The completely packed O(1) loop model can be viewed as an interface representation of critical bond percolation on the square lattice. Romik’s rationality conjecture predicts the striking fact that certain connectivity probabilities are rational numbers even in the infinite-volume limit.
In this talk, I will present joint work with Nathanaël Berestycki and Sunil Chhita establishing rationality for a particular family of connectivity events. Our approach is based on Kasteleyn theory and reduces the problem to the analysis of a dimer model associated with totally symmetric self-complementary plane partitions (TSSCPPs).