HAL Id: jpa-00209601
https://hal.archives-ouvertes.fr/jpa-00209601
Submitted on 1 Jan 1983
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
A new Monte-Carlo approach to the critical properties of self-avoiding random walks
C. Aragão de Carvalho, S. Caracciolo
To cite this version:
C. Aragão de Carvalho, S. Caracciolo. A new Monte-Carlo approach to the critical prop- erties of self-avoiding random walks. Journal de Physique, 1983, 44 (3), pp.323-331.
�10.1051/jphys:01983004403032300�. �jpa-00209601�
A new Monte-Carlo approach to the critical properties
of self-avoiding random walks
C. Aragão de Carvalho (*)
Laboratoire de Physique Théorique et Hautes Energies (**),
Université de Paris-Sud, Centre d’Orsay, Bât. 211, F-91405 Orsay, France and S. Caracciolo
Scuola Normale Superiore, Piazza dei Cavalieri 1, Pisa and INFN, Sezione di Pisa, Italy (Reçu le 22 juillet 1982, accepté le 18 novembre)
Résumé. 2014 On étudie les propriétés critiques des marches aléatoires sans auto-intersection sur des réseaux hyper- cubiques en dimensions trois et quatre. On considère l’ensemble statistique de toutes ces marches comme fonction
d’une température inverse 03B2 et on associe à chaque marche le poids statistique 03B2L, où L est la longueur de la marche.
Cela nous permet d’utiliser une nouvelle et très efficace simulation de Monte-Carlo. On présente une nouvelle interprétation de l’exposant 03B3, très convenable pour des calculs numériques. En quatre dimensions, les violations
logarithmiques prévues par le groupe de renormalisation sont très bien vérifiées.
Abstract
-We investigate the critical properties of self-avoiding random walks on hypercubic lattices in dimen- sions three and four. We consider the statistical ensembles of all such walks as a function of an inverse tempe- rature 03B2 and associate to each walk the statistical weight 03B2L, where L is its length. This allows us to use a novel and very efficient Monte-Carlo procedure. A new interpretation of the exponent 03B3, suitable for numerical inves-
tigations, is presented. In dimension four, the logarithmic violations predicted by the perturbative renormalization group are very well verified.
Classification Physics Abstracts
05.40
-05.50
-11.10
-75.40D
1. Introduction.
-In this paper we present results concerning a numerical analysis of the statistical
properties of self-avoiding random walks (SAW).
Such walks have been extensively studied in the context of polymer physics. The self-avoiding con-
straint can be viewed as the idealized mathematical realization of the excluded volume effect felt by
monomers along the chain. In order to describe the critical behaviour of these walks, in t:1e asymptotic regime of very large number of steps, a fruitful con- nection has been established with a ferromagnetic
vector model in the limit in which the number of
spin components is sent to zero [1, 2]. In this way
the extremely powerful apparatus of field theory and
renormalization group ideas has become available : in particular the so-called s-expansion [3], in which
the parameter s
=4 - d (d being the number of dimensions) takes, for these physical applications,
the value one.
Furthermore, it has become more and more
interesting, in the field theoretic area, to go in the
opposite direction : the representation of an Euclidean
scalar lattice quantum field theory as a gas of walks, interacting only at their intersections, already con-
sidered in [4], has provided a useful tool in a non-
perturbative approach. Thus, correlation inequalities
have been obtained [5] and the approach to the
continuum (= scaling) limit analysed. J. Frohlich [6]
has rigorously proved that the one- and the two- component A I (o 14 theories and the non-linear o-
model, in five or more dimensions, approach, in the continuum, free (= Gaussian) field theories - a
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01983004403032300
324
consequence of the fact that, in such dimensions, two
random walks almost never intersect. Critical expo- nents take, then, their mean field values.
However, the border case of dimension four, in which the h I + 14 theory becomes renormalizable,
needed extra analysis. In reference [7] a connection
has been established between the triviality of the
continuum limit and the presence of violations to the mean field scaling laws. These violations are
expected to be present, in the form of logarithmic
terms, from the perturbation theory analysis of the
renormalization group equations [8, 9].
Here we shall describe in detail a Monte-Carlo simulation in the space of random walks used to test these violations. Such a procedure turns out to be accurate enough to confirm, with excellent agree- ment, the renormalization group predictions. In addition, a similar technique had not yet been attempt- ed for studying the critical behaviour of polymer chains, so it seemed to us equally interesting to test
it in dimension three.
