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) 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 . This polynomial satisfies several recurrence relations which we solve independently in terms of Jacobi-Trudi like determinants. Thus we provide a few determinant expr..