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Minimal spanning tree approach to percolation and conductivity threshold
Volume 139, number 1,2
PHYSICS LETTERS A
24 June 1989
MINIMAL SPANNING TREE APPROACH TO PERCOLATION AND CONDUCTIVITY THRESHOLD C. DTJSSERT, G. RASIGNI and M. RASIGNI Département de Physique des Interactions Photons—Matière, Case ECJ, Faculté des Sciences de St Jérôrne, 13397 Marseille Cedex 13, France Received 8 February 1989; revised manuscript received 19 May 1989; accepted for publication 22 May 1989 Communicated by A.A. Maradudin
The minimal spanning tree (MST) method to analyse order and disorder in distributions of objects is used to investigate the percolation transition on a triangular lattice. It is shown numerically that the value of the threshold probability is retrieved, simultaneously with a new geometrical parameter: the mean edge length. This is expected to be useful in the conductivity studies of percolation networks such as dielectric breakdown, conductivity and other propagation phenomena. The potentialities of the method in dealing with an extensive study of percolation on continuous as well as lattice systems are also exhibited.
The standard formalism for studying percolation is based on the size of the clusters through its first moments. This formalism makes it possible to exhibit critical exponents, and thus to relate this problem directly to the field of general phase transitions [1—31.For some years now, it has appeared that other geometrical properties of the clusters (and particularly the percolation cluster) are of interest. The perimeter [4,1], backbone [5], elastic backbone, dangling ends [6], were then investigated, leading to
shown in this paper that the minimal spanning tree analysis (MST) recently developed by Dussert et al. [17] to study order and disorder in sets of points may be successfully implemented to tackle the percolation problem and the above-mentioned concepts. The main definitions related to the graph theory have previously been given [17]. It should be remembered that an edge-weighted linear graph G= (X, E) is composed of a set of points X= (x1, x2, ...) called nodes and a set of node pairs
various fractal dimensions and new critical exponents. Another approach was to study random walks on the percolation cluster [7,2]. These studies were extended to various deterministic walks in order to determine the shortest paths [8], diameters of clusters [1,3], cutting bonds [5,91, etc. The usual justification given for these studies is their possible use in conductivity of random (or dilute) systems and in flows of fluids in porous media because of their wide range of applications. The problem of the conductivity transition at the percolation threshold was soon investigated both theoretically [10], numerically [111 and experimentally [121. More recently, a large number of miscellaneous concepts appeared about this subject, which include electrical breakdown [13] and minimum gap [14], hopping conductivity [15,16] and shortest path length [81. It is
E= [(x,, xi.)] called edges, with a number called weight (in this paper the Euclidian distance) assigned to each edge. A treeis a connected graph without closed 1oops. A MST is a tree which contains all the nodes and where the sum of the edge weights is minimal. Depending on the starting point there may be more than one MST for a given set of points, but all the MSTs have the same edge-length histogram. It follows that statistical information deduced from the histogram, such as the average edge length m and the standard deviation a may be used as characteristics for the corresponding distribution. In order to be able to compare different distributions regardless the particles density and the sampling window, the value of m and a must be normalized. For this purpose the process described by Hofmann and Jam [181 has been used. It may be summarized as fol-
0375-9601/89/s 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
35
Volume
139. number
1.2
PHYSICS
lows. The area A of the sampling window is defined as the area H of the convex hull of the data normalized by means of the relation
A=Hl(l-fin) >
(1)
