Three symmetry classes, three exponents, and nothing else in between
Eigenvalues push each other apart. The natural next question is: by how much? And the answer turns out not to be a matter of degree. There are exactly three classical ensembles, labelled by a single number β that takes the values 1, 2 and 4 and nothing else — no 1.5, no 3, no continuum. The gap distribution near zero behaves like sβ, so the three classes repel like s, like s² and like s⁴, and a spectrum tells you which one it belongs to.
What decides the class is not the size of the matrix or the size of its entries. It is symmetry, and specifically what happens when you run time backwards. A quantum system with time-reversal symmetry and integer spin can always be written with a real symmetric Hamiltonian: that is β = 1. Break time-reversal symmetry — put the system in a magnetic field, say — and reality is no longer available, so the Hamiltonian is complex Hermitian: β = 2. Keep time-reversal symmetry but give the particles half-integer spin and the natural entries become quaternions: β = 4.
This is Dyson’s classification, written down in the early 1960s in a paper titled The Threefold Way. It says something stronger than “here are three interesting ensembles”. It says that once you demand that a family of Hamiltonians be closed under a symmetry group, these three are all there is.
Put the three spacing laws on one axis, together with the Poisson curve that describes points scattered independently with no repulsion at all, and the ordering is immediate. Poisson peaks at zero: for independent points, the most likely gap is no gap. The three ensembles all vanish at zero, and the higher β, the harder they vanish — the hump shifts right, from 0.798 at β = 1 to 0.886 at β = 2 to 0.940 at β = 4, and it narrows, from variance 0.273 to 0.178 to 0.104 against Poisson’s 1.
Sampling… 0 matrices, 0 gaps so far.
The data comes from the tridiagonal β-ensemble at β = 1, which is exactly the gaussian orthogonal ensemble — real symmetric matrices. Whichever β you sample, the smallest distance falls on the curve with the matching index and on no other. That is what makes this a classification rather than a family: the data tells you its own symmetry class.
Try it: Sample each β in turn and watch the winning distance. At β = 1 the histogram lands on the β = 1 curve at a distance around 0.05 while the β = 2 curve sits at about 0.25; at β = 2 the two swap. The data does not merely say “something repels here” — it names which of the three classes it came from.
The clean way to see an exponent is to take logarithms, because a power law becomes a straight line whose gradient is the power. Plot the three spacing densities on log axes, look at the small-gap end, and you get three lines of visibly different steepness.
Least-squares gradient of log p(s) against log s, fitted over 1.0e-2 ≤ s ≤ 1.0e-1:
Try it: Slide the window down towards zero. The three fitted gradients converge on 1.000, 2.000 and 4.000 — measured, not drawn in. Slide it up towards s = 1 and they all read low, because that far out the exponential factor has taken over and the curve is no longer a pure power. Poisson stays near zero throughout: it has no repulsion to measure.
Three curves, three gradients, and nothing in between them. A power law sβ is a straight line on these axes with gradient β, so the picture is the classification: the exponent is an integer you can read off a graph, and it takes exactly the values 1, 2 and 4.
Try it: The gradients on screen are fitted by least squares to the curves as drawn, not hard-coded. Put the fit window at 10−4 ≤ s ≤ 10−3 and they read 1.000, 2.000 and 4.000. Slide it up to 0.1 ≤ s ≤ 1 and they sag to 0.74, 1.57 and 3.24 — honest arithmetic, not a broken law: out there the exponential factor dominates and the density is no longer a pure power.
The practical consequence is a very sharp suppression of near collisions. The chance that two adjacent levels sit closer than a tenth of the mean spacing is 9.52% for independent points, but only 0.78% at β = 1, 0.107% at β = 2 and 0.0023% at β = 4. Going from β = 1 to β = 4 makes such a near-collision about 340 times rarer; against independent points it is rarer by a factor of about 4,000.
None of this needs to be taken on trust. For 2×2 matrices you can derive the whole thing by counting, and the counting is short. A degeneracy is a coincidence, and the question is how many things have to coincide. Real symmetric matrices need two, complex Hermitian ones three, quaternionic ones five — and each extra condition costs one more power of s.
The general 2×2 matrix of this class is [[a, b], [b, d]], with a and d real — that is forced, since the diagonal of a self-adjoint matrix has to be real — and the off-diagonal entry carrying 1 real number b.
So the whole class is described by 2 + 1 = 3 real numbers. Draw each of them from a Gaussian and you have a random matrix of this symmetry class.
Sampling 2×2 real symmetric matrices… 0 of 120,000.
For 2×2 matrices the surmise is not an approximation — it is the exact answer, which is why the histogram lies on the curve rather than near it. The remarkable part is that the same curve survives to large matrices, where the derivation above no longer applies at all and the true answer needs a Fredholm determinant. Even there the surmise is wrong by under two percent.
Try it: Step through all four stages for each class and watch the two numbers at the bottom move together: the codimension goes 2, 3, 5 and the exponent goes 1, 2, 4. The fitted exponent of P(gap < s) beneath the histogram is measured from 120,000 sampled matrices and comes out at the codimension, not at β. The exponent is one less because differentiating a power drops it by one.
The pattern to hold on to is codimension = β + 1. The degeneracy conditions are one for the diagonal difference plus one for each real component of the off-diagonal entry, and it is that count of off-diagonal components — 1 for a real number, 2 for a complex one, 4 for a quaternion — that is β. The three ensembles are three number systems, and the reason there are three is that the reals, the complexes and the quaternions are the only finite-dimensional associative division algebras over the reals — Frobenius’s theorem.
Once β is written down as a number rather than a symmetry, it is hard not to ask what happens at β = 3. For a long time the answer was that the question was meaningless: there is no third division algebra, so there is no matrix to write down. Then it turned out there is. A random tridiagonal matrix with normal entries on the diagonal and chi entries below it has exactly the eigenvalue density of the β-ensemble — and the chi distribution takes its degrees of freedom as a real number, not an integer.
Sampling… 0 gaps.
Try it: Drag β slowly from one end to the other. The hump slides right and narrows without ever jumping — the measured variance of the gaps falls from about 0.41 at β = 0.5 to about 0.07 at β = 6, matching the predicted value to within a percent or two from β = 1 upwards. Below β = 1 the two separate a little: the surmise is a 2×2 formula, and weak repulsion is where it is worst. Park the dial at β = 3 and watch the classification give up — the data sits between the GUE and GSE curves, about 0.13 from one and 0.15 from the other.
Nothing here needed β to be 1, 2 or 4. The tridiagonal construction puts β into a chi distribution as a number of degrees of freedom, and that number can be anything positive — the matrix model for general β was written down in the early 2000s, decades after the classification. So β is best read as a repulsion strength that happens to be pinned to three values by physics. The values 1, 2 and 4 are the ones a symmetry can produce; every other β on this slider is a legitimate probability distribution with no symmetry behind it.
Try it: Set β = 3 and check the two nearest classical curves. The data is not close to either; it lies between them, exactly where the classification says nothing lives. The distribution is perfectly well defined — it simply is not the spacing law of any symmetry class.
So there are two different statements tangled together, and it is worth keeping them apart. Mathematically, β is a continuous knob controlling how hard eigenvalues repel, and every positive value gives a genuine ensemble. Physically, only three values arise, because only three kinds of self-adjoint matrix are compatible with the way time reversal can act on a quantum system. The threefold way is a statement about physics that happens to be visible in a graph.