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

Understand

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 $L$.

• 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 $z_1,..,z_L$.
• This polynomial satisfies several recurrence relations which we solve independently in terms of Jacobi-Trudi like determinants.