Currents in the dilute $O(n=1)$ model

Abstract

In the framework of an inhomogeneous solvable lattice model, we derive exact expressions for a boundary-to-boundary current on a lattice of finite width. The model we use is the dilute $O(n=1)$ loop model, related to the Izergin-Korepin spin-1 chain and the critical site percolation on the triangular lattice. Our expressions are derived based on solutions of the $q$-Knizhnik-Zamolodchikov equations, and recursion relations.