Classical correlations of defects in lattices with geometrical frustration in the motion of a particle

We map certain highly correlated electron systems on lattices with geometrical frustration in the motion of added particles or holes to the spatial defect-defect correlations of dimer models in different geometries. These models are studied analytically and numerically. We consider different coverings for four different lattices: square, honeycomb, triangular, and diamond. In the case of a hard-core dimer covering, we verify the existing results for square and triangular lattices and obtain new ones for the honeycomb and diamond lattices while in the case of a loop covering we obtain new numerical results for all the lattices and use the existing analytical Liouville field theory for the case of a square lattice. The results show power-law correlations for the square and honeycomb lattices, while exponential decay with distance is found for the triangular lattice and exponential decay with the inverse distance on the diamond lattice. We relate this fact to the lack of bipartiteness of the triangular lattice and in the latter case to the three dimensionality of the diamond. The connection of our findings to the problem of fractionalized charge in such lattices is pointed out.