The dimer $m_0$ and hole $n$ densities, as well as the columnar order parameter $m$, can be expressed straightforwardly in terms of the fields $\phi_i$ and $\chi_{ij}$. From this effective theory one can compute an effective potential $\Gamma$ and the configurations of the observables of interest, $n$, $m_0$ and $m$, as functions of $z$ and $V$, and determine the phase diagram.

As a function of hole density (or hole fugacity) and $u$, we find that the phase diagram has two phases (shown qualitatively in Fig. \[fig:sketch\]), namely a dimer-hole liquid phase and a hole-poor phase with long-range columnar order. The nature of the transitions between these phases is incorrectly described by the mean-field theory, particularly at zero doping and near the tricritical point. The correct behavior is the subject of a detailed analysis in the subsequent sections. Nevertheless, the mean-field phase diagram correctly predicts that at low hole densities and moderate values of $u$ the transition between the dimer-hole liquid and the columnar solid phase is continuous, that for large $u$ the transition is first order, and that there is a tricritical point at $$u_{tr} \simeq 2.733, \qquad z_{tr} \simeq 0.075 \;.$$ Remarkably, the Monte Carlo simulations presented in Section 