  Electric and thermal transport properties of a $\nu=2/3$ fractional quantum
Hall junction are analyzed. We investigate the evolution of the electric and
thermal two-terminal conductances, $G$ and $G^Q$, with system size $L$ and
temperature $T$. This is done both for the case of strong interaction between
the 1 and 1/ 3 modes (when the low-temperature physics of the interacting
segment of the device is controlled by the vicinity of the strong-disorder
Kane-Fisher-Polchinski fixed point) and for relatively weak interaction, for
which the disorder is irrelevant at $T=0$ in the renormalization-group sense.
The transport properties in both cases are similar in several respects. In
particular, $G(L)$ is close to 4/3 (in units of $e^2/h$) and $G^Q$ to 2 (in
units of $\pi T / 6 \hbar$) for small $L$, independently of the interaction
strength. For large $L$ the system is in an incoherent regime, with $G$ given
by 2/3 and $G^Q$ showing the Ohmic scaling, $G^Q\propto 1/L$, again for any
interaction strength. The hallmark of the strong-disorder fixed point is the
emergence of an intermediate range of $L$, in which the electric conductance
shows strong mesoscopic fluctuations and the thermal conductance is $G^Q=1$.
The analysis is extended also to a device with floating 1/3 mode, as studied in
a recent experiment [A. Grivnin et al, Phys. Rev. Lett. 113, 266803 (2014)].