In section 2 we shall describe the main advantages
of our approach compared with previous ones. We
also give, following reference [7], a new interpretation
of the critical exponent y, which is particularly
suitable for numerical evaluations. Section 3 is devoted to a description of the Monte-Carlo algorithm
we used and in section 4 we present our numerical data.
2. The statistical ensembles and their critical pro-
perties.
-Up to now, Monte-Carlo techniques have
been applied to generate SAW with fixed number of steps, taken to be as large as possible, in order to be
in the critical region [10]. Satisfactory results have, thus, been obtained for the critical exponent v [11, 12],
associated to the behaviour, for large number of steps L, of the mean end-to-end distance :
where the subscript L denotes a mean value over the
ensemble of walks with L steps. The authors of reference [13] claim to detect, in dimension four, the logarithmic violation associated with this quan-
tity (1). Work has also been done to determine the
asymptotic behaviour of the total number of SAW that one can draw on various lattices :
but, as far as we know, only p has been evaluated
(at least in two and three dimensions), through its
connection with the so-called attrition number À., which is defined as
( 1 ) We thank B. Derrida for bringing this reference to our attention.
where No(L) is the total number of random paths
of L steps, i.e. qL,
Iwith q the coordination number of the lattice.
On the other hand, pursuing the analogy with the
Euclidean lattice field theory, we prefer to choose
our statistical ensemble as the space of all SAW and to associate with each walk a statistical weight pL,
which is a function only of the length of the walk.
fl is, in fact, the inverse temperature of the associated lattice field theory, where it plays the role of the renormalization field strength. The essential quanti-
ties in such a context are the correlation functions.
The two point function is defined as :
where co is a SAW, as denoted by the index on the sign of sum, and 4w) is its length in lattice space units. It is sometimes suggestive to replace fl by e - b.
For b > bc, the correlation decays exponentially at large distances, with a coefficient which is the inverse correlation length :
This coefficient will vanish at the critical point and,
in its neighbourhood, will scale according to (d # 4)
with
In addition, for d
=4 :
where v
=1/2 is the mean field value. It can be shown that the critical exponents for m-1 (b) are the
same as for the mean end-to-end distance.
Another interesting quantity is the susceptibility :
whose critical behaviour is easily related to that of N(L) :
so that Pc =e-bc =jl-l.0ncemoreind=4(2.10)
should be modified to
where y
=1, as given by mean field theory.
As first discussed in reference [7], there is another
way of characterizing the exponent y, which provides
an interesting pictorial insight and makes it more
accessible from a numerical point of view. Let us
consider the probability that, at a given temperature,
two SAW starting at the same point z and arriving respectively at points y, and Y2 do not intersect :
where Xt/J(Wl’ OJ2) is one when mi and W2 do not
intersect and zero otherwise. We suppose that Q. has
a scaling behaviour near the critical temperature independent at large distances on how yl and Y2 scale and introduce a related quantity Qp, which is the proba- bility that, at inverse temperature p, two SAW starting
at the same point never intersect. One can easily
convince oneself that :
On the other hand Qo is related to 0. by the relation :
which implies, under our assumptions, that 0,
behaves, in the critical region, exactly like QfJ. More-
over, as the space of SAW with fixed endpoints is considerably smaller than the whole space of SAW,
it is clear that the evaluation of 0, through a Monte-
Carlo algorithm, which samples the space of acces-
sible configurations, will be more accurate than that
of QfJ. So, in order to compute the exponents v and y,
we can restrict ourselves to consider only SAW with
fixed endpoints.
Another advantage of our formulation is that the introduction of a temperature forces the curves, in the disordered phase, to remain of finite length. We
are not at the critical point, where enormous fluc-
tuations will be present, but we study only its neigh-
bourhood. Eventually, the behaviour at the critical
point is obtained as the limit of a procedure of scaling transformations, the renormalization group, in which
one must connect quantities defined on lattices of different lattice spacings. A relation among them is then established through the renormalization condi-
tions, which assure that the physical content of the
lattice approximations is preserved. In our case we
shall demand that the physical correlation length be kept constant. This means that, if 0 is the parameter which changes the scales, b must acquire a 8-depen-
dence in such a way that :
As a matter of fact, at least in the scaling limit, we
can invert the functional relation between b and 0 :
so that measuring m at different temperatures will induce the knowledge of the function 0
=0(b), and
so of b
=b(0), which contains the information of how to approach the critical points
3. The Monte-Carlo procedure.
-Our problem
was to approximate physical quantities which, as
seen in the previous section, could all be obtained from :
Both expressions can be viewed as partition functions
on the spaces of all self-avoiding random walks on
the lattice, going from : (a) x to y ; (b) the origin to
any lattice point. The quantities of interest could all be expressed as averages on such spaces, with pro-
bability distribution :
Z stands for either sum appearing in (3 .1 ).
