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Influence of the boundary on the connective constant of branching structures
Physica A 203 (1994) 261-268 North-Holland SSDZ
0378-4371(93)E0422-B
Influence of the boundary on the connective constant of branching structures Moraal fiir
1+
for the
sponding infinite
1. Introduction has been for many that the of a tree has large influence the thermodynamics spin systems on such [l-4]. This due to fact that finite fraction the vertices such a are at boundary. In phase transitions continuous order in contrast the situation the corresponding lattice (Bethe [3]. In article, it shown that presence of a boundary branching structures as Cayley cactus trees ref. [4] a general of a structure as recursive site sequence) also the problem the enumeration self-avoiding walks on such profoundly. It expected that number f(n) SAWS on a structure as
f(n) = cCLn
(1.1)
for large 12 in the thermodynamic limit. (The thermodynamic limit for spin systems does exist on these structures, as is shown in ref. [4].) Here p is called 0378-4371/94/$07.00 0 1994 - Elsevier Science B.V. All rights reserved
262
H. Moraal
I Connective
constant
of branching
structures
the connective constant of the structure. For real lattice systems, eq. (1.1) contains, in general, an extra factor proportional to a power of IZ with characteristic exponent. In section 2, it is shown that eq. (1.1) is not strictly appropriate for the case of a Cayley tree; instead, there occurs a characteristic even-odd (in n) effect, i.e., the forefactor C in eq. (1.1) depends on the parity of n. With this taken into account, the connective constant can be calculated and is found to be the square root of the value for the corresponding Bethe lattice. In section 3, the SAW generating function for the triangular cactus tree is calculated and it is shown that eq. (1.1) holds in this case. The calculated value of the connective constant again differs markedly from the value for the corresponding infinite graph. A short discussion follows as section 4.
2. Cayley trees and Bethe lattices The Bethe lattice with coordination number z is the unique locally finite graph without circuits for which every vertex has valency t = m + 1. Fig. la shows a portion of this graph for the case z = 4. Because there are no closed circuits, every walk on this graph is a SAW. Since there are z = m + 1 directions to start a walk from an arbitrary vertex and m directions after a walk of at least one step has been performed, the number of n-step SAWS starting at a given vertex is simply s(n) = (m + l)m”-’
.
(2.1)
a Fig. 1. (a) Part of the Bethe m=3.
b lattice
with valency
four.
(b) A three-generation
Cayley
tree with
H. Moraal I Connective constant of branching structures
The generating qx)
=
263
function for these walks is
(m+ 1)x 1-mx
(2.2)
’
Eq. (2.1) implies that eq. (1.1) is satisfied with the connective
Am) = m
(Bethe) .
constant (2.3)
A Cayley tree with branching ratio m and k generations is obtained from m Cayley trees with k - 1 generations by connecting the top vertices of these subtrees to a new top vertex by means of m edges. Fig. lb shows a threegeneration Cayley tree with m = 3, corresponding to the Bethe lattice of fig. la. Now the SAWS on a Cayley tree are of three different kinds: (i) SAWS starting at the top vertex. If T(n, k) is the number of these with n steps in a k-generation tree, then the recursion relation
T(n, k)
=
m&,, + mT(n - 1, k - 1)
follows immediately. satisfies
Therefore,
h(4 = m-d1+ 4-1(X)1>
the generating
(2.4) function
tl(x)=mx,
t,(x) of such SAWS
(2.5)
which has the solution
W-3 Strictly speaking, this equation is only valid for mx # 1, but by continuity it can be extended. (ii) SAWS not starting at, but visiting the top vertex. The number S(n, k) of these is given in terms of the T-values: S(n, k) =$m(m - 1)
c
T(n,, k - l)T(n,,
k -
1)
“I,“2
n,+n2=n-2 +
m(m - l)T(n - 2, k - 1) + im(m - 1)6,,, .
