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. APPENDIX 34 A The dilute O(n=1) model and the site percolation on a triangular lattice 34 B Normalization of the transfer-matrix 36 C L = 1 ground state elements and X current 38 D Proof of the fusion equation 39 E Construction of the open boundary K-matrix from the closed boundary K-matrix via insertion of a rapidity line 41

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

“…In particular, the ψ e element is symmetric in the full set of parameters {x ±1 l , z ±1 1 , .., z ±1 L , x ±1 r } and the recurrence (1) can be also applied to x l and x r . The proof of this statement can be found in [7].…”

“…Finally, we would like to stress that our results are important in the study of the correlation functions of the open boundary dTL model. The first progress in this direction has already been made [7].…”

Fusion hierarchies, T-systems and Y-systems for the dilute $\boldsymbol{A_2^{(2)}}$ loop models

“…However, it is a straightforward exercise to obtain similar results for these three values by renormalizing the face operators and taking the proper limits. We note that, for the important case λ = π 3 which is DLM(2, 3), there is a mapping [46] between configurations of critical site percolation on the triangular lattice and loop configurations of the A 3 The periodic dilute Temperley-Lieb algebra…”

Groundstate finite-size corrections and dilogarithm identities for the twisted $A_1^{(1)}$, $A_2^{(1)}$ and $A_2^{(2)}$ models

“…This is the limiting case DLM(2, 3) [37] of the A (2) 2 loop model with λ → π 3 , where the limit is taken after the face operator is renormalised to remove the factors sin 3λ in the denominators. This mapping of the usual lattice model of site percolation to the loop model is described in [70]. The analysis of finite excitations and conformal partition functions of site percolation would provide a highly nontrivial test of the meaning of universality in the setting of these logarithmic CFTs.…”

“…Specifically, the vertex models are the 6-vertex model [4][5][6][7][8][9], 15-vertex model [10][11][12][13][14][15][16] and Izergin-Korepin 19-vertex model [17][18][19][20][21] respectively. The corresponding loop models are the dense Temperley-Lieb loop model [22][23][24][25][26][27][28], fully packed Temperley-Lieb loop model [29][30][31] and dilute Temperley-Lieb loop model [32][33][34][35][36][37] respectively. Algebraically, these loop models are described by the dense and dilute Temperley-Lieb algebras [38][39][40][41][42].…”