  We consider a quantum system of large size $N$ and its subsystem of size $L$
assuming that $N$ is much larger than $L$, which can also be sufficiently
large, i.e., $1 \ll L \lesssim N $. A widely accepted mathematical version of
this heuristic inequality is the asymptotic regime of successive limits: first
the macroscopic limit $N \to \infty$, then an asymptotic analysis of the
entanglement entropy as $L \to \infty$. In this paper, we consider another
version of the above heuristic inequality: the regime of asymptotically
proportional $L$ and $N$, i.e., the simultaneous limits $L \to \infty,\; N \to
\infty, L/N \to \lambda >0$. Specifically, we consider the system of free
fermions which is in its ground state and such that its one-body Hamiltonian is
a large random matrix, that is often used to model the long-range hopping. By
using random matrix theory, we show that in this case, the entanglement entropy
obeys the volume law known for systems with short-ranged hopping but described
either by a mixed state or a pure strongly excited state of the Hamiltonian. We
also give a streamlined proof of Page's formula for the entanglement entropy of
the black hole radiation for a wide class of typical ground states, thereby
proving the universality of the formula.
