Distance sequences and percolation thresholds in Archimedean tilings

Mathl. Comput. Modelling Vol. 26, No. 8-10, pp. 317-320, 1997 Copyright@1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0895-7177197 $17.00 + 0.00

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SO8957177(97)00216-l

Distance Sequences and Percolation Thresholds in Archimedean Tilings P. PRBA Laboratoire d’lnformatique de Marseille, CNRS URA 1787 Facultk des Sciences de Luminy, 13288 Marseille Cedex 9, France and Ecole Nationale Supkrieure de Physique de Marseille 13397 Marseille Cedex 20, France preaQlim.univ-mrs.fr Abstract-Given

a graph G, a vertex x of G, and an integer n 2 0, the circle C, of center x and of radius n is the set of all the vertices at distance n from x, and the circumference cn is the cardinality of C,,. The distance sequence of G and center x is the sequence (co, ~1, ~2,. . , b,. . ). When G is vertex-t,ransitive, we can talk about the distance sequence of G. In this note, we give formulae for calculating distance sequences in Archimedean tilings (which can be seen as vertex-transitive graphs) and we can see that, for these tilings, these sequences are linked with the percolation thresholds.

Keywords-Tilings,

Circles, Percolation.

1. INTRODUCTION edge-to-edge tilings which are tilings of the plane where all the tiles are regular and every edge of a polygon is also an edge of another polygon (thus, all edges have the same length). Let there be k polygons incident on a given vertex. Denote by ei the number of edges of the ith polygon (1 2 i 5 k). Then the type of the vertex is the sequence (el,e2, . . . , ek). For example, in the three monohedral tilings of the plane, vertices are of the same type which are:

We consider polygons

(474,474) (3,3,3,3,3,3) (6,676). We shall denote these sequences

(44), (36), and (63).

An Archimedean

tiling is an edge-to-edge tiling such that all vertices are of the same type. There exist exactly eleven such Archimedean tilings, those with vertex types: (44), (36), (S3), (32,4,3,4), (34, 6), (33,42), (3,4,6,4), (3,6,3,6), (3, 122), (4,6,12), and (4,82). In addition, the tilings are equal to tiling (34,6) occurs in two enantiomorphic forms. The other Archimedean their mirror image. A proof of this result, and drawings of these tilings can be found in [l]. Given a graph G, a vertex zo of G, and an integer n 2 0, the circle C,, of center 20 and of radius n is the set of all vertices at distance n from 20, and the circumference c,, is the cardinality of C,. The distance sequence of G with respect to center 20 is the sequence (cg, cl, ~2,. . . , G, . . . ). When G is vertex-transitive (this is the case for Archimedean tilings), we can talk about the distance sequence of G. I would like to thank G. Resign& M. Rasigni, and C. d’Iribarne

for valuable and helpful discussions.

M-=-t 317

by A.M-%F

P. PtiA

318

In Section 2 of this note, we shall give the distance sequences of these Archimedean tilings. In Section 3, we shall see that, in these tilings, percolation thresholds are linked to distance sequences. We denote

1x1 the floor of the real x and we define {a; b}C by

{a;bjc = { {a1,a2,. ..,a,;

1,

ifb-a(modc)andb>a,

0,

otherwise,

b}c = {a~;b},+

{~2;b}c+~~~+{~n;b~c-

2. THE DISTANCE The following

formulae

give the distance

SEQUENCES

sequences

proofs can be found in [2] for the three monohedral The distance

sequences

for the triangular tiling. The distance sequence

are (4n) for the square of the tiling

(32, 4,3,4)

(ci)~r tilings

tiling,

for all Archimedean

(3n) for the hexagonal

is given by

ci = 4i + 1 - (0; i}s + 2 ([;j+[yJ). The distance

sequence

ci = {8,9,10;

of the tiling

(6, 34) is given by

i}s + 2{3,4; i}s + {8,9; i}is - {2,4,7;

i}rs

+{18,23,26,28,31,33,34,36,38,39,41,42,43,44;i}4o+

The distance

sequence

of the tiling

(4, 82) is given by

The distance

sequence

of the tiling

(33, 42) is Ci = 5i.

