Sampling on Lattices with Free Boundary Conditions Using Randomized Extensions

Many statistical physics models are defined on an infinite lattice by taking appropriate limits of finite lattice regions, where a key consideration is how the boundaries are defined. For several models on planar lattices, such as 3-colorings and lozenge tilings, efficient sampling algorithms are known for regions with fixed boundary conditions, where the colors or tiles around the boundary are pre-specified [14], but much less is known about how to sample when these regions have free boundaries, where we want to include all configurations one could see within a finite window. We introduce a method using randomized extensions of a lattice region to relate sampling problems on regions with free boundaries to a constant number of sampling problems on larger regions with fixed boundaries. We demonstrate this principled approach to sample 3-colorings of regions of Z and lozenge tilings of regions of the triangular lattice, building on arguments for the fixed boundary cases due to Luby et al. [14]. Our approach also yields an efficient algorithm for sampling 3-colorings with free boundary conditions on regions with one reflex corner, the first such result for a nonconvex region. This approach can also be generalized to a broad class of mixed boundary conditions. Sampling for these families of regions is significant because it allows us to establish self-reducibility, giving the first algorithm to approximately count the total number of 3-colorings of rectangular lattice regions.