Blume-Capel model on directed and undirected Small-World Voronoi-Delaunay random lattices arXiv:1001.2918v1 [cond-mat.stat-mech] 17 Jan 2010 F.P. Fernandes $^{1}$, F.W.S. Lima$^{1}$, and J. A. Plascak2 , 3 $^{1}$Departamento de F´ısica, Universidade Federal do Piau´ı , 64049-550, Teresina, PI, Brazil $^{2}$Departamento de F´ısica, Universidade Federal de Minas Gerais, C. P. 702, 30123-970, Belo Horizonte, MG, Brazil and 3 Center for Simulational Physics, University of Georgia, Athens, GA 30602, USA The critical properties of the spin-1 two-dimensional Blume-Capel model on directed and undi- rected random lattices with quenched connectivity disorder is studied through Monte Carlo simula- tions. The critical temperature, as well as the critical point exponents are obtained. For the undi- rected case this random system belongs to the same universality class as the regular two-dimensional model. However, for the directed random lattice one has a second-order phase transition for q < q$_{c}$ and a first-order phase transition for q > q$_{c}$ , where q$_{c}$ is the critical rewiring probability. The critical exponents for q < q$_{c}$ was calculated and they do not belong to the same universality class as the regular two-dimensional ferromagnetic model. PACS numbers: 05.70.Ln, 05.50.+q, 75.40.Mg, 02.70.Lq There has been recently an increasing interest in the study of networks which are not usual regular lat- tices. This growing interest, principally among physi- cists, comes mainly from the fact that they can model complex systems in real world networks $^{1}$. In special, small-world (SW) networks 2 have been much studied as they furnish a simple way to attempt to describe complex topologies such as social and economic organizations, the speed of disease spread, connectivities of computers, just to cite a few of them $^{1}$. As is usual, a rigorous treatment to get the properties of a real physical realization on such complex systems is a quite difficult task. However, one expects that simple theoretical models, such as the discrete spin Ising model, could be able to capture the main features of the actual complicated processes occurring on real networks. In this direction, the spin-1 / 2 Ising model has been studied on SW networks and, for exchange interactions which are independent of the length distance of the spins, one gets a second-order phase transition at a finite critical tem- perature T$_{c}$ $^{3–5}$. S´ anches et al. 6 introduced the asymmetric directed SW networks and for an Ising like spin system they showed that a first-order phase transition takes place when the rewiring probability q is greater than a critical value q$_{c}$ . The same result has been achieved by Lima et al. when treating the spin-1 , 3 / 2 and 2 Ising model on the same networks $^{7}$. In most of the above theoretical models the undirected, as well as the directed SW networks, have been con- structed by taking, as an initial step, a regular pure hy- percubic lattice $^{6}$. It will be very interesting, thus, to see what should be the behavior of SW networks built up over an already random system, such as the Voronoi- Delaunay random lattices. So, in this paper, we study rected SW networks constructed from an initial Voronoi- Delaunay random lattice. The purpose here is indeed two-fold: first, we would like to see the effect of the dis- order itself on the spin-1 model on this undirected lattice (since, for the spin-3 / 2 model 8 there has been a change in its universality class) and, secondly, the behavior of spin models on SW networks obtained from random number of nearest neighbors. In the next section we present the SW networks, the model and the simulation background. The results and discussions are presented in the last sec- tion. Undirected Voronoi-Delaunay random lattice - The Voronoi construction, or tessellation, for a given set of points in the plane can be defined as follows $^{9}$. Initially, for each point one determines the polygonal cell consist- ing of the region of space nearer to that point than any other point. Then one considers that the two cells are neighboring when they possess an extremity in common. From the Voronoi tessellation the dual lattice can be ob- tained by the following procedure: ( i ) when two cells are neighbors, a link is placed between the two points located in the cells; ( ii ) from the links one obtains the triangu- lation of space that is called the Delaunay lattice; ( iii ) the Delaunay lattice is dual to the Voronoi tessellation in the sense that points corresponding to cells link to edges, and triangles to the vertices of the Voronoi tessellation. Directed Voronoi-Delaunay SW network - We use here the same process defined by S´ anchez et al. 6 for direct- ing the Voronoi-Delaunay random lattice. In this case, we start with the Voronoi construction, or tessellation, as described in the previous paragraph, by creating the ran- dom two-dimensional lattice consisting of sites linked to their nearest neighbors by both outgoing and incoming II. SW NETWORKS, MODEL AND SIMULATION I. INTRODUCTION