  We discuss two types of quantum mechanical models that couple large numbers
of Majorana fermions and have orthogonal symmetry groups. In models of vector
type, only one of the symmetry groups has a large rank. The large $N$ limit is
taken keeping $gN=\lambda$ fixed, where $g$ multiplies the quartic Hamiltonian.
We introduce a simple model with $O(N)\times SO(4)$ symmetry, whose energies
are expressed in terms of the quadratic Casimirs of the symmetry groups. This
model may be deformed so that the symmetry is $O(N)\times O(2)^2$, and the
Hamiltonian reduces to that studied in arXiv:1802.10263. We find analytic
expressions for the large $N$ density of states and free energy. In both vector
models, the large $N$ density of states varies approximately as
$e^{-|E|/\lambda}$ for a wide range of energies. This gives rise to critical
behavior as the temperature approaches the Hagedorn temperature $T_{\rm H} =
\lambda$. In the formal large $N$ limit, the specific heat blows up as $(T_H-
T)^{-2}$, which implies that $T_H$ is the limiting temperature. However, at any
finite $N$, it is possible to reach arbitrarily large temperatures. Thus, the
finite $N$ effects smooth out the Hagedorn transition. We also study models of
matrix type, which have two $O(N)$ symmetry groups with large rank. An example
is provided by the Majorana matrix model with $O(N)^2\times O(2)$ symmetry,
which was studied in arXiv:1802.10263. In contrast with the vector models, the
density of states is smooth and nearly Gaussian near the middle of the
spectrum.
