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

TL;DR

This paper conjectures exact formulas for correlation probabilities in the dense O(1) loop model on cylinders, linking them to combinatorial objects and providing asymptotic insights via Coulomb gas methods.

Contribution

It introduces conjectured exact expressions for correlations in the dense O(1) loop model, connecting statistical physics with combinatorial enumeration and asymptotic analysis.

Findings

Conjectured exact formulas for loop correlation probabilities.

Connection between loop probabilities and binomial determinants.

Asymptotic behavior derived using Coulomb gas techniques.

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.