2004/01/31 by Saibal Mitra, S. Mitra, B. Nienhuis +5 · 1 citation
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Algorithm #Boundary (topology) #Boundary value problem #Combinatorics #Conjecture #Geometry #Grid #Ground state #Loop (graph theory) #Mathematical analysis #Mathematics #Periodic boundary conditions #Physics #Pure mathematics #Quantum many-body systems #Quantum mechanics #Square (algebra) #Square tiling #State (computer science) #Theoretical and Computational Physics #cond-mat.stat-mech #math-ph #math.MP
paper · pdf · doi:10.1088/1742-5468/2004/09/p09010
published as JSTAT (2004) P09010 · 25 pages, version accepted by JSTAT
arxiv created 2004/09/29 · openalex publication_date 2004/10/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.