to the bell curve.
| Tail | Measured | A Gaussian would give |
|---|---|---|
| below mean − 2 sd | — | 2.28% |
| above mean + 2 sd | — | 2.28% |
| below mean − 3 sd | — | 0.13% |
| above mean + 3 sd | — | 0.13% |
The dashed curve is not a fitted guess at the Tracy–Widom density — it is a Gaussian, given the sample’s own mean and variance so that it is the most flattering bell curve available. It still fails: the sample leans right, with a long thin tail above the mean and an abrupt wall below it. That wall is the pressure from the eigenvalues underneath, which makes an unusually small largest eigenvalue much harder to arrange than an unusually large one. Samples come from the tridiagonal β-ensemble at n = 200.
Try it: Toggle the bell curve on and off. The two agree near the peak and part company on both flanks — and they part in opposite directions, which is what makes this a shape difference rather than a scale difference.
Over 12,000 samples at n = 200 and β = 2, the measured tails came out asymmetric in exactly the direction the skewness predicts: 1.65% of samples fell more than two standard deviations below the mean against 2.77% more than two above, where a Gaussian would put 2.28% on each side. Three deviations out the gap is starker still — 0.050% below against 0.300% above, six times as many on the right. A largest eigenvalue that is unusually large only has to escape upward into empty space. One that is unusually small has to be pushed down into the crowd, and the crowd pushes back.
The bulk of the spectrum comes in three flavours according to the symmetry of the matrix, and so does the edge. Stronger repulsion holds the top eigenvalue more tightly, so the distribution narrows as β rises.
| measured | expected | |
|---|---|---|
| mean | — | -1.206 |
| variance | — | 1.608 |
| skewness | — | 0.293 |
Published Tracy–Widom values for β = 1.
| measured | expected | |
|---|---|---|
| mean | — | -1.771 |
| variance | — | 0.813 |
| skewness | — | 0.224 |
Published Tracy–Widom values for β = 2.
| measured | expected | |
|---|---|---|
| mean | — | -2.055 |
| variance | — | 0.411 |
| skewness | — | 0.166 |
Published GSE values divided by 2^(1/6): the classical β = 4 normalisation is not the one this sampler uses.
Reading across, the variance falls as β rises — the more strongly the eigenvalues repel one another, the less room the top one has to move. The skewness falls too, but far more slowly, and it never reaches zero: all three laws lean the same way. 0 spectra of size 200 sampled per class, all from the tridiagonal β-ensemble in a single run.
A normalisation to be careful about: the published β = 4 constants (−2.3069, 0.5177, 0.1655) are quoted in the classical convention for the symplectic ensemble, which is not the one the tridiagonal sampler in this module uses. Measured over 12,000 samples at n = 400, the sampler gives mean −2.0376 and variance 0.4115 — and 0.4115 × 21/3 = 0.5185 against the published 0.5177, so the two conventions differ by a factor of 21/6 in scale and by nothing else. That factor is measured here rather than derived, and the test suite asserts it. The skewness, which no choice of scale can change, agrees without any adjustment; it is the statistic to trust when conventions are in doubt.