Theorem 18.44.27840E247F9D5

Proof. The proof reduces the kernel to a probability for a sum of independent variables, one for each factor, and applies the one-variable transition of section 18.42 to each of them. In each variable, the affine coefficient of a global section of $L^ m$ is entire, and its expression in the frame at infinity must be holomorphic, so it is a polynomial of degree at most $d_ jm$. Hence the monomials $z^\ell e^ m$, $0\leq \ell _ j\leq d_ jm$, form a basis of the global sections. A polynomial vanishes to order at least $k_ m$ at $x$ exactly when all its monomials have total degree at least $k_ m$, so the monomials with $\sum _ j\ell _ j\geq k_ m$ form a basis of $V_ m$. Once $k_ m\leq \min _ j(d_ jm)+1$, which holds for large $m$, all monomials of total degree below $k_ m$ are global, and counting them gives (18.44.4).

Invariance under the rotation of each coordinate makes these monomials orthogonal. Their squared norms are $\prod _ jJ_{j,m,\ell _ j}$, where

These integrals converge even for $\ell =d_ jm$. Write $K_{j,m}=\sum _{\ell =0}^{d_ jm}w_{j,m,\ell }$, and let $\mathsf L_1,\ldots ,\mathsf L_ q$ be independent random variables with $\Pr \{ \mathsf L_ j=\ell \} =w_{j,m,\ell }(t_ j)/K_{j,m}(t_ j)$ for $0\leq \ell \leq d_ jm$. Then

Each factor $(\mathbb P^1,\mathcal O(d_ j))$, with the metric $\mathrm e^{-\psi _ j(|z_ j|^2)}$ and the volume $a_ j\, \mathrm dA_ j$, satisfies the hypotheses of theorem 18.42.15, and $K_{j,m}$ is its full density. Put $V_ j=mG_ j(t_ j)$. Since $\partial _{z_ j}\partial _{\bar z_ j}\psi _ j(|z_ j|^2)=g_ j(t_ j)$, the full density (18.42.25) gives $K_{j,m}=mg_ j(t_ j)/(\pi a_ j(t_ j))(1+o(1))$. For any fixed $y\in \mathbb R$, apply the radial formula (18.42.33) to the integer order

This order tends to infinity and is sublinear. The selected kernel of the factor, divided by its full kernel, is $\Pr \{ \mathsf L_ j\geq b_{j,m}\} $, and it tends to $\Phi (-y)$. Hence

That is, these normalized variables converge in distribution to a standard Gaussian variable. Here $\mu _ j$ and $\sqrt{V_ j}$ serve only as centering and scale; we do not need the exact mean or variance of $\mathsf L_ j$. Since $V_ j/V\to r_ j\alpha _ j/\sum _ i r_ i\alpha _ i>0$, independence gives a standard Gaussian limit for $(\sum _ j\mathsf L_ j-\sum _ j\mu _ j)/\sqrt V$. To see this, use the product of the limiting Gaussian laws on bounded rectangles, and then tightness and the fact that the boundary of a half-space has zero Gaussian measure. Consequently the probability in (18.44.5) tends to $\Phi (\tau )$, which proves the local assertion for every sequence $k_ m\to \infty $ with $k_ m=o(m)$.

For the global assertion we also need the full kernel at a fixed $t>0$, where the formula for shrinking radii does not apply. We suppress the index of the factor and set $F(u)=\psi (\mathrm e^ u)$. Then $F'(u)=H(\mathrm e^ u)$ and $F''(u)=G(\mathrm e^ u)>0$. For $\ell /m$ in a compact subinterval of $(0,d)$, the saddle $s=H^{-1}(\ell /m)$ stays compact, and Laplace expansion gives uniformly

Indeed, the phase in the logarithmic variable has a uniformly positive second derivative there, and outside a fixed neighborhood of the saddle point, convexity and the slopes at the endpoints bound the tails exponentially. For $\ell =mH(t)+O(\sqrt m)$, evaluation yields

For indices with saddle points in a compact interval, the loss in the phase, as a function of $\ell $, has second derivative $1/(mG(s))\geq c/m$, which gives summable Gaussian tails. For the remaining indices, fix small numbers $\delta ,\sigma >0$, and bound the norm $J_{m,\ell }$ from below by the integral over an interval of length $1/m$ about $t\mathrm e^{-\sigma }$ or $t\mathrm e^{\sigma }$. If $|\ell /m-H(t)|\geq \delta $, the appropriate choice gains at least $m(\delta \sigma -C\sigma ^2)\geq cm$ in the exponent. Each such weight $w_{m,\ell }(t)$ is then at most $Cm\mathrm e^{-cm}$, and there are $dm+1$ indices. Gaussian summation therefore gives $K_ m(t)/m\to g(t)/(\pi a(t))$.

In the product chart, $\theta ^ q/q!=\prod _ j(g_ j(t_ j)/\pi )\mathrm dA_ j$, and its mass is $D$ because $\int _0^\infty g_ j(t)\, \mathrm dt=H_ j(\infty )-H_ j(0)=d_ j$. The normalized full density $m^{-q}\prod _ jK_{j,m}$ therefore tends to the density of $\theta ^ q/q!$ with respect to $\nu _ Y$ almost everywhere; the coordinate divisors have zero volume. The total mass of the normalized full kernel measure is $m^{-q}\prod _ j(d_ jm+1)\to D$. For the nonnegative densities $f_ m,f$ of these measures with respect to $\nu _ Y$, dominated convergence applied to $\min (f_ m,f)$ gives

Finally, the difference between the full and the selected kernel is nonnegative and has integral $\binom {k_ m+q-1}{q}=o(m^ q)$. This proves the total-variation convergence. $\square$