Correlations in the $n\rightarrow 0$ limit of the dense O(n) loop model

Abstract

The two-dimensional dense O(n) loop model for $n=1$ is equivalent to the bond percolation and for $n=0$ to the dense polymers or spanning trees. We consider the boundary correlations on the half space and calculate the probability $P_b$ that a cluster of bonds has a single common point with the boundary. In the limit $n\rightarrow 0$, we find an analytical expression for $P_b$ using the generalized Kirchhoff theorem.