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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 $Δ^2$ and with homogeneous long-range interactions, which decay with the distance as a power $r^{-(1+σ)}$, $σ>0$. 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 $σ< σ_c(Δ) \le 0.5$, while the correlation length becomes divergent at the transition point for $σ_c(Δ) < σ< 1$. In the latter regime the average magnetization is continuous for small enough $Δ$, but for larger $Δ$ it is discontinuous at the transition point, thus the phase-transition is of mixed order.