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

TL;DR

This paper analyzes boundary correlations in the dense O(n) loop model, deriving an analytical expression for the probability of boundary contact in the limit as n approaches zero, linking to spanning trees and dense polymers.

Contribution

It provides the first analytical expression for boundary cluster probabilities in the n→0 limit of the dense O(n) loop model using the generalized Kirchhoff theorem.

Findings

Derived an explicit formula for boundary cluster probability P_b as n→0.

Connected the model to spanning trees and dense polymers.

Enhanced understanding of boundary correlations in 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.