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Séminaire Lotharingien de Combinatoire, B73b (2015), 13 pp.

# Ron Peled and Dan Romik

# Bijective Combinatorial Proof of the Commutation of Transfer Matrices in the Dense *O*(1) Loop Model

**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.

Received: February 17, 2015.
Accepted: February 24, 2015.

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