Here the first term comes from two branches, the second from one branch only and the third vertex. The generating function
(2.7)
SAWS extending more than one generation in SAWS extending more than one generation in term from n = 2 SAWS folding over the top corresponding to eq. (2.7) is
264
H. Moraal
I Connective
constant
of branching
structures
Sk(X) = +z(m - l)X”[l + tk_i(X)12 .
(2.8)
(iii) SAWS not visiting the top vertex at all. The number U(n, k) of these is easily seen to be given by
u(n, k) = m[T(n, k -
1) + S(n, k - 1) + U(n, k - 1)] ,
(2.9)
where the expression in square brackets is simply the total number F(n, k - 1) of n-step SAWS for a (k - 1)-generation tree. Eq. (2.9) leads to a recursion relation for the corresponding generating functions:
Uk(4= m[t,-,(x) + Sk-1(4+ Uk-I(41= mfk-l(x) >
q(x) = 0.
(2.10)
This has the solution k-l uk(x)
=
!Tl
m’[tk-,(x)
+
sk-f(x)I
(2.11)
.
Eqs. (2.6), (2.8) and (2.11) imply for the total generating
function fk(x): (2.12)
so that, since the number of vertices of a k-generation
Cayley tree is
k+l WI
=
“,
the generating
_;
’
(2.13)
>
function per vertex is, in the limit k+ 00, given by
fktx) f(x) =$lil N(k) = 3x
(m+l)x+2 1-mx2
(2.14)
’
This shows that the number of SAWS per vertex in the thermodynamic F(n) =
ms
+(m+l)ms
for IZodd , forneven,
n=2.Y+1,
limit is (2.15)
n=2s+2,
showing an even-odd effect. If one defines the connective constant by eq. (l.l), but with different prefactors for odd and even terms as given above, then p(m) = v5i
follows:
(2.16)
(WW
the presence
of the boundary
changes
the connective
constant
265
H. Moraal I Connective constant of branching structures
drastically, since eq. (2.16) is the square root of the Bethe lattice value of eq. (2.3).
3. Triangular cactuses As a further example, the infinite triangular cactus, of which fig. 2a shows a part (this is the largest part fitting on the triangular lattice without distortion) and the corresponding tree, shown in fig. 2b for k = 4, are considered. To find the connective constant of the infinite graph, it is remarked that there are two ways to start the SAW at a new triangle corresponding to whether one or two sides of the previous triangle have been included. Therefore, the number t(n) of SAWS of length 12 on the infinite graph which start with an edge of a prescribed triangle, satisfies the recursion relation t(n) = 2t(n - 1) + 2t(n - 2) + 2s,,, ) The corresponding
t(x) =
generating
t(0) = t(-1)
= 0.
(3.1)
function is, therefore,
2x
(3.2)
1-2X-2X2’
Since eq. (3.2) implies that the pole nearest the origin is at X, = +(fi connective constant of the infinite triangular cactus is &tri) = 1 + X4.
a
- l), the
(3.3)
b
Fig. 2. (a) Part of the infinite triangular cactus. (b) A four-generation
triangular cactus tree.
266
H. Moraal
/ Connective
constant
of branching
structures
The SAWS for the finite tree are of the same three different types as for the Cayley tree of the previous section: (i) The number of SAWS starting at the top vertex is Z’(n, k) = 2T(n - 1,
k -
1) + 2T(n - 2,
k -
2) + 26,,, ,
(3.4)
where the first two terms are due to SAWS not including or including the “horizontal” edge of the top triangle, respectively. The corresponding generating functions satisfy the recursion relation
fk(X)= 2x + 2x(1 +X)&,(X)
t1@)=2X(l+x),
)
(3.5)
with solution 1-(2x+2X2)k l_2x_2x2
tk(X)=2x2(2X+2X2)k-*+2.X
.
