  It has recently been demonstrated that the large N limit of a model of
fermions charged under the global/gauge symmetry group $O(N)^{q-1}$ agrees with
the large $N$ limit of the SYK model. In these notes we investigate aspects of
the dynamics of the $O(N)^{q-1}$ theories that differ from their SYK
counterparts. We argue that the spectrum of fluctuations about the finite
temperature saddle point in these theories has $(q-1)\frac{N^2}{2}$ new light
modes in addition to the light Schwarzian mode that exists even in the SYK
model, suggesting that the bulk dual description of theories differ
significantly if they both exist. We also study the thermal partition function
of a mass deformed version of the SYK model. At large mass we show that the
effective entropy of this theory grows with energy like $E \ln E$ (i.e. faster
than Hagedorn) up to energies of order $N^2$. The canonical partition function
of the model displays a deconfinement or Hawking Page type phase transition at
temperatures of order $1/\ln N$. We derive these results in the large mass
limit but argue that they are qualitatively robust to small corrections in
$J/m$.
