state. This mapping works strictly speaking only for the fully packed case. Holes are sites that don’t belong to any dimer and thus represent violations of the full packing rule. They play the role of topological defects (“vortices”) in the (dual) height representation.

The exact solution of the non-interacting fully packed dimer model on the square (and other) lattice has been known for a long time[@fisher63]. In particular the long-distance behavior of the dimer density correlation functions and the hole density correlation functions are known explicitly[@fisher63; @youngblood80]. These correlation functions obey power law behaviors and show that this is a critical system. Here we will use the standard approach to map the exact long distance behavior of two-dimensional critical systems to the behavior of the simplest critical system, the Gaussian or free boson model[@kadanoff77]. This approach is consistent for the free dimer model on an even lattice with periodic boundary conditions since its central charge (or conformal anomaly) is also $c=1$.

We will use as reference states (“ideal states” in the terminology of Kondev and Henley[@kondev-henley96]) the four columnar states of the dimer coverings, which have the largest number of flippable plaquettes, and use them to define an effective field 