Contact probability scaling of the Hilbert curve

Using Hi-C experiments, it has become possible to measure contact probability scalings for genomic polymers. However, the theoretical analysis of such scalings remains in its infancy. Here, we prove that contact probability scales with linear distance for lattice approximations of the Hilbert curve. These results point to the potential for new theoretical approaches to the study of contact probability, and shed light on the analysis behind the fractal globule, a recent model for the three-dimensional structure of the human genome.