On some ground state components of the O(1) loop model

TL;DR

This paper investigates specific ground state components of the O(1) loop model, using integral formulas related to the quantum Knizhnik-Zamolodchikov equation to connect to plane partition enumeration.

Contribution

It introduces new integral formulae for polynomial solutions of the quantum Knizhnik-Zamolodchikov equation and relates these to conjectures about the O(1) loop model's ground state.

Findings

Computed two series of partial sums of the model's entries.

Connected these sums to enumeration problems of plane partitions.

Provided evidence supporting conjectures about the model's ground state components.

Abstract

We address a number of conjectures about the ground state O(1) loop model, computing in particular two infinite series of partial sums of its entries and relating them to the enumeration of plane partitions. Our main tool is the use of integral formulae for a polynomial solution of the quantum Knizhnik-Zamolodchikov equation.