  Extending our recent work (\arXiv{\tt hep-th/0310106}) we study the
nonsinglet sector of $c=1$ matrix model by renormalization group analysis for a
gauged matrix quantum mechanics on circle with an appropriate gauge breaking
term to incorporate the effect of world-sheet vortices. The flow equations
indicate BKT phase transition around the self-dual radius and the nontrivial
fixed points of the flow exhibit black hole like phases for a range of
temperatures beyond the self-dual point. One class of fixed point interpolate
between $c=1$ for $R > 1$ and $c=0$ as $R \to 0$ via black hole phase that
emerges after the phase transition. The other two classes of nontrivial fixed
points also develop black hole like behavior beyond R=1. From a thermodynamic
study of the free energy obtained from the Callan-Symanzik equations we show
that all these unstable phases do have negative specific heat. The
thermodynamic quantities indicate that the system does undergo a first order
phase transition near the Hagedorn temperature, around which the new phase is
formed, and exhibits one loop finite energy correction to the Hagedorn density
of states. The flow equations also suggest a deformation of the target space
geometry through a running of the compactification radius where the scale is
given by the dilaton. Remarkably there is a regime where cyclic flow is
observed.
