Percolation Thresholds on 2D Voronoi Networks and Delaunay Triangulations

Percolation thresholds on 2D Voronoi networks and Delaunay triangulations Adam M. Becker∗ Department of Physics, University of Michigan, Ann Arbor MI 48109-1040 Robert M. Ziffy Center for the Study of Complex Systems and Department of Chemical Engineering, University of Michigan, Ann Arbor MI 48109-2136 (Dated: August 5, 2021) The site percolation threshold for the random Voronoi network is determined numerically for the first time, with the result pc = 0:71410 ± 0:00002, using Monte-Carlo simulation on periodic systems of up to 40000 sites. The result is very close to the recent theoretical estimate pc ≈ 0:7151 of Neher, Mecke, and Wagner. For the bond threshold on the Voronoi network, we find pc = 0:666931±0:000005, implying that for its dual, the Delaunay triangulation, pc = 0:333069±0:000005. These results rule out the conjecture by Hsu and Huang that the bond thresholds are 2/3 and 1/3 respectively, but support the conjecture of Wierman that for fully triangulated lattices other than the regular triangular lattice, the bond threshold is less than 2 sin π=18 ≈ 0:3473. PACS numbers: I. INTRODUCTION The Voronoi diagram [1] for a given set of points on a plane (Fig. 1) is simple to define. Given some set of points P on a plane R2, the Voronoi diagram divides the plane R2 into polygons, each containing exactly one member of P . Each point's polygon cordons off the portion of R2 that is closer to that point than to any other member of P . More precisely, the Voronoi polygon around pi 2 P 2 contains all locations on R that are closer to pi than to any other element of P . The total Voronoi diagram is the set of all the Voronoi polygons for P on R2; the Voronoi network is the set of vertices and edges of the Voronoi diagram. The dual to the Voronoi diagram is interesting in its own right. Known as the Delaunay triangulation [2] (see Fig. 2), it can be defined independently of the Voronoi diagram for the same set of points P on R2: it is simply the set of all possible triangles formed from triples chosen out of P whose circumscribed circles do not contain any other members of P (Fig. 3). The Delaunay triangulation FIG. 1: Voronoi diagram with a Poisson distribution of gen- and the Voronoi diagram for the same set of points can erating points. be seen in Fig. 4. Note that while the members of P are sites in the Delaunay triangulation, they are not sites in the Voronoi network, whose sites are the vertices of the polygons; also note that the edges of the Voronoi works. The fastest ones run in O(n log n) time for general diagram lie along the perpendicular bisectors of the edges distributions of points [3, 4, 5, 6], where n is the num- arXiv:0906.4360v3 [cond-mat.dis-nn] 1 Sep 2009 of the Delaunay triangulation | however, the edges of ber of generating sites, and this has been proven to be the Voronoi diagram do not always intersect the edges the optimal worst-case performance [4]. For a Poisson of the Delaunay triangulation, as seen in Fig. 4. The distribution of points on the plane, there are many O(n) Delaunay triangulation represents the connectivity of the expected-time algorithms [7, 8, 9, 10]. Voronoi tessellation of the surface. In addition to being theoretically interesting [11, 12, There are many algorithms for constructing these net- 13, 14, 15], both the Voronoi diagram and the Delaunay triangulation are widely used in modeling and analyzing physical systems. They have seen use in lattice field theory and gauge theories [16], analyzing molecular dy- ∗Electronic address: [email protected] namics of glassy liquids [17], detecting galaxy clusters yElectronic address: rziff@umich.edu [18], modeling the atomic structure and folding of pro- 2 FIG. 2: Delaunay triangulation for the same set of generating FIG. 3: The Delaunay triangulation for a set of five points, points as in Fig. 1. The generating points become the vertices along with the associated circumcircles. in this network. teins [19, 20], modeling plant ecosystems and plant epi- demiology [21], solving wireless signal routing problems [22, 23], assisting with peer-to-peer (P2P) network con- struction [24], the finite-element method of solving dif- ferential equations [25], game theory [26, 27], modeling fragmentation [28], and numerous other areas [29, 30]. Percolation theory is used to describe a wide variety of natural phenomena [31, 32]. In the nearly seventy years since the first papers on percolation appeared [33, 34], it has become a paradigmatic example of a continuous phase transition. For a given network, finding the crit- ical probability, pc, at which the percolation transition occurs is a problem of particular interest. pc has been