(3.6)
(ii) SAWS folding over the top vertex cannot contain the third edge of the top triangle, so that eqs. (2.7) and (2.8) for m = 2 hold here as well; for the generating function s&) this means Sk(X) = X2[1 + tk_I(X)]2 .
(3.7)
(iii) The SAWS not visiting the top vertex now consist not only of the ones of eqs. (2.9) and (2.10), but also ones containing the “horizontal” top triangle edge: U(n, k) = 2[T(n, +
k -
n;,
1) + s(n,
k -
1) + U(n,
k -
l)] + 22-(n - 1,
~(~,,k-1)~(~,,k-l),
k -
1) + S,,, (3.8)
n1+n2=n-1
from which the corresponding
generating
function follows as
%f(x)= 2[tk-*(X)+ Sk-l(X)+ %-l(X)1+ 2X&l(X) u*(x) =x . + W~,-,(412 +x 7
(3.9)
Eqs. (3.5), (3.7) and (3.9) give the following recursion relation for their sum
f&): fk(x)=2fk-*(x)
with solution
+3x+x2+4x(1
+x)t,_,(X)+x(2+x)[tk-1(X)]2,
(3.10)
H. Moraal
I Connective constant of branching structures
267
k-l fk(x)
=
2k-1fI(X) + lgI 2k-‘-‘[3x +x2 + 4x(1 + x)t,(x) +X(X + 2)t,($]
,
(3.11) (3.12)
fi (X) = 3X(1 + x) .
Since the functions t,(x) are known explicitly, eq. (3.6), this can be evaluated completely; for the limit of infinitely many generations, this gives, pro triangle,
=+ 4x+3x2+ (
X(X+ 2)(2X2 + 2x + 1)(4x2 + 2.X+ 1) 1 - 2X2(1 + x)’ >.
(3.13)
The connective constant is given by the inverse of x,, the value of x nearest to the origin, where the denominator of the third term of eq. (3.13) vanishes. This gives &cac) = $(VXZ
+ ti)
,
(3.14)
which is approximately 2.090658 . . . , whereas the numerical value of the infinite graph case is 2.732050 . . . , eq. (3.3). Explicit evaluation of the coefficients of a series expansion of eq. (3.13) shows that eq. (1.1) is satisfied for large 12 with forefactor approximately 4.65090 . . . .
4. Discussion As is clear from the examples studied above, the presence of a boundary containing a finite fraction of the sites of a branching structure modifies the connective constant as compared to the value of this for the corresponding locally finite graph. From the way in which a recursive site graph sequence is generally defined [4], i.e., by connecting a number of copies of the (k - l)generation graphs to a new top vertex by a prescribed k-generation subgraph, it is clear that the breakup of the class of SAWS into the three classes used in the above is always possible. The solution then always proceeds in the same way: (i) Find the recursion relation for the class of SAWS starting at the top vertex; this recursion relation is, apart from the indexing by generations,
268
H. Moraal
I Connective
constant
of branching
structures
identical to the recursion relation for the number of SAWS on the corresponding locally finite graph. (ii) The class of SAWS folding over the top vertex is completely given in terms of those starting at the top vertex. (iii) The number of SAWS not visiting the top vertex satisfies a recursion relation containing the solutions of (i) and (ii). The solution of this recursion relation solves the problem completely. In ref. [4], another class of recursively generated pseudo-lattices, the recursive bond graph sequences, has also been generally defined. These are useful in obtaining approximate renormalization relations for spin systems. The problem of the enumeration of SAWS on such pseudo-lattices has been studied [5]. For this case, however, there is no locally finite corresponding infinite graph to compare the results with.
References [l] [2] [3] [4] [5]
L.K. Runnels, J. Math. Phys. 8 (1967) 2081. T.P. Eggarter, Phys. Rev. B 9 (1974) 2928. E. Miiller-Hartmann and J. Zittartz, Phys. Rev. Lett. 33 (1974) 893. H. Moraal, Classical, Discrete Spin Models (Springer, Berlin, 1984). J. Melrose, J. Phys. A 18 (1985) L 17.
