  A convenient way to study phase transitions of finite spins systems of linear
size $L$ is to fix boundary conditions that impose the presence of a
system-size interface. In this paper, we study the statistical properties of
such an interface in a disordered Potts ferromagnet in dimension $d=2$ within
Migdal-Kadanoff real space renormalization. We first focus on the interface
free-energy and energy to measure the singularities of the average and random
contributions, as well as the corresponding histograms, both in the
low-temperature phase and at criticality. We then consider the critical
behavior of the interfacial adsorption of non-boundary states. Our main
conclusion is that all singularities involve the correlation length
$\xi_{av}(T) \sim (T_c-T)^{-\nu}$ appearing in the average free-energy $\bar{F}
\sim (L/\xi_{av}(T))^{d_s}$ of the interface of dimension $d_s=d-1$, except for
the free-energy width $\Delta F \sim (L/\xi_{var}(T))^{\theta}$ that involves
the droplet exponent $\theta$ and another correlation length $\xi_{var}(T)$
which diverges more rapidly than $\xi_{av}(T)$. We compare with the spin-glass
transition in $d=3$, where $\xi_{var}(T)$ is the 'true' correlation length, and
where the interface energy presents unconventional scaling with a chaos
critical exponent $\zeta_c>1/\nu$ [Nifle and Hilhorst, Phys. Rev. Lett. 68,
2992 (1992)]. The common feature is that in both cases, the characteristic
length scale $L_{ch}(T)$ associated with the chaotic nature of the
low-temperature phase, diverges more slowly than the correlation length.
