Bijective combinatorial proof of the commutation of transfer matrices in   the dense O(1) loop model
  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. 
Ron Peled and Dan Romik
math-ph math.CO math.MP math.PR