Potts model on directed small-world Voronoi–Delaunay lattices

• q=3q=3 state in Potts model has a different universality class for the second order phase transition when p=0.01p=0.01.
• For p>0.01p>0.01 in the q=3q=3 state, the phase transition is first order.
• For the q=4q=4 state the phase transition is first order for any value of the rewiring probability pp.

The critical properties of the Potts model with q=3q=3 and 44 states in two-dimensions on directed small-world Voronoi–Delaunay random lattices with quenched connectivity disorder are investigated. This disordered system is simulated by applying the Monte Carlo update heat bath algorithm. The Potts model on these directed   small-world random lattices presents in fact a second-order phase transition with new critical exponents for q=3q=3 and value of the rewiring probability p=0.01p=0.01, but for q=4q=4 the system exhibits only a first-order phase transition independent of pp (0