Monte-Carlo simulations are a standard technique
in sampling configuration spaces with a given pro-
bability distribution. Averages obtained via this sampling can be shown to converge to those in the
given probability, provided each configuration belongs
to a Markov chain, characterized by a transition probability W, satisfying :
where (0, co’ are self-avoiding walks and 5(.) is the
desired probability. The problem of finding a W that
fulfills (3.3) is, in general, solved by means of the
detailed balance condition. It suffices (although it is
not necessary) to impose :
In order to generate self-avoiding walks with a
probability distribution related to (3.2), we adapted
a technique introduced in reference [14]. First, let us
treat the case of walks with fixed endpoints : one
defines a set of elementary local deformations of the
walks, as shown schematically in figure 1. Then, at
every iteration, one makes a random choice of a step s of the walk and a random choice of e.1, one of the
2(d - 2) oriented directions orthogonal to that of s.
Choosing e.1 defines the displacement of s and this
allows us to determine which of the situations in
326
Fig. 1.
-Elementary deformations for the Monte-Carlo
procedure.
figure 1 will occur, by inspecting the nearest neigh-
bour steps. This local deformation takes us to a new curve co’
=m’(s, e 1.), whose length differs from that of w by ± 2.0 units. Let us denote this change by
Then, we go from a (self-avoiding) w to w’ by :
where the factor [L(w)] - ’ comes from the random choice of step and XSAW(CO) is defined equal to unity if m is self-avoiding, and zero otherwise.
The probabilities p(b), chosen independent of s
and e 1.’ were obtained from two requirements : a) That W should satisfy the o detailed balance » condition for a modified distribution
replacing with (T, W with W in (3.4) we derive,
b) That p(6) should minimize the probability, W(w --+> m), of null transitions.. To make this idea clearer, let us consider the full transition probability
for the Markov chain. Using (3.3a), we obtain :
Note that, given a link s and a direction e,, we have to examine the curve a) both locally (through inspec-
tion of nearest neighbour steps) and globally (due to
the self-avoiding character) to determine W(w --.. m’).
Let us introduce a set { co(s; co) I such that :
and rewrite (3.9) in the form :
The c’s were introduced so as to account for both the local and global dependence of (3.10) on the
curve to, thus making the sum over e 1. unconstrained.
In order to satisfy (3.11), for every step s the curly
bracket has to be equal to one. Thus, if we know the independent { c(s, e; co) }, we can find a number of
relations between { p(6) I and { co(s ; to) }. It is clear that :
where the inequality has been obtained by considering
the case in which the self-avoiding constraint does not act; i.e. Brownian motion. So we reduce the
problem to minimizing null transitions for that case, thus a local problem. The determination of p(b) can
then be accomplished by considering the four inde-
pendent nearest neighbour arrangements described in figure 2, which yield :
Fig. 2.
-Independent arrangements of a step s and its
nearest neighbors leading to (3.13).
since we must respect (3.8), we may use two other
equations, i.e. two of co’s, to determine p( + 2), p(O), p( - 2). Equations (3 .13a) and (3 .13b), with co(I) = co(II)
=0 are the ones, since they minimize W(m - (1)) (in fact, any other choice would lead to f3 > 1, which is unsatisfactory). One then finds :
The new curve w’ will be accepted if : (i) for a
random number, r E [0, 1], the Monte-Carlo test
is fulfilled; (ii) m’ satisfies the constraint of being self-avoiding.
B,
For walks with a free endpoint, as in the case of (3.1b), a few modifications are needed. The set of deformations has to be enlarged to include the addi- tion of a step to the walk. Furthermore, one should
select a point (rather than a step) to implement the deformation, which makes the denominator in (3.6) change to [m + 1]. Finally, an extra relation emerges from (3.11), (3.12) :
choosing co(V)
=0 and using (3.8) yields, besides (3.14), the formulae :
Although the ergodicity requirement, (3.3b), is
fulfilled a special difficulty arises in three dimensions.
