Pfaffian Algorithms for Sampling Routings on Regions with Free Boundary Conditions
Sets of non-intersecting, monotonic lattice paths, or fixed routings, provide a common representation for several combinatorial problems and have been the key element for designing sampling algorithms. Markov chain algorithms based on routings have led to efficient samplers for tilings, Eulerian orientations  and triangulations , while an algorithm which successively calculates ratios of determinants has led to a very fast method for sampling fixed routings . We extend Wilson’s determinant algorithm  to sample free routings where the number of paths, as well as the endpoints, are allowed to vary. The algorithm is based on a technique due to Stembridge for counting free routings by calculating the Pfaffian of a suitable matrix  and a method of Colbourn, Myrvold and Neufeld  for efficiently calculating ratios of determinants. As an application, we show how to sample tilings on planar lattice regions with free boundary conditions.
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