in which n is the number of points in the window and f is the number of faces on the convex hull. Given that the expected length of a randomly chosen edge of a MST related to n uniformly distributed points in a sampling window A, is asymptotically proportional [ 191 to (nA)“‘/(n-l),
(2)
the normalized values of m and fl of the MST constructed from a given set of data are obtained by dividing the original ones by expression (2). All distributions can be plotted in the (m, a) plane and easily compared with well-characterized distributions (for instance, perfectly ordered or random ones) due to the normalization process. For example, consider the set of points ordered at the nodes of a lattice (triangular for instance). The edge-length histogram of the corresponding MST provides the normalized values o=O and m= 1.075 (fig. 1). The arrangement may be randomized by giving each point a new position deduced from its previous position using a Gaussian distribution with a standard deviation of w. Then a trajectory is obtained (full line in fig. 1)) each point of the trajectory corresponding
24 June I989
LETTERS A
to a certain degree of order. For a certain value of o. the random distribution (0~0.300, m=0.650) is reached. We explained in refs. [ 171 and [ 201 how to quantize accurately the (m, a) plane and how to determine the degree of order for any distribution by taking a simple reading in the ( IM, a) plane. A survey of its main properties shows that the MST is a data structure well suited to retrieve the usual properties of the percolation network. Indeed, classification and pattern recognition techniques which use the MST [ 2 1 ] immediately bring out the clusters (and denombrate them), as well as the backbone. dangling ends and paths of various characteristics. All these parameters can be linearly determined and given that some algorithms exist which are able to compute an MST in a time proportional to n log n. where II is the number of occupied sites on the lattice, the procedure seems very efficient for an cxtensive study of the percolation phenomenon. .4ctually the MST, in addition to the easy computation of these parameters, lead to the definition of new ones which are presented in the following. By way of illustration fig. 2 shows for a very sim-
P-1 ..’ ;..
.
.
‘..
_.. L.
.
.I. ,‘:
E Perfect 0.5
0.6
order
0.7 Mean
0.8 value
0.9
1
.
.,\.
.,I. . . . *~ .?~~_ . . Jo
./I’ . I) .
.
.
‘.
_-r’
.
0.3
1.1
m
Fig. 1. (m, 0) diagram. The triangles represent the percolation trajectory, the full line is for the trajectory ofthe triangular lattice progressively disordered [ 171 and the dashed line is the hardcore trajectory.
36
.
P :
0
.
‘, ._ji.
.
Fig. 2. Minimal spanning tree associated to a simplified set of points (10x 10 triangular lattice) for p= 1 and p=O.3, where p is the probability for a site of the lattice to be occupied. (0) Occupied site. (a ) Unoccupied site. Dashed line: convex hull related to the set of occupied sites forp=0.3. It can be seen that for this simplified casef=9.
Volume 139, number 1,2
PHYSICS LETTERS A
plified set of points (lOx 10 triangular lattice) the MST corresponding to p 1 and p=0.3, where p is the probability for a site of the lattice to be occupied. The actual computations were performed on a 300 x 300 triangular lattice. For various values of p the MSTs were computed and the corresponding values of m and a were retrieved and reported in the (m, a) plane (fig. 1). The (m, a) diagram reveals an inflexion point of the trajectory for a value of (m, a) corresponding to p=p~ =0.5 which is known to be the percolation threshold for the triangular lattice. It is to be noticed that both m and a have a particular behaviour at p=p~=O.5O,which is revealed on their derivatives (fig. 3). Thus the MST analysis clearly exhibits the threshold percolation (critical probability) as do various other methods. Moreover, for p=p~,we have m=m~=0.78.This critical value of m which is linked to the geometry of the systern, may be considered as a new characteristic of the percolation system related to the triangular lattice and should be of interest in the studies on conductivity, The trajectory related to the percolation system maybe compared with the one determined by taking
°~Thi~
• (a)~ b E
•
O.2~
O8~
i.
