Contact process on a Voronoi triangulation

We study the continuous absorbing-state phase transition in the contact process on the Voronoi-Delaunay lattice. The Voronoi construction is a natural way to introduce quenched coordination disorder in lattice models. We simulate the disordered system using the quasistationary simulation method and determine its critical exponents and moment ratios. Our results suggest that the critical behavior of the disordered system is unchanged with respect to that on a regular lattice, i.e., that of directed percolation.

Figure 1

(Color online) (a) A patch of a Voronoi diagram. (b) The corresponding dual lattice to the diagram shown in (a).

Figure 2

Degree distribution $P\left(q\right)$ of the Voronoi-Delaunay lattice, for system size $L=2560$.

Figure 3

Quasistationary density of active sites $\rho$ as a function of the control parameter $\lambda$. System sizes: $L=20$, 40, 80, and 160, from top to bottom.

Figure 4

QS density of active sites $\rho \left(q\right)$ vs $\lambda$, for sites with $q=3,4,\dots ,10$, from bottom to top. Circles: average over all sites. System size $L=160$.

Figure 5

QS order parameter $\rho$ vs system size $L$ at criticality $(\lambda =1.54266)$.

Figure 6

Critical lifetime of the QS state $\tau$ vs $L$.

Figure 7

(Color online) Decay of the order parameter starting from a full lattice of size $L=2560$. $\lambda =1.5428$, 1.5427, and 1.5426, from top to bottom.

Figure 8

(Color online) Quasistationary moment ratio $m$ vs $\ln L$, for $\lambda =1.54256$, 1.542 60, 1.542 64, 1.542 68, and 1.542 80, from top to bottom. System size: $L=640$. Inset: Quasistationary moment ratio $m$ vs $\rho$, system sizes: $L=20$, 40, 80, and 160.

Figure 9

Survival probability vs time, in the critical CP on a Voronoi lattice (solid line), and for the critical CP on a square lattice with random dilution of 2% (dotted) and 5% (dashed). System sizes: $L=640$ (left curves) and $L=1280$ (right).