(i,5) – (i,6) ;]{} iin [-0.5,2.5,3.5,6.5,7.5,8.5]{} [(i,3) – (i,4) ;]{} at (0,-0.5) \[\] [$1$]{}; at (2,-0.5) \[\] [$2$]{}; at (4,-0.5) \[\] [$3$]{}; at (6,-0.5) \[\] [$4$]{}; at (8,-0.5) \[\] [$5$]{}; at (0,1.5) \[\] [$B$]{}; at (0,5.5) \[\] [$B$]{}; at (3,1.5) \[\] [$U$]{}; at (3,5.5) \[\] [$U$]{}; at (7,1.5) \[\] [$U$]{}; at (7,5.5) \[\] [$U$]{}; at (1,3.5) \[\] [$U$]{}; at (5,3.5) \[\] [$U$]{}; at (8,3.5) \[\] [$\overline{B}$]{}; at (-1.5,1.5) \[\] [${{\mathbb U}}^{e}$]{}; at (-1.5,5.5) \[\] [${{\mathbb U}}^{e}$]{}; at (-1.5,3.5) \[\] [${{\mathbb U}}^{o}$]{};

Once again the model is said to be integrable because the Markov matrix $M$ commutes with a whole family of operators generated by the transfer matrix $$[M,t(z)]=0.$$ The goal is now to use the integrable machinery to handle exact computations for these models.

Stationary distribution in a matrix product form. \[subsec:stationary\_state\]
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It turns out that the integrability framework can be efficiently exploited to construct exactly in a matrix product form the stationary state of the models. The general method has been proposed in [@Sasamoto2; @CRV] and we sketch here the main points. We will focus on the open boundaries case but the periodic boundaries case can be treated similarly and we just provide the solution at the end 