The distance

sequence

of the tiling

(3,6,3,6)

is given by

Cl =4, ci = The distance

sequence

4i + 2,

if i is odd and greater

5i - 2,

if i is even and greater

of the tiling

(3,4,6,4)

than

is given by c+ = 4i.

The distance

sequence

of the tiling

(3, 122) is given by 5i -(“gi + 13)

if i E 0 (mod4),

-,

if i zs 1 (mod4),

1

ci =

4

I 2i + 2,

I

if i E 2 (mod4),

(9i - 3) 4

’

tilings.

Their

and in [3] for the other ones.

if i G 3 (mod4).

than

1, 0.

tiling,

and (67~)

Archimedean Tilings

The distance

sequence

of the tiling

(4,6,12)

319

is given by

Cl = 3,

for i > 2:

+ 2{3; i}ac + {2,4,7,10,12,16,17,22; + 4{14,19,20;

i}ss + 3{8,9,13,15,18,21;

i}as

i}ss,

3. DISTANCE SEQUENCES AND PERCOLATION THRESHOLDS Given a connected graph G, a spanning tree of G is a tree with the same vertex set as G and an edge set contained in the edge set of G. When G is valuated, a minimal spanning tree is a spanning tree such that the sum of the lengths of its edges is minimal. In 1986, the minimal spanning tree was used for studying order and disorder [4] and, in 1989, for the study of the site percolation [5]. In 1991, percolation thresholds for nine Archimedean tilings (all except (6, 34) were estimated with the same technique [6]. Very recently, the technique was and (32,4,3,4)) improved [7] and now, estimations of site percolation thresholds are known for all these tilings except (6, 34); they are as shown in Table 1. Table 1. Tiling

Threshold

0.

(44)

1.

(3Y

,590 ,500

2.

(S3)

,698

3.

(33,42)

.549

4.

(3,4,6,4)

.620

5.

(3,6,3,6)

.650

6.

(3,12?

,807

7.

(4,6,12)

,746

8.

(4,8’)

,729

9.

(4,32,4,3)

,550

The percolation thresholds for the tilings (44), (36), (S3), and (3,6,3,6) were already studied by computer simulation [8-lo]. Actually, the percolation threshold is known to be exactly l/2 for the triangular lattice (36). Intuitively, Figure 1 suggests that there exists a link between percolation thresholds and distance sequences. Percolation thresholds are on the z-axis and liminf(c,/n) on the y-axis. It would be interesting to formally establish this link; unfortunately, we only have experimental results. According to these and 0.6 for (6, 34).

values,

we can conjecture

that

the percolation

threshold

is between

0.55

320

P.

+ 0

PRkA

-I-

+

4

5

+ 2

+ 8

I

.5

I

.6

I

.7 percolation threshold

+

I

b

.8

Figure 1.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

B. Griinbaum and G.C. Sheppard, Tilings and Patterns, Freeman, (1987). P. P&a, Distance sequences in infinite regular tessellations, Discrete Math. (to appear). P. Prka, Distance sequences in Archimedean tilings, F&search Report #19, LIM, (1994). C. Dussert, G. F&sign& M. Rasigni, J. Palmari and A. Llebaria, Minimal spanning tree: A new approach for studying order and disorder, Physical Review B 34 (5), 3528-3531 (1986). C. Dussert, G. Rasigni and M. Rasigni, Minimal spanning tree approach to percolation and conductivity threshold, Physics Letters A 139 (1,2), 35-38 (1989). B. Lebrun, Utilisation de l’arbre de longueur minimum pour l’btude de l’ordre et du d&ordre B l’aide de param~tres statistiques. Applications B la percolation, Projet de fin d’kudes, ENSPM, (1991). C. d’Iribarne, G. Ftasigni and M. Rasigni, Determination of site percolation transitions for 2D mosaics by means of minimal spanning tree approach, Physics Letters A (to appear). G. Grimmet, Percolation, Springer Verlag, (1989). M. Sahimi, Applications of Percolation Theory, Taylor & Francis, (1994). D. Stauffer, Zntrodzlction to Percolation Theory, Taylor & Francis, (1985).