0.1, 0.6
___________
(b)~
~
~~::L~•L~ 8~
‘
—
/
where the step of the lattice is equal to unity. The calculation of the mean edge length is made of two terms: the first involves the subtrees constructed on each cluster (of which the mean edge length is unity), and the second involves the inter-cluster links. A priori those links have the form l(s, s’, p) where s and s’ are the sizes of the clusters linked by 1. However the building of the MST may be viewed as a dynamical process where the sites are progressively ineluded in the growing tree. From this point of view the relevant parameter is the length I~=l(s, p) of the
(~
201
b ~ 1~
‘~
m=p-112 (s-1)n~+~ ~ (4) where ~ is the number of s-clusters divided by the
~
I
a full lattice and then moving the sites according to the process previously described [20]. It can be seen that the trajectory related to the percolation system is below this latter trajectory, which is satisfactory, given that the percolation system is much more ordered, due to the lattice structure, than a continuous one. Another simulation may be plotted, which consists in a system of hard disks of which the diameter increases [11,22]. The resulting trajectory starts from the random distribution and goes asymptotically to the triangular lattice. These curves reveal the very influence of the lattice as compared to the continuous case, as well as the potentiality of the method of analysis which is efficient in both those cases, and which makes possible a comparison between them. Let us also remark that for n large the normalization relation (2) may be written “n/A 1/2 1/2 (3
edges which link the s-clusters to the MST. Then, the parameter m may be written
0 •
O6~
24 June 1989
~ 0 0
P
02
04
06
0.8
1
p
Fig. 3. (a) Plot of the mean edge length m ofthe MST built on the triangular percolating lattice versus the probability of occupancy of the Sites p. (b) and (c) Plots of dm/dp and d2m/dp2 respectively versus p. (d) Plot of the edge length standard deviation a ofthe MST built on the triangular percolating lattice versusp. (e) and (f) Plots of da/dp and d2a/dp2 respectively ver-
number of sites and the summations are on the size of the clusters. The first term of eq. (4) is well-known in the classical theory of percolation [1—3] and its singular behaviour nearp~may be extracted from the two critical exponents a and /1:
[
1
n~ (~)j
cc p—vt
,
(5)
sn 5(p)
susp. The m and a values are normalized,
2—a
sing
S
cc (p—pr)
.
(6)
sing
37
Volume 139,number 1.2
PHYSICSLETTERSA
But the second term is new, and particularly the parameter l~.Obviously, this parameter is strongly related to the problems of conductivity. The possibility of a dielectric breakdown, for example, of the network before percolation, depends on the value of this parameter which is related to the minimum gap order parameter [14]. A discussion of the hopping conduction in quantum percolation may be related to an adaptation of the same concept. Before percolation, the conductivity variation involves the evolution of 1~,and thus of m, if micro-internalbreakdowns are to be taken into account as would be the case in a realistic simulation of matter [13]. The minimal spanning tree (MST) method to analyse order and disorder in distributions of objects was used to investigate the percolation transition on a triangular lattice. It was shown numerically that the value of the threshold probability is retrieved, simultaneously with a new geometrical parameter: the mean edge length. This quite new approach should be very useful to tackle the conductivity studies of percolation networks such as dielectric breakdown and hopping conductivity. Moreover it must be emphasized that the MST makes it possible to determine this percolation order parameter in time n log n which is computationally more efficient than many of the more complex order parameters ofpercolation which require of order n2 steps to compute.
