A CONJECTURE ABOUT THE SIZE OF A PARTICULAR CELLULAR AUTOMATON :
THE PERFECTLY GROWING CRYSTAL
H. Dreyssé and R. Riedinger
Laboratoire de Physique du Solide, Faculté des Sciences et Techniques, 4 Rue des Frères Lumière,
68093 Mulhouse Cedex, France
(Reçu le 17 dicembre 1986, rivisi le 25 fivrier 198?’, accepti le 10 mars 1987)
Résumé.- Pour minimiser l’occupation mémoire dans la mise en oeuvre de la méthode récursive, proposée
initialement par Heine, Haydock et Kelly, nous avons été amenés à construire un amas à partir d’un
ensemble de départ par applications successives d’un ensemble de générateurs sur un réseau. Cet amas
est organisé en couronnes et nous donnons une conjecture sur sa taille. Cette conjecture est testée sur
des réseaux avec des sites de connectivité différente où elle semble rester exacte ; cette conjecture devrait
être comprise à partir de l’aspect auto-similaire de l’amas en croissance. Nous relions cet amas à d’autres aspects : automate cellulaire, animaux, croissance d’un cristal parfait et analyse combinatoire. Nous illustrons également la génération de certain joints de grains par ce procédé.
Abstract.-In order to optimize a computer implementation of the recursion method, (initially proposed
by Heine, Haydock and Kelly), we build a cluster from an initial seed of points by adjoining the new sites
obtained from the actual cluster by translation of a given set of vectors (the generators), on a lattice. This
cluster is organized in shells and a conjecture is given about its size. This conjecture is checked on lattices
with inequivalent sites and seems to remain valid this case. The conjecture about the size seems to be related to a self-similarity of the growing cluster. We relate the construction of these clusters to other
topics : cellular automata, animals, growth of a crystal and combinatories. We show also how the growing
process may be used to generate cluster with grain boundaries.
Classification Physics Abstracts
05.50 - 61.50C - 02-10
In this letter we present a conjecture about the size
of clusters built by a geometrical recursion law. This
problem can be related to the study of cellular au-
tomata, the partition of an integer, animals, and per- fectly growing crystals [1]. Let us first define these
particular clusters, which we call "zebra", due to their special organization in shells or strips.
1. Definition
The recursion method [2] is a powerful method to cal-
culate many physically interesting quantities in real
space. We implemented it in our previous studies of
the electronic structure of impurities near surfaces [3].
In order to minimize the computer memory require- ments, we built an optimal cluster of sites which avoids any useless site for a given number of levels of the as-
sociated continued fraction. Let us briefly recall the
essence of the recursion method : starting from an
initial function 10) (atomic or molecular), we build a
sequence of orthonormalized states in) given by the
recursion relation :
where, at stage n(n > 1), n), [n - I) and bn are
known ; H is the Hamiltonian of the system, an =
(n IH I n) . The initial conditions are to select a nor-
malized vector 10) and to assume that bo = 0.
In a tight-binding formalism, the natural basis is the whole set of atomic states located on all atomic sites,
and the action of H on an atomic state li) can be writ-
ten
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01987004806091500
This algorithm may also be transposed to generate the cluster of sites ; one has to replace the Hamiltonian H
by the connectivity matrix :
Let us assume that the set of the sites is part of a per- fect ordered crystal. The direct neighbourhood of each
atom is exactly the same. If Rs denotes the position
of the site i and Tri the position of an interacting site
(his 0 0), Rï - Ri is a vector of a finite set vectors
(g), called generators.
We can define paths and steps : one step corresponds
to a single application of the operation T onto a site.
(go from the site Ri to the site Tri = 1ii Rï = R, +
ga, ga E fg}). It is similar to the presence of a "time".
The optimal cluster, the "zebra", needed for an exact
determination of n stages of the continued fractions, is
built by the following recursion laws :
1. The starting seeds is given ; it forms the shell 0 and may be either a single site or a finite cluster.
2. From the shell, n one builds the shell (n + 1) , which
is the set of all the sites connected in one step to the full cluster at stage n (in fact, one needs only to con-
sider the shell n) sites which does not yet belong to the
cluster (in fact, one has only to check a finite number of preexisting shells (n, n -1, ...)).
3. The "zebra" at stage n is the reunion of all shells,
up to n.
