Height fluctuations of the full- plane dimer model (with hints at the GFF)

Abstract: In previous sessions, we have seen how to define Gibbs measures for the dimer model on an infinite periodic bipartite graph. I will show in the concrete case of the honeycomb lattice that, under this measure, the variance of the height difference between two faraway faces grows logarithmically with the distance between the faces. This is the first step towards the proof (that I will not do) that the height function converges to the Gaussian Free Field, in a suitable sense.