Eigenvalues refuse to sit close together, and random points have no such manners
Random points clump together and leave holes. The eigenvalues of a random matrix are just as random, yet they space themselves out. This push is called level repulsion.
One row is eigenvalues; the other is independent random points with the same average gap. Press Roll continuously and watch, then hide the labels and guess which is which.
The eigenvalue row is a β = 1 tridiagonal ensemble at N = 600, unfolded to mean spacing 1 and cropped to a run in the bulk; the other row pins its two ends to the same interval and scatters the rest uniformly, so both rows have 40 gaps with the same total length. Nothing is stretched to fit. Over long runs at β = 1, about 7% of eigenvalue gaps come in under 0.3 against about 26% of the random ones — 6.8% and 25.9% are the closed-form values — and the widest gap in a run of 40 averages 2.4 for eigenvalues against 4.2 for random points.
Scatter forty points at random along a line and you will get knots and bald patches. That is not bad luck, it is what random means: each point is placed without regard for the others, so nothing stops two of them landing almost on top of each other. The most likely gap between neighbours is zero.
The eigenvalues of a random matrix are as random as anything in mathematics, and they do not behave like this at all. They space themselves out. Two of them almost never sit close together, the gaps come in a narrow band around the average, and the whole arrangement looks combed rather than sprinkled. Level repulsion is the name for it, and it is the fact that makes random matrix theory a subject rather than an exercise.
The reason it matters is that the spacing law is far more rigid than the semicircle it lives inside. Change the matrix, change the entries, change the size, and the semicircle stretches and shifts — but rescale to the local average and the spacing distribution does not move at all. It is the same curve for a uranium nucleus, a chaotic billiard, and, apparently, the zeros of the Riemann zeta function.
No statistics yet. One row is the eigenvalues of an actual matrix, rescaled so the average gap is 1. The other row is that many independent random numbers on the same interval — same count, same span, same average gap. Every difference you can see is a difference in arrangement alone.
Try it: Press Roll continuously and just watch for thirty seconds. Then hide the labels and guess: most people get it right almost every time within a few pairs, and it is worth noticing that you can do this reliably before anyone has told you what to look for.
Drag the sample size from 2 matrices to 100 and watch the three distances on the right. Only the distance to the correct curve falls.
Spacings come from the tridiagonal β-ensemble, unfolded across the middle 60% of the spectrum where the density is well behaved; the edges belong to a different limit. Numbers on the right are L1 distances between the histogram and each curve, so smaller is better; binning alone puts a floor of roughly 0.003 on them, well below anything you will see here. At β = 1 and N = 300 the distance to the β = 1 curve measures about 0.10 at 8 matrices, 0.05 at 30 and 0.035 at 100 — while the distance to the β = 2 curve sits near 0.25 and the distance to Poisson near 0.55 no matter how much data you throw at them. A wrong model does not get better with more evidence.
Collect every gap, rescale so the mean is 1, and histogram them. For independent points the answer is e−s: highest at zero, decaying, no structure. For eigenvalues of a real symmetric matrix it is Wigner’s surmise, (π/2)·s·e−πs²/4, which starts at zero, rises to a hump near s = 0.8 and falls away faster than an exponential. Both ends of that description are the story: tiny gaps are suppressed, and so are huge ones.
Wigner wrote his curve down in the mid-1950s as a guess, derived from the smallest case he could solve exactly — the 2×2 below. It is not the true large-matrix answer, which needs Fredholm determinants, but it sits within a percent or two of it, which is why it is still the curve everyone plots.
Try it: Drag the sample size from 2 matrices to 100 and watch the three numbers on the right. Only one of them moves. The distance to the correct curve falls from around 0.10 to 0.035 while the distances to the other two sit exactly where they were — a wrong model does not improve with more evidence, and that asymmetry is worth more than any eyeball fit.
