The dilute Temperley–Lieb O(n  =  1) loop model on a semi infinite strip: the sum rule

This is the second part of our study of the ground state eigenvector of the transfer matrix of the dilute Temperley–Lieb loop model with the loop weight n  =  1 on a semi infinite strip of width L (Garbali and Nienhuis 2017 J. Stat. Mech. 043108). We focus here on the computation of the normalization (otherwise called the sum rule) Z_L of the ground state eigenvector, which is also the partition function of the critical site percolation model. The normalization Z_L is a symmetric polynomial in the inhomogeneities of the lattice Z₁,..,Z_L. This polynomial satisfies several recurrence relations which we solve independently in terms of Jacobi–Trudi like determinants. Thus we provide a few determinant expressions for the normalization Z_L.