Bijective combinatorial proof of the commutation of transfer matrices in the dense O(1) loop model

TL;DR

This paper presents a new combinatorial bijection proof for the commutation of transfer matrices in the dense O(1) loop model, connecting statistical physics, combinatorics, and integrable systems.

Contribution

It introduces a novel combinatorial bijection proof for the transfer matrices' commutation, replacing the traditional algebraic Yang-Baxter approach.

Findings

Establishes a bijective proof of transfer matrix commutation

Connects combinatorial structures with statistical physics models

Provides insights into the algebraic and combinatorial interplay

Abstract

The dense O(1) loop model is a statistical physics model with connections to the quantum XXZ spin chain, alternating sign matrices, the six-vertex model and critical bond percolation on the square lattice. When cylindrical boundary conditions are imposed, the model possesses a commuting family of transfer matrices. The original proof of the commutation property is algebraic and is based on the Yang-Baxter equation. In this paper we give a new proof of this fact using a direct combinatorial bijection.