0378-4371(93)E0422-B
Influence of the boundary on the connective constant of branching structures Moraal fiir
1+
for the
sponding infinite
1. Introduction has been for many that the of a tree has large influence the thermodynamics spin systems on such [l-4]. This due to fact that finite fraction the vertices such a are at boundary. In phase transitions continuous order in contrast the situation the corresponding lattice (Bethe [3]. In article, it shown that presence of a boundary branching structures as Cayley cactus trees ref. [4] a general of a structure as recursive site sequence) also the problem the enumeration self-avoiding walks on such profoundly. It expected that number f(n) SAWS on a structure as
f(n) = cCLn
(1.1)
for large 12 in the thermodynamic limit. (The thermodynamic limit for spin systems does exist on these structures, as is shown in ref. [4].) Here p is called 0378-4371/94/$07.00 0 1994 - Elsevier Science B.V. All rights reserved
262
H. Moraal
I Connective
constant
of branching
structures
the connective constant of the structure. For real lattice systems, eq. (1.1) contains, in general, an extra factor proportional to a power of IZ with characteristic exponent. In section 2, it is shown that eq. (1.1) is not strictly appropriate for the case of a Cayley tree; instead, there occurs a characteristic even-odd (in n) effect, i.e., the forefactor C in eq. (1.1) depends on the parity of n. With this taken into account, the connective constant can be calculated and is found to be the square root of the value for the corresponding Bethe lattice. In section 3, the SAW generating function for the triangular cactus tree is calculated and it is shown that eq. (1.1) holds in this case. The calculated value of the connective constant again differs markedly from the value for the corresponding infinite graph. A short discussion follows as section 4.
2. Cayley trees and Bethe lattices The Bethe lattice with coordination number z is the unique locally finite graph without circuits for which every vertex has valency t = m + 1. Fig. la shows a portion of this graph for the case z = 4. Because there are no closed circuits, every walk on this graph is a SAW. Since there are z = m + 1 directions to start a walk from an arbitrary vertex and m directions after a walk of at least one step has been performed, the number of n-step SAWS starting at a given vertex is simply s(n) = (m + l)m”-’
.
(2.1)
a Fig. 1. (a) Part of the Bethe m=3.
b lattice
with valency
four.
(b) A three-generation
Cayley
tree with
H. Moraal I Connective constant of branching structures
The generating qx)
=
263
function for these walks is
(m+ 1)x 1-mx
(2.2)
’
Eq. (2.1) implies that eq. (1.1) is satisfied with the connective
Am) = m
(Bethe) .
constant (2.3)
A Cayley tree with branching ratio m and k generations is obtained from m Cayley trees with k - 1 generations by connecting the top vertices of these subtrees to a new top vertex by means of m edges. Fig. lb shows a threegeneration Cayley tree with m = 3, corresponding to the Bethe lattice of fig. la. Now the SAWS on a Cayley tree are of three different kinds: (i) SAWS starting at the top vertex. If T(n, k) is the number of these with n steps in a k-generation tree, then the recursion relation
T(n, k)
=
m&,, + mT(n - 1, k - 1)
follows immediately. satisfies
Therefore,
h(4 = m-d1+ 4-1(X)1>
the generating
(2.4) function
tl(x)=mx,
t,(x) of such SAWS
(2.5)
which has the solution
W-3 Strictly speaking, this equation is only valid for mx # 1, but by continuity it can be extended. (ii) SAWS not starting at, but visiting the top vertex. The number S(n, k) of these is given in terms of the T-values: S(n, k) =$m(m - 1)
c
T(n,, k - l)T(n,,
k -
1)
“I,“2
n,+n2=n-2 +
m(m - l)T(n - 2, k - 1) + im(m - 1)6,,, .