found analytically for certain 2D networks [35, 36, 37]; however, most networks remain analytically intractable. Numerical methods have been used to find pc for many such networks, e.g., [31, 38, 39, 40, 41, 42, 43]. In this paper we consider the percolation thresholds of the Voronoi and Delaunay networks for a Poisson distribution of generating points, as represented in Figs. 1, 2, and 4. There are four percolation thresholds related FIG. 4: The Delaunay triangulation (dotted lines) superposed to these two networks: the site and bond percolation on the Voronoi diagram (solid lines), its dual graph, for a set on each. Being a fully triangulated network, the site of Poisson-distributed generating points. percolation threshold of the Delaunay network is exactly site;Del 1 pc = 2 [35, 44, 45, 46]. This result has recently been proven rigorously by Bollobásand Riordan [47]. Some- of this value, either analytically or numerically. (There what surprisingly, a search of the literature revealed no are places in the literature (e.g. [29]) where the Voronoi prior calculation of the site percolation threshold of the \site" threshold is listed as 1/2. This is true if the \sites" Voronoi network at all, despite the widespread use of are taken to be the generating points from which the dia- such networks. A prediction for its value has recently gram is created, rather than the vertices of the diagram. been made by Neher, Mecke and Wagner [48]; they use Thus, this is actually the Voronoi tiling threshold, i.e. site;Vor an empirical formula to predict pc = 0:7151, but the percolation threshold of the Voronoi polygons, which they too were unable to find any previous calculation is in turn equivalent to the Delaunay site threshold, well 3 known to be 1/2.) third point in the Delaunay triangle by looking at The bond thresholds of the Voronoi and Delaunay net- the radii of the circumcircles of the triangles formed works are complementary, by that edge with each point in its bin and all of the neighboring bins. bond;Vor bond;Del pc = 1 − pc ; (1) 4. Look at the other Delaunay triangles that have because these networks are dual to one another [49]. The been found in that bin and neighboring bins to first numerical measurement of the bond threshold for make sure this new triangle is not a duplicate of either network seems to be that of Jerauld et al. [50], one that has already been found. If it is not, find bond;Del who in 1984 found pc = 0:332. Shortly there- which of its neighbors have already been discovered after, Yuge and Hori [51] performed a renormalization and mark them as its neighbors, and vice versa. bond;Del group calculation which yielded pc = 0:3229. In bond;Del 1999, Hsu and Huang [52] found pc = 0:3333(1) 5. From the list of neighbors of the current triangle, bond;Vor and pc = 0:6670(1) through Monte Carlo meth- figure out which of its edges are not already shared ods. (The numbers in parentheses represent the errors with neighbors (if any), and, if there are any un- in the last digits.) These values led them to make the shared edges, whether the missing neighbor should intriguing conjecture that the thresholds are exactly 1=3 be above or below the edge. and 2=3 respectively. There is, however, no known theoretical reason to believe that this conjecture is true. In 6. Repeat steps 3-5 until there are no unprocessed order to test this conjecture, and to find the site perco- edges left, at which point the Delaunay triangulation threshold of the Voronoi network, we have carried lation is finished. Because the neighbors of each out extensive numerical simulations, as detailed below. triangle are known, this algorithm also yields an In section II, we describe our methods, and in section III adjacency list of the sites on the Voronoi diagram we discuss our results and compare them to the thresh- (because the Voronoi diagram is dual to the Delau- olds of several related lattices, and also discuss the cov- nay triangulation). ering graph and further generalizations of the Voronoi system. Conclusions are given in section IV. This algorithm is significantly easier to implement with periodic boundary conditions, because every triangle is guaranteed to have exactly three neighbors. Further- II. GENERATING ALGORITHMS AND more, the imposition of periodic boundary conditions ANALYSIS TECHNIQUES also gives the Delaunay and Voronoi networks a very use- ful property: there are always exactly twice as many De- A. Delaunay/Voronoi generation algorithm launay triangles (Voronoi sites) as there are generating points (vertices of the Delaunay network or polygons in In order to avoid edge effects in the networks when the Voronoi network) for a given diagram.

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