Around the Razumov-Stroganov conjecture: proof of a multi-parameter sum rule

TL;DR

This paper proves a conjecture linking the sum of groundstate vector entries in an O(1) loop model to the count of n x n alternating sign matrices, using a connection to the six-vertex model.

Contribution

It establishes a new proof of the Razumov-Stroganov conjecture by relating the O(1) loop model to the six-vertex model with domain wall boundary conditions.

Findings

Sum of normalized groundstate vector entries equals the number of n x n alternating sign matrices.

Identifies a correspondence between the state sum of an inhomogeneous O(1) model and the six-vertex model partition function.

Provides a multi-parameter sum rule proof for the Razumov-Stroganov conjecture.

Abstract

We prove that the sum of entries of the suitably normalized groundstate vector of the O(1) loop model with periodic boundary conditions on a periodic strip of size 2n is equal to the total number of n x n alternating sign matrices. This is done by identifying the state sum of a multi-parameter inhomogeneous version of the O(1) model with the partition function of the inhomogeneous six-vertex model on a n x n square grid with domain wall boundary conditions.