References [l]D. Stauffer, Phys. Rep. 54 (1979)1. [2] J.W. Essam, Rep. Prog. Phys. 43 (1980) 833. [3] D. Stauffer, Introduction to percolation theory (Taylor and Francis, London, 1985). [4] P.L. Leath and G.R. Reich, J. Phys. C 11(1978) 4017. [5]H.E.Stanley,J.Phys.Al0(l977)L21l:
38
24June 1989
HE. Stanley and A. Coniglio. in: Percolation structures and processes (Ann. Israel Phys. Soc.). Vol.5, eds. G. Deutscher, R. Zallen and J. Adler (Huger. Bristol, 1983), HE. Stanle~yand N. Ostrowsky. in: On growth and form (Nijhoff. The Hague, 1986), and references therein. [6] H.J. Herrmann. D.C. Hong and H.E. Stanley. J. Phys. A 17 (1984) L26 I: T. Ohtsuki and T. Keyes, J. Phys. A 17(1984) L267. [7] Gennes, Phys. A 38 (1972) [8] PG. KM. Dc Middlemiss. S.G.Lett. Whittington and339. D.S. Gaunt. J. Phys. A 13 (1980) 1835; M. Barma and P. Ray, Phys. Rev. B 34 (1986)3403. [9]T. Nagatani, J. Phys. A 19 (1986) LI 165. [10] S. Kirkpatrick, Rev. Mod. Phys. 45 (1973) 574; J.P. Straley, AlP Conf. Proc. 40 (1978) 118, and other thisC.H. volume of AlP Conf. Proc. [II] particles G.E. Pikeinand Seager, Phys. Rev. B 10 (1974) 1421, 1435. [12] J. Clerc, G. Giraud and J. Roussenq, Compt. Rend. Acad. Sci. B 281(1975) 227: H. Ottavi, J. Clerc, G. Giraud, J. Roussenq, E. Guyon and C.D.Mitescu,J.Phys.C 11(1978)1311. [13] P.M. Duxbury, P.D. Beale and P.L. Leath. Phys. Rev. Lett. 57 (1986) 1052. [14] R.B. Stinchcombe, P.M. Duxbury and P. Shukla, J. Phys. A 19 (1986) 3903. [15] V. B4Ambegaokar, (1971) 2612.B.1. Halperin and J.S. Langer. Phys. Rev. [16] P.G. de Gennes. Recherche 7 (1976) 919: J. Phys. (Paris) Lelt. 37 (1976) LI: G. Deutscher. Y. Levy and B. Souillard, Europhys. Lett. 4 (1987) 577. [17]C. Dussert, G. Rasugni, M. Radigni, ~. Palmari and A. Llebaria. Phys. Rev. B 34 (1986) 3528. [18] R. Hofmann and AK. Jam, Patt. Recognit. Lett. 1 (1983) [75. [19]J. Bearwood, J.H. Halton and J.M. Hammersley. Proc. Cambridge Philos. Soc. 55 (1959) 299. [20] C. Dussert, M. Rasigni, J. Palmari, G. Rasigni. A. Llebaria and F. Marty, J. Theor. Biol. 125 (1987) 317. [21] C.T. Zahn, IEEE Trans. Comput. C 20 (1971) 68. [221 M. Ahmadzadeh and W. Simpson, Phys. Rev. B 25 (1982) 4633; G. Amarendra and G. Ananthakrishna. Solid State Commun.58(l986)873.
PHYSICS LETTERS A
24 June 1989
MINIMAL SPANNING TREE APPROACH TO PERCOLATION AND CONDUCTIVITY THRESHOLD C. DTJSSERT, G. RASIGNI and M. RASIGNI Département de Physique des Interactions Photons—Matière, Case ECJ, Faculté des Sciences de St Jérôrne, 13397 Marseille Cedex 13, France Received 8 February 1989; revised manuscript received 19 May 1989; accepted for publication 22 May 1989 Communicated by A.A. Maradudin
The minimal spanning tree (MST) method to analyse order and disorder in distributions of objects is used to investigate the percolation transition on a triangular lattice. It is shown numerically that the value of the threshold probability is retrieved, simultaneously with a new geometrical parameter: the mean edge length. This is expected to be useful in the conductivity studies of percolation networks such as dielectric breakdown, conductivity and other propagation phenomena. The potentialities of the method in dealing with an extensive study of percolation on continuous as well as lattice systems are also exhibited.