Drag the handle to set a 2×2 matrix and try to close the gap between its eigenvalues — only the exact centre works. Then switch the off-diagonal to complex and try again.
λ₂ − λ₁ = √((a−d)² + 4b²) = 1.421
Drag anywhere in the plane. The gap is the straight-line distance from the origin, exactly — that is what √((a−d)² + (2b)²) says. To make two eigenvalues of a real symmetric 2×2 collide you have to land on a single point of a two-dimensional plane, not cross a line, and hitting a point by accident is much harder than crossing a line. Turn on the complex off-diagonal and the target becomes a point in three dimensions: harder again, which is why the exponent goes from 1 to 2. The measured exponents come out at 2.00 and 2.98 against theoretical 2 and 3.
Why should small gaps be rare at all? The answer is visible on a 2×2 matrix, which is where Wigner got his curve. For [[a, b], [b, d]] the two eigenvalues are (a + d)/2 ± ½√((a − d)² + 4b²), so the gap between them is
λ₂ − λ₁ = √((a − d)² + (2b)²)
which is a distance — the distance from the origin in a plane with coordinates a − d and 2b. To make the eigenvalues collide you must land on a single point of that plane: a must equal d, and b must be zero. Two conditions, not one. Near-collisions therefore occupy a small disc rather than a strip, so the probability of a gap below s falls like s² and the density of gaps rises like s from zero. That linear rise is the entire content of the exponent β = 1.
Make the off-diagonal complex and the plane becomes a three-dimensional space — the real part, the imaginary part, and a − d must all vanish together. Three conditions, probability falling like s³, density rising like s². That is β = 2, and it is why the exponent takes the values it does and not others.
Try it: Drag the handle onto the horizontal axis — you have set b = 0, and the gap is still |a − d|. Drag it onto the vertical axis — now a = d, and the gap is still |2b|. Only the exact centre gives zero. Then switch the off-diagonal to complex, nudge the imaginary part off zero, and the entire plane becomes degeneracy-free: no amount of dragging can close the gap.
Push to 1,200 runs and compare the widest gaps. Random points often leave a hole three spacings wide; eigenvalues almost never do.
Each run takes 40 consecutive unfolded spacings from a β = 1 tridiagonal ensemble at N = 400 and pits them against 40 gaps cut from the same interval by uniformly scattered points, so the two sides always have the same span and the same mean spacing. Measured over 400 runs of 40: the widest eigenvalue gap averages 2.39 and exceeds 3 in 4.3% of runs; the widest random gap averages 4.24 and exceeds 3 in 92.8%. The dashed line is not a fit — it is L·HK/K, the exact mean of the largest of K gaps made by uniform points, which for runs of 40 is 4.28.
Repulsion has a consequence at the other end too. If neighbouring eigenvalues cannot huddle, they also cannot leave a hole — every point that fails to appear in one place has to appear somewhere else, and its neighbours are pushing back. So the largest gap in a stretch of spectrum is not just smaller on average than the largest gap among the same number of random points; it is smaller almost every time, and the two distributions barely overlap.
Try it: Push to 1,200 runs and compare the two tails. Random points open a gap wider than three mean spacings in over 90% of runs of 40; eigenvalues manage it in about 4%. The dashed line is the exact mean for the random side, L·HK/K, and it is worth noting that no such closed form is being used for the eigenvalue side — that histogram is measured.
Everything from here follows from this one behaviour. The exponent β counting the conditions for a degeneracy is the same β that indexes Dyson’s three symmetry classes. Repulsion is why the largest eigenvalue fluctuates on a scale of N−2/3 instead of the N−1/2 a central limit theorem would give, which is the Tracy–Widom law. And it is the spacing distribution, not the semicircle, that Hugh Montgomery described to Freeman Dyson over tea in 1972 and that Dyson recognised on the spot — see also Wigner’s Nuclei for where the idea came from, and Number Theory for the zeta function it collided with.
See also: Wigner’s Nuclei and Number Theory.
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