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 ∆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 ≤ 2048 is calculated exactly by a combinatorial optimization algorithm. The phase-transition stays first-order for σ < σc(∆) ≤ 0.5, while the correlation length becomes diver- gent at the transition point for σc(∆) < σ < 1. In the latter regime the average magnetization is continuous for small enough ∆, but for larger ∆ it is discontinuous at the transition point, thus the phase-transition is of mixed order