The standard formalism for studying percolation is based on the size of the clusters through its first moments. This formalism makes it possible to exhibit critical exponents, and thus to relate this problem directly to the field of general phase transitions [1—31.For some years now, it has appeared that other geometrical properties of the clusters (and particularly the percolation cluster) are of interest. The perimeter [4,1], backbone [5], elastic backbone, dangling ends [6], were then investigated, leading to
shown in this paper that the minimal spanning tree analysis (MST) recently developed by Dussert et al. [17] to study order and disorder in sets of points may be successfully implemented to tackle the percolation problem and the above-mentioned concepts. The main definitions related to the graph theory have previously been given [17]. It should be remembered that an edge-weighted linear graph G= (X, E) is composed of a set of points X= (x1, x2, ...) called nodes and a set of node pairs
various fractal dimensions and new critical exponents. Another approach was to study random walks on the percolation cluster [7,2]. These studies were extended to various deterministic walks in order to determine the shortest paths [8], diameters of clusters [1,3], cutting bonds [5,91, etc. The usual justification given for these studies is their possible use in conductivity of random (or dilute) systems and in flows of fluids in porous media because of their wide range of applications. The problem of the conductivity transition at the percolation threshold was soon investigated both theoretically [10], numerically [111 and experimentally [121. More recently, a large number of miscellaneous concepts appeared about this subject, which include electrical breakdown [13] and minimum gap [14], hopping conductivity [15,16] and shortest path length [81. It is
E= [(x,, xi.)] called edges, with a number called weight (in this paper the Euclidian distance) assigned to each edge. A treeis a connected graph without closed 1oops. A MST is a tree which contains all the nodes and where the sum of the edge weights is minimal. Depending on the starting point there may be more than one MST for a given set of points, but all the MSTs have the same edge-length histogram. It follows that statistical information deduced from the histogram, such as the average edge length m and the standard deviation a may be used as characteristics for the corresponding distribution. In order to be able to compare different distributions regardless the particles density and the sampling window, the value of m and a must be normalized. For this purpose the process described by Hofmann and Jam [181 has been used. It may be summarized as fol-
0375-9601/89/s 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
35
Volume
139. number
1.2
PHYSICS
lows. The area A of the sampling window is defined as the area H of the convex hull of the data normalized by means of the relation
A=Hl(l-fin) >
(1)
in which n is the number of points in the window and f is the number of faces on the convex hull. Given that the expected length of a randomly chosen edge of a MST related to n uniformly distributed points in a sampling window A, is asymptotically proportional [ 191 to (nA)“‘/(n-l),
(2)
the normalized values of m and fl of the MST constructed from a given set of data are obtained by dividing the original ones by expression (2). All distributions can be plotted in the (m, a) plane and easily compared with well-characterized distributions (for instance, perfectly ordered or random ones) due to the normalization process. For example, consider the set of points ordered at the nodes of a lattice (triangular for instance). The edge-length histogram of the corresponding MST provides the normalized values o=O and m= 1.075 (fig. 1). The arrangement may be randomized by giving each point a new position deduced from its previous position using a Gaussian distribution with a standard deviation of w. Then a trajectory is obtained (full line in fig. 1)) each point of the trajectory corresponding
24 June I989
LETTERS A
to a certain degree of order. For a certain value of o. the random distribution (0~0.300, m=0.650) is reached. We explained in refs. [ 171 and [ 201 how to quantize accurately the (m, a) plane and how to determine the degree of order for any distribution by taking a simple reading in the ( IM, a) plane. A survey of its main properties shows that the MST is a data structure well suited to retrieve the usual properties of the percolation network. Indeed, classification and pattern recognition techniques which use the MST [ 2 1 ] immediately bring out the clusters (and denombrate them), as well as the backbone. dangling ends and paths of various characteristics. All these parameters can be linearly determined and given that some algorithms exist which are able to compute an MST in a time proportional to n log n. where II is the number of occupied sites on the lattice, the procedure seems very efficient for an cxtensive study of the percolation phenomenon. .4ctually the MST, in addition to the easy computation of these parameters, lead to the definition of new ones which are presented in the following. By way of illustration fig. 2 shows for a very sim-
P-1 ..’ ;..
.
.
‘..
_.. L.
.
.I. ,‘:
E Perfect 0.5
0.6
order
0.7 Mean
0.8 value
0.9
1
.
.,\.
.,I. . . . *~ .?~~_ . . Jo
./I’ . I) .
.
.
‘.
_-r’
.
0.3
1.1
m
Fig. 1. (m, 0) diagram. The triangles represent the percolation trajectory, the full line is for the trajectory ofthe triangular lattice progressively disordered [ 171 and the dashed line is the hardcore trajectory.
36
.
P :
0
.
‘, ._ji.
.
Fig. 2. Minimal spanning tree associated to a simplified set of points (10x 10 triangular lattice) for p= 1 and p=O.3, where p is the probability for a site of the lattice to be occupied. (0) Occupied site. (a ) Unoccupied site. Dashed line: convex hull related to the set of occupied sites forp=0.3. It can be seen that for this simplified casef=9.