Here the first term comes from two branches, the second from one branch only and the third vertex. The generating function
(2.7)
SAWS extending more than one generation in SAWS extending more than one generation in term from n = 2 SAWS folding over the top corresponding to eq. (2.7) is
264
H. Moraal
I Connective
constant
of branching
structures
Sk(X) = +z(m - l)X”[l + tk_i(X)12 .
(2.8)
(iii) SAWS not visiting the top vertex at all. The number U(n, k) of these is easily seen to be given by
u(n, k) = m[T(n, k -
1) + S(n, k - 1) + U(n, k - 1)] ,
(2.9)
where the expression in square brackets is simply the total number F(n, k - 1) of n-step SAWS for a (k - 1)-generation tree. Eq. (2.9) leads to a recursion relation for the corresponding generating functions:
Uk(4= m[t,-,(x) + Sk-1(4+ Uk-I(41= mfk-l(x) >
q(x) = 0.
(2.10)
This has the solution k-l uk(x)
=
!Tl
m’[tk-,(x)
+
sk-f(x)I
(2.11)
.
Eqs. (2.6), (2.8) and (2.11) imply for the total generating
function fk(x): (2.12)
so that, since the number of vertices of a k-generation
Cayley tree is
k+l WI
=
“,
the generating
_;
’
(2.13)
>
function per vertex is, in the limit k+ 00, given by
fktx) f(x) =$lil N(k) = 3x
(m+l)x+2 1-mx2
(2.14)
’
This shows that the number of SAWS per vertex in the thermodynamic F(n) =
ms
+(m+l)ms
for IZodd , forneven,
n=2.Y+1,
limit is (2.15)
n=2s+2,
showing an even-odd effect. If one defines the connective constant by eq. (l.l), but with different prefactors for odd and even terms as given above, then p(m) = v5i
follows:
(2.16)
(WW
the presence
of the boundary
changes
the connective
constant
265
H. Moraal I Connective constant of branching structures
drastically, since eq. (2.16) is the square root of the Bethe lattice value of eq. (2.3).
3. Triangular cactuses As a further example, the infinite triangular cactus, of which fig. 2a shows a part (this is the largest part fitting on the triangular lattice without distortion) and the corresponding tree, shown in fig. 2b for k = 4, are considered. To find the connective constant of the infinite graph, it is remarked that there are two ways to start the SAW at a new triangle corresponding to whether one or two sides of the previous triangle have been included. Therefore, the number t(n) of SAWS of length 12 on the infinite graph which start with an edge of a prescribed triangle, satisfies the recursion relation t(n) = 2t(n - 1) + 2t(n - 2) + 2s,,, ) The corresponding
t(x) =
generating
t(0) = t(-1)
= 0.
(3.1)
function is, therefore,
2x
(3.2)
1-2X-2X2’
Since eq. (3.2) implies that the pole nearest the origin is at X, = +(fi connective constant of the infinite triangular cactus is &tri) = 1 + X4.
a
- l), the
(3.3)
b
Fig. 2. (a) Part of the infinite triangular cactus. (b) A four-generation
triangular cactus tree.
266
H. Moraal
/ Connective
constant
of branching
structures
The SAWS for the finite tree are of the same three different types as for the Cayley tree of the previous section: (i) The number of SAWS starting at the top vertex is Z’(n, k) = 2T(n - 1,
k -
1) + 2T(n - 2,
k -
2) + 26,,, ,
(3.4)
where the first two terms are due to SAWS not including or including the “horizontal” edge of the top triangle, respectively. The corresponding generating functions satisfy the recursion relation
fk(X)= 2x + 2x(1 +X)&,(X)
t1@)=2X(l+x),
)
(3.5)
with solution 1-(2x+2X2)k l_2x_2x2
tk(X)=2x2(2X+2X2)k-*+2.X
.
(3.6)
(ii) SAWS folding over the top vertex cannot contain the third edge of the top triangle, so that eqs. (2.7) and (2.8) for m = 2 hold here as well; for the generating function s&) this means Sk(X) = X2[1 + tk_I(X)]2 .
