A formula for crossing probabilities of critical systems inside polygons

In this article, we generalize known formulas for crossing probabilities. Prior crossing results date back to J. Cardy's prediction of a formula for the probability that a percolation cluster in two dimensions connects the left and right sides of a rectangle at the percolation critical point in the continuum limit. Here, we predict a new formula for crossing probabilities of a continuum limit loop-gas model on a planar lattice inside a $2N$-sided polygon. In this model, boundary loops exit and then re-enter the polygon through its vertices, with exactly one loop passing once through each vertex, and these loops join the vertices pairwise in some specified connectivity through the polygon's exterior. The boundary loops also connect the vertices through the interior, which we regard as a crossing event. For particular values of the loop fugacity, this formula specializes to FK cluster (resp. spin cluster) crossing probabilities of a critical $Q$-state random cluster (resp. Potts) model on a lattice inside the polygon in the continuum limit. This includes critical percolation as the $Q=1$ random cluster model. These latter crossing probabilities are conditioned on a particular side-alternating free/fixed (resp. fluctuating/fixed) boundary condition on the polygon's perimeter, related to how the boundary loops join the polygon's vertices pairwise through the polygon's exterior in the associated loop-gas model. For $Q\in\{2,3,4\}$, we compare our predictions of these random cluster (resp. Potts) model crossing probabilities in a rectangle ($N=2$) and in a hexagon ($N=3$) with high-precision computer simulation measurements. We find that the measurements agree with our predictions very well for $Q\in\{2,3\}$ and reasonably well if $Q=4$.

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