Volume 139, number 1,2
PHYSICS LETTERS A
plified set of points (lOx 10 triangular lattice) the MST corresponding to p 1 and p=0.3, where p is the probability for a site of the lattice to be occupied. The actual computations were performed on a 300 x 300 triangular lattice. For various values of p the MSTs were computed and the corresponding values of m and a were retrieved and reported in the (m, a) plane (fig. 1). The (m, a) diagram reveals an inflexion point of the trajectory for a value of (m, a) corresponding to p=p~ =0.5 which is known to be the percolation threshold for the triangular lattice. It is to be noticed that both m and a have a particular behaviour at p=p~=O.5O,which is revealed on their derivatives (fig. 3). Thus the MST analysis clearly exhibits the threshold percolation (critical probability) as do various other methods. Moreover, for p=p~,we have m=m~=0.78.This critical value of m which is linked to the geometry of the systern, may be considered as a new characteristic of the percolation system related to the triangular lattice and should be of interest in the studies on conductivity, The trajectory related to the percolation system maybe compared with the one determined by taking
°~Thi~
• (a)~ b E
•
O.2~
O8~
i.
0.1, 0.6
___________
(b)~
~
~~::L~•L~ 8~
‘
—
/
where the step of the lattice is equal to unity. The calculation of the mean edge length is made of two terms: the first involves the subtrees constructed on each cluster (of which the mean edge length is unity), and the second involves the inter-cluster links. A priori those links have the form l(s, s’, p) where s and s’ are the sizes of the clusters linked by 1. However the building of the MST may be viewed as a dynamical process where the sites are progressively ineluded in the growing tree. From this point of view the relevant parameter is the length I~=l(s, p) of the
(~
201
b ~ 1~
‘~
m=p-112 (s-1)n~+~ ~ (4) where ~ is the number of s-clusters divided by the
~
I
a full lattice and then moving the sites according to the process previously described [20]. It can be seen that the trajectory related to the percolation system is below this latter trajectory, which is satisfactory, given that the percolation system is much more ordered, due to the lattice structure, than a continuous one. Another simulation may be plotted, which consists in a system of hard disks of which the diameter increases [11,22]. The resulting trajectory starts from the random distribution and goes asymptotically to the triangular lattice. These curves reveal the very influence of the lattice as compared to the continuous case, as well as the potentiality of the method of analysis which is efficient in both those cases, and which makes possible a comparison between them. Let us also remark that for n large the normalization relation (2) may be written “n/A 1/2 1/2 (3
edges which link the s-clusters to the MST. Then, the parameter m may be written
0 •
O6~
24 June 1989
~ 0 0
P
02
04
06
0.8
1
p
Fig. 3. (a) Plot of the mean edge length m ofthe MST built on the triangular percolating lattice versus the probability of occupancy of the Sites p. (b) and (c) Plots of dm/dp and d2m/dp2 respectively versus p. (d) Plot of the edge length standard deviation a ofthe MST built on the triangular percolating lattice versusp. (e) and (f) Plots of da/dp and d2a/dp2 respectively ver-
number of sites and the summations are on the size of the clusters. The first term of eq. (4) is well-known in the classical theory of percolation [1—3] and its singular behaviour nearp~may be extracted from the two critical exponents a and /1:
[
1
n~ (~)j
cc p—vt
,
(5)
sn 5(p)
susp. The m and a values are normalized,
2—a
sing
S
cc (p—pr)
.
