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The fusion hierarchy, $T$-system and $Y$-system of functional equations arethe key to integrability for 2d lattice models. We derive these equations forthe generic dilute $A_2^{(2)}$ loop models. The fused transfer matrices areassociated 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 closurerelations and derive closed finite $T$- and $Y$-systems. The TBA diagrams ofthe $Y$-systems and associated Thermodynamic Bethe Ansatz (TBA) integralequations are not of simple Dynkin type. They involve $p'+2$ nodes if $p$ iseven 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 originatesfrom the addition of crossing symmetry. In an appropriate regime, the knowncentral 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 loopfugacity $\beta=0$ and critical site percolation on the triangular lattice${\cal DLM}(2,3)$ with $c=0$, $\lambda=\frac{\pi}{3}$ and $\beta=1$. Solvingthe TBA equations for the conformal data will determine whether these modelslie in the same universality classes as their $A_1^{(1)}$ counterparts. Morespecifically, it will confirm the extent to which bond and site percolation liein the same universality class as logarithmic conformal field theories.

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