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

Abstract

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) ZL of the ground state eigenvector, which is also the partition function of the critical site percolation model. The normalization ZL is a symmetric polynomial in the inhomogeneities of the lattice . 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 ZL.