Exact expressions for correlations in the ground state of the dense O(1) loop model

Abstract

Conjectures for analytical expressions for correlations in the dense O$(1)$ loop model on semi infinite square lattices are given. We have obtained these results for four types of boundary conditions. Periodic and reflecting boundary conditions have been considered before. We give many new conjectures for these two cases and review some of the existing results. We also consider boundaries on which loops can end. We call such boundaries ''open''. We have obtained expressions for correlations when both boundaries are open, and one is open and the other one is reflecting. Also, we formulate a conjecture relating the ground state of the model with open boundaries to Fully Packed Loop models on a finite square grid. We also review earlier obtained results about this relation for the three other types of boundary conditions. Finally, we construct a mapping between the ground state of the dense O$(1)$ loop model and the XXZ spin chain for the different types of boundary conditions.

25 pages, version accepted by JSTAT

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• Inhomogeneous loop models with open boundaries
• Raise and Peel Models of fluctuating interfaces and combinatorics of Pascal's hexagon
• Short-range correlations in percolation at criticality
• Ground-state properties of a supersymmetric fermion chain
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• Koornwinder polynomials and the XXZ spin chain
• Exact finite size groundstate of the O(n=1) loop model with open boundaries
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• Critical dense polymers with Robin boundary conditions, half-integer Kac labels and ${\mathbb{Z}}_4$ fermions
• Path representation of maximal parabolic Kazhdan-Lusztig polynomials
• Connectivity patterns in loop percolation I: the rationality phenomenon and constant term identities
• Proof of Razumov-Stroganov conjecture for some infinite families of link patterns
• $U_q{\mathfrak{sl}}_2$-invariant non-compact boundary conditions for the XXZ spin chain
• A_k Generalization of the O(1) Loop Model on a Cylinder: Affine Hecke Algebra, q-KZ Equation and the Sum Rule
• The dilute Temperley-Lieb O($n=1$) loop model on a semi infinite strip: the sum rule
• The dilute Temperley-Lieb O($n=1$) loop model on a semi infinite strip: the ground state
• Symmetry classes of alternating sign matrices in the nineteen-vertex model
• Currents in the dilute $O(n=1)$ model
• Fully packed loop models on finite geometries
• Non-contractible loops in the dense O(n) loop model on the cylinder
• Exact densities of loops in O(1) dense loop model and of clusters in critical percolation on a cylinder
• Finite size lattice results for the two-boundary Temperley--Lieb loop model
• Bijective combinatorial proof of the commutation of transfer matrices in the dense O(1) loop model
• A Positive integral property on the ground state of the two-boundary Temperley--Lieb Hamiltonian
• Correlations in the $n\to 0$ limit of the dense O(n) loop model
• Exact densities of loops in O(1) dense loop model and of clusters in critical percolation on a cylinder II: rotated lattice
• The ghost algebra and the dilute ghost algebra
• The scalar product of XXZ spin chain revisited. Application to the ground state at $\Delta =-1/2$
• Density profiles in the raise and peel model with and without a wall. Physics and combinatorics
• Avalanches in the Raise and Peel model in the presence of a wall