  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.
