Phase-transitions of the random bond Potts chain with long-range interactions

We study phase-transitions of the ferromagnetic $q$-state Potts chain with random nearest-neighbour couplings having a variance $\Delta^2$ and with homogeneous long-range interactions, which decay with the distance as a power $r^{-(1+\sigma)}$, $\sigma>0$. In the large-$q$ limit the free-energy of random samples of length $L \le 2048$ is calculated exactly by a combinatorial optimization algorithm. The phase-transition stays first-order for $\sigma < \sigma_c(\Delta) \le 0.5$, while the correlation length becomes divergent at the transition point for $\sigma_c(\Delta) < \sigma < 1$. In the latter regime the average magnetization is continuous for small enough $\Delta$, but for larger $\Delta$ it is discontinuous at the transition point, thus the phase-transition is of mixed order.