The Page–Sen calculation used in the proxy

We record the finite calculation to make the imported random-state input explicit. The result is due to Page and Sen PageAverage,Sen; no priority is claimed.

Let $m\le n$, $D=mn$, and let $G$ be an $m\times n$ standard complex Gaussian matrix. Put

Gaussian radius and direction are independent, $T$ is Gamma with shape $D$, and $\rho$ is the Haar reduction. Hence, in natural entropy units,

because $\mathrm{Tr}(W\ln W)=T\ln T+T\mathrm{Tr}(\rho \ln \rho )$ and $\mathbb{E}(T\ln T)=D{\psi }_0(D+1)$.

Write $\alpha =n-m$. The singular-value Jacobian gives the eigenvalue density proportional to

Orthogonalizing the Vandermonde with generalized Laguerre polynomials gives the one-point density

These statements follow by the singular-value change of variables and expansion of the squared orthogonal-polynomial determinant.

Rodrigues's formula and integration by parts give the norms and orthogonality. The identities

follow respectively from coefficient comparison and differentiation of the generating function

Consequently

Indeed, differentiate

the polynomial derivative cross term is $-j{h}_j$. Differentiation under the integral is justified locally for $\alpha >-1$ by the exponential weight and integrability at zero.

Summing the displayed logarithmic integrals gives

For example, substitute ${\psi }_0(r)={\psi }_0(n+1)-{{\sum }_{\ell =r}}^n{\ell }^{-1}$ and use

to evaluate the finite double sum. Equation (eq:gamma-page) becomes

Using ${\psi }_0(\ell +1)={\mathsf{H}}_{\ell }-{\gamma }_{\mathrm{E}}$ and dividing by $\ln 2$ proves Lemma lem:pagesen, including $m=1$.