Fusion hierarchies, $T$-systems and $Y$-systems for the dilute $A_2^{(2)}$ loop models

The fusion hierarchy, $T$-system and $Y$-system of functional equations are the key to integrability for 2d lattice models. We derive these equations for the generic dilute $A_2^{(2)}$ loop models. The fused transfer matrices are associated with nodes of the infinite dominant integral weight lattice of $s\ell(3)$. For generic values of the crossing parameter $\lambda$, the $T$- and $Y$-systems do not truncate. For the case $\frac{\lambda}{\pi}=\frac{(2p'-p)}{4p'}$ rational so that $x=\mathrm{e}^{\mathrm{i}\lambda}$ is a root of unity, we find explicit closure relations and derive closed finite $T$- and $Y$-systems. The TBA diagrams of the $Y$-systems and associated Thermodynamic Bethe Ansatz (TBA) integral equations are not of simple Dynkin type. They involve $p'+2$ nodes if $p$ is even and $2p'+2$ nodes if $p$ is odd and are related to the TBA diagrams of $A_2^{(1)}$ models at roots of unity by a ${\Bbb Z}_2$ folding which originates from the addition of crossing symmetry. In an appropriate regime, the known central charges are $c=1-\frac{6(p-p')^2}{pp'}$. Prototypical examples of the $A_2^{(2)}$ loop models, at roots of unity, include critical dense polymers ${\cal DLM}(1,2)$ with central charge $c=-2$, $\lambda=\frac{3\pi}{8}$ and loop fugacity $\beta=0$ and critical site percolation on the triangular lattice ${\cal DLM}(2,3)$ with $c=0$, $\lambda=\frac{\pi}{3}$ and $\beta=1$. Solving the TBA equations for the conformal data will determine whether these models lie in the same universality classes as their $A_1^{(1)}$ counterparts. More specifically, it will confirm the extent to which bond and site percolation lie in the same universality class as logarithmic conformal field theories.