  We investigate numerically the low temperature equilibration of glassy
systems via non-local Monte Carlo methods. We re-examine several systems that
have been studied previously and investigate new systems in order to test the
performance of such methods near the putative Kauzmann temperature, $T_K$,
where the configurational entropy is presumed to vanish. Our results suggest
that previous numerical claims in favor of and against a thermodynamic
transition at a finite $T_K$ must be re-evaluated. Our work provides some
guidelines and suggestions for future numerical investigations of disordered
systems at high densities and low temperatures.
