Take the measure

$$ d\mu(x)=Z^{-1}\frac{dx}{\xi(1/2+x)} $$

and build its \(n\)-particle orthogonal-polynomial ensemble. This creates a random cloud \(X_1,\ldots,X_n\), together with the zeros of the corresponding orthogonal polynomial \(p_n\).

At first these are two different objects: one random, one deterministic.

After scaling by the \(n\)-th Jacobi coefficient \(b_n\), both clouds converge to the same explicit law on \([-2,2]\):

$$ d\nu_{\mathrm U}(x) =\frac1{2\pi} \operatorname{arcosh}\left(\frac2{|x|}\right) \mathbf 1_{\{0<|x|<2\}}\,dx. $$

It has a logarithmic little mountain at the origin. I find it extremely cute.

Where the scale comes from#

The Jacobi coefficients grow on a Lambert-\(W\) scale:

$$ b_n\sim \frac{\pi n}{W_0(2n/e)}. $$

This is almost linear, but slowed by a logarithm. More importantly, fixed shifts become negligible:

$$ \frac{b_{n+k}}{b_n}\longrightarrow 1 $$

for every fixed integer \(k\).

That turns the rescaled Jacobi matrices into the same local tridiagonal object, with symbol

$$ w+w^{-1}. $$

Once that right limit appears, several asymptotic theorems click into place at once.

The moments are central binomial coefficients wearing tiny hats#

The limiting odd moments vanish. The even moments are

$$ \int x^{2m}\,d\nu_{\mathrm U}(x) =\frac{\binom{2m}{m}}{2m+1}. $$

There is a lovely probabilistic model for this. If \(T\) is uniform on \([0,1]\) and \(\Theta\) is uniform on \([0,2\pi]\), independently, then

$$ 2T\cos\Theta $$

has exactly this law.

Condition on \(T\), and you get an arcsine distribution on \([-2T,2T]\). Then average over all radii \(T\). The factor \(1/(2m+1)\) is just the radial moment. The central binomial coefficient comes from the closed nearest-neighbor walks hidden in \((w+w^{-1})^{2m}\).

This is one of those derivations where every symbol suddenly admits it was geometry the whole time.

The factor of two ambush#

The source conventions do not all normalize the recurrence coefficient in the same way. If you translate them carelessly, you get support \([-1,1]\) instead of \([-2,2]\).

The correct parameter map doubles the off-diagonal recurrence radius. The local arcsine law lives on \([-2,2]\), and the uniform radial mixture keeps that support.

The verifier checks this normalization directly, reconstructs the exact Ullman moments through moment \(24\), pins the primary-source artifacts, and verifies the Lambert-\(W\) regular variation.

What this does and does not remember#

The random empirical measure converges weakly in probability. The zero-counting measure converges weakly as well. The particle cloud and polynomial-zero cloud therefore share one macroscopic shape.

But that shape is universal for a much larger recurrence class. You can modify the underlying entire function to introduce explicit off-axis zeros and preserve the same asymptotic cloud.

So this is real structure, not a disguised criterion for the location of the zeros of \(\xi\). It tells us what the reciprocal-\(\Xi\) ensemble looks like from very far away. It deliberately forgets the fine arithmetic information.

Sometimes forgetting exactly the right thing produces a beautiful theorem.

Notebook references: C-0027, C-0028, C-0031, K-0002