  We study the behavior of a simple string bit model at finite temperature. We
use thermal perturbation theory to analyze the high temperature regime. But at
low temperatures we rely on the large $N$ limit of the dynamics, for which the
exact energy spectrum is known. Since the lowest energy states at infinite $N$
are free closed strings, the $N=\infty$ partition function diverges above a
finite temperature $\beta_H^{-1}$, the Hagedorn temperature. We argue that in
these models at finite $N$, which then have a finite number of degrees of
freedom, there can be neither an ultimate temperature nor any kind of phase
transition. We discuss how the discontinuous behavior seen at infinite $N$ can
be removed at finite $N$. In this resolution the fundamental string bit degrees
of freedom become more active at temperatures near and above the Hagedorn
temperature.

High Energy Physics - Theory