Abstract
We present conjectured exact expressions for two kinds of correlations in the denseO (n = 1) loop model on square lattices with periodic boundary conditions. These are the probability that a point is surrounded bym loops and theprobability that k consecutive points on a row are on the same or on different loops. The denseO (n = 1) loop model is equivalent to the bond percolation model at the critical point. The formerprobability can be interpreted in terms of the bond percolation problem as giving theprobability that a vertex is on a cluster that is surrounded by clusters and dual clusters. The conjectured expression for this probability involvesa binomial determinant that is known to give weighted enumerations ofcyclically symmetric plane partitions and also of certain kinds of families ofnonintersecting lattice paths. By applying Coulomb gas methods to the denseO (n = 1) loop model, we obtain new conjectures for the asymptotics of this binomial determinant.
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