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We present conjectured exact expressions for two types of correlations in thedense O$(n=1)$ loop model on $L\times \infty$ square lattices with periodicboundary conditions. These are the probability that a point is surrounded by$m$ loops and the probability that $k$ consecutive points on a row are on thesame or on different loops. The dense O$(n=1)$ loop model is equivalent to thebond percolation model at the critical point. The former probability can beinterpreted in terms of the bond percolation problem as giving the probabilitythat a vertex is on a cluster that is surrounded by $\floor{m/2}$ clusters and$\floor{(m+1)/2}$ dual clusters. The conjectured expression for thisprobability involves a binomial determinant that is known to give weightedenumerations of cyclically symmetric plane partitions and also of certain typesof families of nonintersecting lattice paths. By applying Coulomb gas methodsto the dense O$(n=1)$ loop model, we obtain new conjectures for the asymptoticsof this binomial determinant.Comment: 17 pages, replaced by version accepted by JSTA

Exact conjectured expressions for correlations in the dense O(1) loop model on cylinders