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A convenient way to study phase transitions of finite spins systems of linearsize $L$ is to fix boundary conditions that impose the presence of asystem-size interface. In this paper, we study the statistical properties ofsuch an interface in a disordered Potts ferromagnet in dimension $d=2$ withinMigdal-Kadanoff real space renormalization. We first focus on the interfacefree-energy and energy to measure the singularities of the average and randomcontributions, as well as the corresponding histograms, both in thelow-temperature phase and at criticality. We then consider the criticalbehavior of the interfacial adsorption of non-boundary states. Our mainconclusion 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 forthe free-energy width $\Delta F \sim (L/\xi_{var}(T))^{\theta}$ that involvesthe droplet exponent $\theta$ and another correlation length $\xi_{var}(T)$which diverges more rapidly than $\xi_{av}(T)$. We compare with the spin-glasstransition in $d=3$, where $\xi_{var}(T)$ is the 'true' correlation length, andwhere the interface energy presents unconventional scaling with a chaoscritical exponent $\zeta_c>1/\nu$ [Nifle and Hilhorst, Phys. Rev. Lett. 68,2992 (1992)]. The common feature is that in both cases, the characteristiclength scale $L_{ch}(T)$ associated with the chaotic nature of thelow-temperature phase, diverges more slowly than the correlation length.

Critical behavior of interfaces in disordered Potts ferromagnets : statistics of free-energy, energy and interfacial adsorption