In figure 1 we show a growing cluster with a seed of two sites.
2. Conjecture over the size of the "zebra"
For a large variety of generators spanning a dspace
(d=2,3,4) we have checked that s(n), the number of sites in the shell n, satisfies to
for n greater than a finite threshold value, nc. Here A is the finite difference operator.
Moreover, the value of the constant T = A’-’s(n) for a
d-dimensional crystal depends only on the generators.
We have listed the values of nc and T for different gen- erators and seeds in table I and figure 2.
So it is possible to determine the number of sites in the full "zebra" at stage n, S(n)
Fig.l.-Example of 2D "zebra". 0 denotes the seed ; and
A denote the points in two successive shells. Full lines connect points in the same shell ; these lines and stripes
are drawn for a better visualization of the different shells. Note the self-similarity of the "zebra".
It is clear from equation (4) that s(n) [S(n) resp.] is
a polynomial of highest order (d - 1) [d resp.] in n,
fully determined from (4) and the knowledge of a finite
number of s(n) as initial data. In a 3d-cubic lattice, S (n) verifies
with S(0) = 1 (the seed is a point), and the values of
T are given in table I. Let us note that the simplicity
of formula (7) stems from the fact that, in the present
examples, the asymptotic threshold nc is attained for
nc = 2. The values S (0) and S (1) also comply with
this formula, due to the empirical law about r :
where Nt} is the number of sites connected to a given
site. As mentioned above, this formula can be used as a check on the size of the minimal cluster for obtain-
ing n exact levels in a continued fraction built by the
recursion method on a lattice. We worked out similar relations for other lattices from (4).
3. Relations of the "zebra" with other topics
1. The "zebra" is a cellular automaton [4,5]. According
to Wolfram [6], the "zebra" is a discrete lattice of sites
Table 1.- Value of the threshold n,, and limit of ð. d-1 s(n) for different systems, for various dimensions d, generators, and seeds. (s.c. : simple cubic, b.c.c. : body centered cubic, f.c.c. : face centered cubic, n.n. : nearest neighbours,
n.n.n. : next nearest neighbours).
which evolves in discrete time steps. Each site takes
two possible values (occupied or not), according to the
same deterministic rules (one site is occupied in the
shell n if, without belonging to the previous shells, this
site is connected to a site in these previous shells).
Thus the rules for the evolution of a site depend only
on a local neighbourhood of sites around it.
2. The "zebra" is connected to combznatorzes and to lattice animals. Suppose we have ng generators gj,
spanning a d-dimension space and the seed is a unique point. Any site x in shell n can be represented as spanning a d-dimension space and the seed is a unique point. Any site x in shell n can be represented as
From the recursion law, we deduce the following prop-
erty : any point of the shell n may be reached from the
source in a number of steps greater or equal to n.
This implies that it is always possible to find a set of integers {njl such that the site is reached in n steps from the seed, or
Equation (8c) defines a partition of the integer n into
ng (or less) non-negative integers.
But even if the generators are linearly dependent, the
statement that the path from the origin to point x is the
shortest one, measured in steps holds. The generators being linearly dependent, there exist p constraints.
Fig.2.-Value of r = A1s(n) for different generators.
where cij and cIJ’, are positive or null integers which
satisfy the following relation
This relation implies that p > (ng - d). This entirely
determines the possible sets of values of nj in equation
(8c) and thus the conditional partition [7]. Equation (9) can also be used as a sieve, as follows
and if all the n’. are greater than or equal to a certain
set ci j it is possible to write x as
and thus,
This point belongs to the shell q of lower index. The
partition resulting from ( l0e) is not allowed. These
conditional partitions may lead to a formal proof of
our conjecture. In the case of d linearly independent generators applied to a punctual seed (the site at ori- gin), our conjecture about the number of sites in the shell n, s(n), allows us to determine the number of di- rected animals, An, of size n on the lattice defined by
the d generators.
Thus, even in the general case (linearly dependent gen-
erators, which is close to the polyominos [8]), any point
of the shell n can be attained in n steps from the origin
but not less and the shell n is the set of the extremities of the stretched animals of size n, the other extrem-
ity being fixed at the origin (this animal is in fact a
worm !).