(6)
sing
37
Volume 139,number 1.2
PHYSICSLETTERSA
But the second term is new, and particularly the parameter l~.Obviously, this parameter is strongly related to the problems of conductivity. The possibility of a dielectric breakdown, for example, of the network before percolation, depends on the value of this parameter which is related to the minimum gap order parameter [14]. A discussion of the hopping conduction in quantum percolation may be related to an adaptation of the same concept. Before percolation, the conductivity variation involves the evolution of 1~,and thus of m, if micro-internalbreakdowns are to be taken into account as would be the case in a realistic simulation of matter [13]. The minimal spanning tree (MST) method to analyse order and disorder in distributions of objects was used to investigate the percolation transition on a triangular lattice. It was shown numerically that the value of the threshold probability is retrieved, simultaneously with a new geometrical parameter: the mean edge length. This quite new approach should be very useful to tackle the conductivity studies of percolation networks such as dielectric breakdown and hopping conductivity. Moreover it must be emphasized that the MST makes it possible to determine this percolation order parameter in time n log n which is computationally more efficient than many of the more complex order parameters ofpercolation which require of order n2 steps to compute.
References [l]D. Stauffer, Phys. Rep. 54 (1979)1. [2] J.W. Essam, Rep. Prog. Phys. 43 (1980) 833. [3] D. Stauffer, Introduction to percolation theory (Taylor and Francis, London, 1985). [4] P.L. Leath and G.R. Reich, J. Phys. C 11(1978) 4017. [5]H.E.Stanley,J.Phys.Al0(l977)L21l:
38
24June 1989
HE. Stanley and A. Coniglio. in: Percolation structures and processes (Ann. Israel Phys. Soc.). Vol.5, eds. G. Deutscher, R. Zallen and J. Adler (Huger. Bristol, 1983), HE. Stanle~yand N. Ostrowsky. in: On growth and form (Nijhoff. The Hague, 1986), and references therein. [6] H.J. Herrmann. D.C. Hong and H.E. Stanley. J. Phys. A 17 (1984) L26 I: T. Ohtsuki and T. Keyes, J. Phys. A 17(1984) L267. [7] Gennes, Phys. A 38 (1972) [8] PG. KM. Dc Middlemiss. S.G.Lett. Whittington and339. D.S. Gaunt. J. Phys. A 13 (1980) 1835; M. Barma and P. Ray, Phys. Rev. B 34 (1986)3403. [9]T. Nagatani, J. Phys. A 19 (1986) LI 165. [10] S. Kirkpatrick, Rev. Mod. Phys. 45 (1973) 574; J.P. Straley, AlP Conf. Proc. 40 (1978) 118, and other thisC.H. volume of AlP Conf. Proc. [II] particles G.E. Pikeinand Seager, Phys. Rev. B 10 (1974) 1421, 1435. [12] J. Clerc, G. Giraud and J. Roussenq, Compt. Rend. Acad. Sci. B 281(1975) 227: H. Ottavi, J. Clerc, G. Giraud, J. Roussenq, E. Guyon and C.D.Mitescu,J.Phys.C 11(1978)1311. [13] P.M. Duxbury, P.D. Beale and P.L. Leath. Phys. Rev. Lett. 57 (1986) 1052. [14] R.B. Stinchcombe, P.M. Duxbury and P. Shukla, J. Phys. A 19 (1986) 3903. [15] V. B4Ambegaokar, (1971) 2612.B.1. Halperin and J.S. Langer. Phys. Rev. [16] P.G. de Gennes. Recherche 7 (1976) 919: J. Phys. (Paris) Lelt. 37 (1976) LI: G. Deutscher. Y. Levy and B. Souillard, Europhys. Lett. 4 (1987) 577. [17]C. Dussert, G. Rasugni, M. Radigni, ~. Palmari and A. Llebaria. Phys. Rev. B 34 (1986) 3528. [18] R. Hofmann and AK. Jam, Patt. Recognit. Lett. 1 (1983) [75. [19]J. Bearwood, J.H. Halton and J.M. Hammersley. Proc. Cambridge Philos. Soc. 55 (1959) 299. [20] C. Dussert, M. Rasigni, J. Palmari, G. Rasigni. A. Llebaria and F. Marty, J. Theor. Biol. 125 (1987) 317. [21] C.T. Zahn, IEEE Trans. Comput. C 20 (1971) 68. [221 M. Ahmadzadeh and W. Simpson, Phys. Rev. B 25 (1982) 4633; G. Amarendra and G. Ananthakrishna. Solid State Commun.58(l986)873.