  We study the phase transition from a nematic phase to a high-density
disordered phase in systems of long rigid rods of length $k$ on the square and
triangular lattices. We use an efficient Monte Carlo scheme that partly
overcomes the problem of very large relaxation times of nearly jammed
configurations. The existence of a continuous transition is observed on both
lattices for $k=7$. We study correlations in the high-density disordered phase,
and we find evidence of a crossover length scale $\xi^* \gtrsim 1400$, on the
square lattice. For distances smaller than $\xi^*$, correlations appear to
decay algebraically. Our best estimates of the critical exponents differ from
those of the Ising model, but we cannot rule out a crossover to Ising
universality class at length scales $\gg \xi^*$. On the triangular lattice, the
critical exponents are consistent with those of the two dimensional three-state
Potts universality class.
