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