GinOE number variance scaling limits

Let $\mathcal{N}_a$ be the number of eigenvalues in the disc of radius $a$, decomposed as $\mathcal{N}_a=\mathcal{N}_a^\mathbb{C}+\mathcal{N}_a^\mathbb{R}$ into complex and real eigenvalues. For the real Ginibre ensemble, write

$$\operatorname{Var} \mathcal{N}_a^{(1)}=\operatorname{Var} \mathcal{N}_a^\mathbb{C}+\operatorname{Var} \mathcal{N}_a^\mathbb{R}+2\operatorname{Cov}(\mathcal{N}_a^\mathbb{C},\mathcal{N}_a^\mathbb{R}).$$

GinOE number variance scaling limits. As $N\to\infty$, the following limits are expected. For fixed $a\in(0,1)$,

$$\frac{\operatorname{Var}\mathcal{N}_a^\mathbb{C}}{\sqrt{N/\pi}}\sim2\sqrt{2}\,a,\qquad \frac{\operatorname{Var}\mathcal{N}_a^\mathbb{R}}{\sqrt{N/\pi}}\sim(2\sqrt{2}-2)\,a,$$ $$2\frac{\operatorname{Cov}(\mathcal{N}_a^\mathbb{C},\mathcal{N}_a^\mathbb{R})}{\sqrt{N/\pi}}\sim-2(2\sqrt{2}-2)\,a,$$

and hence

$$\frac{\operatorname{Var}\mathcal{N}_a^{(1)}}{\sqrt{N/\pi}}\sim2a.$$

In the edge scaling $a=1-\mathcal{S}/\sqrt{2N}$,

$$\frac{\operatorname{Var}\mathcal{N}_a^{(1)}}{\sqrt{N/\pi}}\sim2f(\mathcal{S}),\qquad f(\mathcal{S})=\sqrt{2\pi}\int_{-\infty}^{\mathcal{S}}\frac{\operatorname{erfc}(t)\operatorname{erfc}(-t)}{4}\,dt.$$

For fixed $a>1$,

$$\frac{\operatorname{Var}\mathcal{N}_a^\mathbb{C}}{\sqrt{N/\pi}}\sim2\sqrt{2}-2,\qquad \frac{\operatorname{Var}\mathcal{N}_a^\mathbb{R}}{\sqrt{N/\pi}}\sim2\sqrt{2}-2,$$ $$2\frac{\operatorname{Cov}(\mathcal{N}_a^\mathbb{C},\mathcal{N}_a^\mathbb{R})}{\sqrt{N/\pi}}\sim-2(2\sqrt{2}-2),\qquad \frac{\operatorname{Var}\mathcal{N}_a^{(1)}}{\sqrt{N/\pi}}\sim0.$$

These predictions describe the bulk, edge, and outside scaling regimes of the number variance for the real Ginibre ensemble. The corresponding limiting behaviour is known for the complex and symplectic Ginibre ensembles, while the stated real-ensemble limits remain conjectural in this source.