There, transitions between walks with and without
« knots » are suppressed. Clearly, because our walks
are not closed, knots can be untied by passing through
the endpoints. For closed walks, the presence of knots (here the concept of knot is well defined) corresponds to the existence of distinct topological
sectors, each of them characterized by a particular
value of a topological invariant, i.e. the number of knots. In such a situation, the rules described above would not permit us to go from one sector to another.
As we work with open walks the problem, as such, does not exist. However, in practice, it is much less
probable to produce a walk with a « knot » from one
without. Once it is produced, the «knot» tends to
stay on for a long time since a rather special sequence of deformations is needed to untie it through the endpoints. The problem is made worse the farther
we take the endpoints. Obviously, in two dimensions
this difficulty does not exist.
The probability distribution to which we tend
asymptotically is (3. 7) rather than (3. 2) (due to the
random choice of step). Nevertheless, we can easily
relate averages in (3.7) to those in (3.2). Denoting
the former by {.]AVE and the latter by . >, an example
of such a relation is :
A few technical remarks on the practical advan- tages of the method are in order : i) since each walk is specified by the points of the lattice that it visits,
we saved considerable computer memory by packing
all coordinates of a given point into the same computer word. The number of points of the walk could, thus,
be as large as the number of available words, allowing
for extremely long walks; (ii) in four dimensions, if
we use 32-bit words, each coordinate can be as large
as 2’ (periodic boundary conditions were imposed).
Even as we went close to the critical point, this gua- ranteed that our walks never touched the boundary.
In three dimensions the situation is, of course, even more favorable; (iii) the characterization of each walk as a set of computer words allowed for a quick
check of the constraint of being self-avoiding ; (iv) by
virtue of (i), (ii) and (iii) our problem, from a practical point of view, reduced to the time it took for the
procedure to sweep reasonably configuration space, at a given value of fl. Typically, we needed tens of
millions of iterations to achieve the desired level of accuracy. In fact, we always made runs of ten or, at
most, twenty million iterations and stored the last walk to be used subsequently; (v) the computer time needed for ten million iterations was always between
15 and 22.5 min. of IBM-168. The rejection rate was roughly 86 % (of which no more than 1-2 % due to
the new curve not being self-avoiding). Thus, for any
given /3, no less than 1.5 million walks per run of ten million iterations would be used in the averaging;
(vi) in all our measurements we threw out the first
one million iterations.
4. The numerical data.
-Here we describe in detail which measurements were made and how they
compare with available theoretical predictions. The
first quantity of interest is the average number of steps for walks withfixed endpoints. It can be comput- ed by using (3.18) and it is related to the behaviour of the inverse correlation length m (in units of lattice
spacing) by :
If we now make use of (2.6)-(2.8), we obtain
328
Table I.
-Values of L((o) > vs. b, in d
=4. Column 1 shows the Monte-Carlo data; column 2, the best fit
with (4. 2a) ; column 3, the number of iterations used in the average. The values of the parameters are given below as well as the variance-covariance matrix.
Table II.
-Values of L(o)) > vs. b, in d
=3. Column 1
shows the Monte-Carlo data; column 2, the best fit
with (4. 2b) ; column 3, the number of iterations used in the average. The values of the parameters are given below as well as the variance-covariance matrix.
Table III.
-Values of L((o) > vs. b, in d
=4, for a
different choice of endpoints. The format is the same as
in table I.
The Monte-Carlo procedure for walks with fixed
endpoints outlined in section 3 allowed us to compute values of ( L(w) >o-+x for several values of p. They
are shown in tables I and II for d
=4, d
=3, res- pectively. We then used expressions (4.2) in least-
squares fits to the data, from which we determined three parameters in each case - (C4, bc, X) for d=4
and (C3, hc, v) for d
=3. As the percentual uncer- tainty over the values in tables I and II was maintained fixed (by suitably varying the number of iterations
with b), we used unit weights in all our fits. The
values thus obtained were :
The uncertainties quoted above correspond to square roots of the diagonal elements of the variance- covariance matrix produced by our least-squares fit.