(3.7)
(iii) The SAWS not visiting the top vertex now consist not only of the ones of eqs. (2.9) and (2.10), but also ones containing the “horizontal” top triangle edge: U(n, k) = 2[T(n, +
k -
n;,
1) + s(n,
k -
1) + U(n,
k -
l)] + 22-(n - 1,
~(~,,k-1)~(~,,k-l),
k -
1) + S,,, (3.8)
n1+n2=n-1
from which the corresponding
generating
function follows as
%f(x)= 2[tk-*(X)+ Sk-l(X)+ %-l(X)1+ 2X&l(X) u*(x) =x . + W~,-,(412 +x 7
(3.9)
Eqs. (3.5), (3.7) and (3.9) give the following recursion relation for their sum
f&): fk(x)=2fk-*(x)
with solution
+3x+x2+4x(1
+x)t,_,(X)+x(2+x)[tk-1(X)]2,
(3.10)
H. Moraal
I Connective constant of branching structures
267
k-l fk(x)
=
2k-1fI(X) + lgI 2k-‘-‘[3x +x2 + 4x(1 + x)t,(x) +X(X + 2)t,($]
,
(3.11) (3.12)
fi (X) = 3X(1 + x) .
Since the functions t,(x) are known explicitly, eq. (3.6), this can be evaluated completely; for the limit of infinitely many generations, this gives, pro triangle,
=+ 4x+3x2+ (
X(X+ 2)(2X2 + 2x + 1)(4x2 + 2.X+ 1) 1 - 2X2(1 + x)’ >.
(3.13)
The connective constant is given by the inverse of x,, the value of x nearest to the origin, where the denominator of the third term of eq. (3.13) vanishes. This gives &cac) = $(VXZ
+ ti)
,
(3.14)
which is approximately 2.090658 . . . , whereas the numerical value of the infinite graph case is 2.732050 . . . , eq. (3.3). Explicit evaluation of the coefficients of a series expansion of eq. (3.13) shows that eq. (1.1) is satisfied for large 12 with forefactor approximately 4.65090 . . . .
4. Discussion As is clear from the examples studied above, the presence of a boundary containing a finite fraction of the sites of a branching structure modifies the connective constant as compared to the value of this for the corresponding locally finite graph. From the way in which a recursive site graph sequence is generally defined [4], i.e., by connecting a number of copies of the (k - l)generation graphs to a new top vertex by a prescribed k-generation subgraph, it is clear that the breakup of the class of SAWS into the three classes used in the above is always possible. The solution then always proceeds in the same way: (i) Find the recursion relation for the class of SAWS starting at the top vertex; this recursion relation is, apart from the indexing by generations,
268
H. Moraal
I Connective
constant
of branching
structures
identical to the recursion relation for the number of SAWS on the corresponding locally finite graph. (ii) The class of SAWS folding over the top vertex is completely given in terms of those starting at the top vertex. (iii) The number of SAWS not visiting the top vertex satisfies a recursion relation containing the solutions of (i) and (ii). The solution of this recursion relation solves the problem completely. In ref. [4], another class of recursively generated pseudo-lattices, the recursive bond graph sequences, has also been generally defined. These are useful in obtaining approximate renormalization relations for spin systems. The problem of the enumeration of SAWS on such pseudo-lattices has been studied [5]. For this case, however, there is no locally finite corresponding infinite graph to compare the results with.
References [l] [2] [3] [4] [5]
L.K. Runnels, J. Math. Phys. 8 (1967) 2081. T.P. Eggarter, Phys. Rev. B 9 (1974) 2928. E. Miiller-Hartmann and J. Zittartz, Phys. Rev. Lett. 33 (1974) 893. H. Moraal, Classical, Discrete Spin Models (Springer, Berlin, 1984). J. Melrose, J. Phys. A 18 (1985) L 17.