Exact conjectured expressions for correlations in the dense O$(1)$ loop model on cylinders

Abstract

We present conjectured exact expressions for two types of correlations in the dense O$(n=1)$ loop model on $L\times \infty$ square lattices with periodic boundary conditions. These are the probability that a point is surrounded by $m$ loops and the probability that $k$ consecutive points on a row are on the same or on different loops. The dense O$(n=1)$ loop model is equivalent to the bond percolation model at the critical point. The former probability can be interpreted in terms of the bond percolation problem as giving the probability that a vertex is on a cluster that is surrounded by $\floor{m/2}$ clusters and $\floor{(m+1)/2}$ dual clusters. The conjectured expression for this probability involves a binomial determinant that is known to give weighted enumerations of cyclically symmetric plane partitions and also of certain types of families of nonintersecting lattice paths. By applying Coulomb gas methods to the dense O$(n=1)$ loop model, we obtain new conjectures for the asymptotics of this binomial determinant.

17 pages, replaced by version accepted by JSTAT

References in corpus (1)

• The quantum symmetric XXZ chain at Delta=-1/2, alternating sign matrices and plane partitions

Cited by in corpus (12)

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• On the weighted enumeration of alternating sign matrices and descending plane partitions
• On polynomials interpolating between the stationary state of a O(n) model and a Q.H.E. ground state
• Short-range correlations in percolation at criticality
• Ground-state properties of a supersymmetric fermion chain
• Exact finite size groundstate of the O(n=1) loop model with open boundaries
• Exact densities of loops in O(1) dense loop model and of clusters in critical percolation on a cylinder
• Nested Closed Paths in Two-Dimensional Percolation
• Universality of closed nested paths in two-dimensional percolation
• Exact asymptotics of the characteristic polynomial of the symmetric Pascal matrix
• Exact densities of loops in O(1) dense loop model and of clusters in critical percolation on a cylinder II: rotated lattice
• Evolving fractal dimensions in iterative bicolored percolation