Refined Razumov-Stroganov conjectures for open boundaries

TL;DR

This paper extends the Razumov-Stroganov conjectures by introducing boundary parameters, relating ground states of Markovian Hamiltonians to refined symmetric alternating-sign matrices and FPL configurations, with implications for integrable models.

Contribution

It introduces boundary parameters into the conjecture, linking ground states to refined matrix classes and FPL configurations, expanding the scope of the original conjecture.

Findings

Parameter-dependent ground states relate to refined symmetric alternating-sign matrices.

Conjecture of a relation between two-boundary Hamiltonian ground states and doubly refined FPL configurations.

Potential applications to O(1), XXZ models, and Raise and Peel models.

Abstract

Recently it has been conjectured that the ground-state of a Markovian Hamiltonian, with one boundary operator, acting in a link pattern space is related to vertically and horizontally symmetric alternating-sign matrices (equivalently fully-packed loop configurations (FPL) on a grid with special boundaries).We extend this conjecture by introducing an arbitrary boundary parameter. We show that the parameter dependent ground state is related to refined vertically symmetric alternating-sign matrices i.e. with prescribed configurations (respectively, prescribed FPL configurations) in the next to central row. We also conjecture a relation between the ground-state of a Markovian Hamiltonian with two boundary operators and arbitrary coefficients and some doubly refined (dependence on two parameters) FPL configurations. Our conjectures might be useful in the study of ground-states of the O(1) and XXZ models, as well as the stationary states of Raise and Peel models.