Page curve for fermionic Gaussian states

In a seminal paper, Page found the exact formula for average entanglement entropy pure random state. We consider analogous problem ensemble of fermionic Gaussian states, which plays crucial role in context free Hamiltonians. Using recent results from matrix theory, we show that states subsystem $N_A$ out $N$ degrees freedom is given by $\langle S_A\rangle_\mathrm{G}\!=\!(N\!-\!\tfrac{1}{2})\Psi(2N)\!+\!(\tfrac{1}{4}\!-\!N_A)\Psi(N)\!+\!(\tfrac{1}{2}\!+\!N_A\!-\!N)\Psi(2N\!-\!2N_A)\!-\!\tfrac{1}{4}\Psi(N\!-\!N_A)\!-\!N_A$, where $\Psi$ digamma function. Its asymptotic behavior thermodynamic limit S_A\rangle_\mathrm{G}\!=\! N(\log 2-1)f+N(f-1)\log(1-f)+\tfrac{1}{2}f+\tfrac{1}{4}\log{(1-f)}\,+\,O(1/N)$, $f=N_A/N$. Remarkably, its leading order agrees with over eigenstates quadratic Hamiltonians number conservation, as Lydzba, Rigol and Vidmar. Finally, compute variance limit, constant $\lim_{N\to\infty}(\Delta S_A)^2_{\mathrm{G}}=\frac{1}{2}(f+f^2+\log(1-f))$.