Geodesic
Bijective Combinatorial Proof of the Commutation of Transfer Matrices in the Dense O(1) Loop Model
Séminaire lotharingien de combinatoire, Tome 73 (2015-2016)
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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. 

@article{SLC_2015-2016_73_a1,
     author = {Ron Peled and Dan Romik},
     title = {Bijective {Combinatorial} {Proof} of the {Commutation} of {Transfer} {Matrices} in the {Dense} {O(1)} {Loop} {Model}},
     journal = {S\'eminaire lotharingien de combinatoire},
     year = {2015-2016},
     volume = {73},
     url = {http://geodesic.mathdoc.fr/item/SLC_2015-2016_73_a1/}
}
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Ron Peled; Dan Romik. Bijective Combinatorial Proof of the Commutation of Transfer Matrices in the Dense O(1) Loop Model. Séminaire lotharingien de combinatoire, Tome 73 (2015-2016). http://geodesic.mathdoc.fr/item/SLC_2015-2016_73_a1/