3. The "zebra" can be considered as a perfectly growing crystal, since it involves no randomness. Conjecture (4)
is valid for any set of generators on a lattice. Let us
note, that generally, our set of generators is compat- ible with a lattice where the possible interaction be-
tween two sites is not necessarily related to a geomet- rical proximity but is associated with the generators.
In the case of d linearly independent generators and
one point as seed, simple arguments of self-similarity
can be invoked to give a proof of conjecture (4). We
can also give a proof of conjecture (4) in the following
case. Let p linearly dependent generators be with the following property : all lattice sites inside the geomet- rical envelope of the generators applied to a point seed
are occupied. then we can tile the whole space into a
set of pyramidal cones, defined with the seed point as a
summit and a base of adjacent vertices of the first shell.
In each cone, the preceding self-similarity property re- mains valid and this leads to a proof of conjecture (4).
More generally, one seems to find such self-similarity properties which are related to conjecture (4). An-
other interesting property of the "zebra" is its ability
to build clusters with some restrictions ; e.g. surfaces with steps or grain boundaries (Fig.3). For instance,
with a judicious choice of generators operating in half-
spaces only, one can build a large family of flexion grain
boundaries. Let us mention that the generators used in this case to build the lattice no longer represent physical Hamiltonian interactions, as in the recursion method.
Finally one can visualize the growing process by using
Fig.3.-Example of 2D crystal with periodical steps with the used generators and one point as seed (·). The first
four shells are represented.
a computer.
One can then see that, when conjecture (4) is satisfied,
all the defects in the crystal are "blocked" i.e. they are just moving in a translational way, and no longer grow.
4. We can extend our conjecture to lattices with sites
having different connectivities. Conjecture (4) remains
valid, but now A-1 s(n) also depends on the origin
and the number of steps n. However, in the cases re- ported in figure 4, we recover the periodic behaviour
Fig.4a.-Rectangular-triangulax lattice with nearest neig-
bour interactions, s(n) is the number of sites in the shell n.
Fig.4b.-Lattice with nearest neigbour interactions, 8 (n)
is the number of sites in the shell n.
with a period v independent of the starting point (here,
in both cases, v=3). Simple geometrical arguments can be used to transform this problem into a linear chain problem and the values of Ad-1S (n) are easily deter-
mined. Moreover one can consider a "supershell" con- sisting of the reunion of v consecutive shells. If E(n)
denotes the number of sites in the "supershell" n, we find again that the remarkable property 6.1 E (n) does
not depend on the starting point but only on the lat-
tice and thus on the generators. (In Fig. 4a, &’E(n) =
14=4+6+4=5+5+4, in Fig. 4b, A’E(n) =16=6+5+
5=4+6+6). Thus it seems that conjecture (4) found
for lattices with sites having all the same direct neigh-
bourhood, can be extended to lattices with sites of different connectivities. Let us note that, in two cases, the lattices can be built by applying v different sets of generators periodically. Let us note general expression
of this conjecture would be : given a lattice built from
a initial set of states by periodical application of v sets
of generators, following the recursion law (each point
of a shell is connected at least to a point of the previous
shells but is not present in these previous shells) ; one
can consider a "supershell" as the reunion of v con-
secutive shells. Thus if E ( n) is the number of sites in
"supershell" n
for n greater than a threshold value nc, depending of
the sets of generators and the seed, r depending only
on the sets of.generators.
4. Conclusion
In spite of our efforts, we were unable to find a general proof of the conjecture on the size of the "zebra". How- ever, it was possible to find connections of the "zebra"
with different topics of statistical physics. In any case this conjecture is useful in the recursion method itself, by giving an estimation of the memory storage needed for a given-number of levels of the continued fraction
to be built buy the recursion method. The ability of
the "zebra" to generate some corrugated surfaces and
grain boundaries has also been emphasized.
Acknowledgments
We whish to thak Profs. G. Rauch and Foata and Dr. J. Vannimenus for stimulating discussions. We
are indebted to Dr. I. Bose and P. Ray for suggesting
the study of lattices with sites having different con- nectivitiet. Computational facilities form Groupement
pour un Centre de Calcul Vectoriel en Recherche (GC- CVR, Palaiseau), Centre Interregional de Recherche en
Calcul Electronique (CIRCE, Orsay) and Institut de
Recherches Polytechniques (IRP, Mulhouse) are grate-
fully acknowledged.
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