by Baxter for the six-vertex model[@baxter73] to fit our MC data and find an accurate estimate of the KT transition coupling in the undoped case.

At low doping, we study the transition from the dimer-hole fluid phase to the columnar state. We confirm that the scaling dimension of the columnar order parameter operator is equal to $1/8$, as predicted by our analytic results of Section \[sec:finite-density\]. We also present a typical set of data that demonstrates how the scaling dimension of the orientational order parameter varies, again in agreement with the analytic results of Section \[sec:finite-density\], and use these results to locate numerically the phase boundary. We then turn to the behavior at larger doping and stronger couplings where the transition becomes first order. We study this regime using grand-canonical Metropolis Monte Carlo simulations. We confirm the first-order nature of the phase transition by means of a careful analysis of the finite-size scaling behavior of the order parameters across the phase boundary and of their susceptibilities. We use these results to locate the phase boundary in the first-order regime as well. In Section \[sec:hole-int\], we use MC simulations to study the effects of a direct hole-hole repulsion which suppresses the 