Although the data shown corresponds to a particular
choice of endpoints, additional tests indicate that
our results are stable under changes of endpoints, provided they are sufficiently far apart; see, for example, table III. One word of caution : there is a
strong correlation among the three parameters. Thus
one is forced to have un’certainties of the order 1 %,
in d
=4, to be able to say something about the
presence of logarithmic deviations. The value of bc,
in d
=4, is smaller than that predicted by extrapo-
lating high-temperature series [15], bc
=1.912 8. If
Fig. 3.
-Plot of In L(w) vs. ln ib in d
=4. The full curve
is our best fit whereas the dotted one corresponds to a mean
field fit (no log corrections) with be
=1.904.
we fix be
=1.912 8, a two-parameter fit to the data yields even larger values for N. On the other hand, setting A’
=0 (mean field theory) yields a bc
=1.896 ± 0.002, much lower than any of the
values quoted in the literature, and a fit not as good
as (4.3a). This illustrates the correlation effect that
we just mentioned and supports the existence of log
violations to the scaling laws in four dimensions. For
comparison, the value of A’ predicted by perturbative
solutions of the renormalization group equations
is 1/8. Figure 3 summarizes the situation in four dimensions. As for d
=3, we did not push the method
too far. Our aim was not to make very precise mea-
Fig. 4.
-Plot of In L(w) vs. In Lb in d
=3. Only the best
fit is shown.
surements but, rather, indicate that with even less computer time than in four dimensions, one could
obtain a reasonable estimate of the critical exponent v.
Inspection of table II shows, in fact, much fewer
iterations than in d
=4 and correspondingly higher
uncertainties. Although the correlation effects men-
tioned before are also present here, dealing with
power laws is certainly an advantage over the logs
of dimension four. For comparison, v is predicted
to be either 3/5 (the Flory value [16]) or 0.59 ± 0.01 (the 8-expansion value [17]). Figure 4 summarizes
the situation in three dimensions.
Another quantity which can, in principle, be investigated is the average number of steps for walks with one free endpoint. It can be related to the sus-
ceptibility x by means of :
From the scaling behaviour of x (2 .11 ) one obtains :
The Monte-Carlo procedure for walks with one free endpoints and use of (3.18) allowed us to collect
the data of table IV. From (4. 5) one then could, in principle, obtain the value of 19 in d
=4. However, for d
=3 the formula analogous to (4. 5) is obtained if we set 9=0. This means that the exponent y is
« lost » in the overall constant in front. In d
=4 the situation is not much better because : a) the procedure
for generating walks with one free endpoint is much
slower since transitions which change the endpoint
are - L(w) -1 times less probable than ordinary
ones : b) the determination of 9 is quite inaccurate
since it appears in a correction term to a leading
power law. Thus it may mix with subleading correc-
tions to scaling. This is different from (4.4a) where
the JV-term multiplies the power.
In view of the preceding paragraph, it is clear that
a new ingredient must come into play to allow for a
determination of 9 and y. This is the quantity Q,
introduced in section 2, whose scaling behaviour is the same as that of Q, a quantity directly expressible
Table IV.
-Values of L(w) > vs. b for walks with
one free endpoint. d
=4.
....
330
Table V.
-Values of Q vs. b, in d
=4. Column 1
shows the Monte-Carlo data ; column 2, the best fit
with (4. 7a) ; column 3, the number of iterations used in the average. The values of the parameters corres- pond to be
=1.904. The variance-covariance matrix is
also shown.
via (2.13) in terms of the susceptibility x. Q can be computed by working with fixed endpoints. One simply generates two sets of walks that start at the Table VI.
-Values of Q vs. b, in d
=3. Column 1
shows the Monte-Carlo data; column 2, the best fit with (4 . 7b) ; column 3, the number of iterations used in the average. The values of the parameters correspond to be
=1.540. The variance-covariance matrix is also shown.
origin and end at x, and X2, respectively. Then, using the notation of (3.18) we have
where the prime denotes an average over walks that do not intersect. The data for Q in d
=4 and d
=3
are shown in tables V and VI, respectively. There we
chose x and x2 to be equidistant of the origin, which
is quite practical as it allows for two independent
measurements of the mean length (fixed end points)
for every measurement of Q (thus, the number of
iterations fort L(w) >o-+x is twice that of Q ). Further-
more, the dependence of Q on the endpoints can be
used to establish that this quantity does vanish in four dimensions, as was done in [7]. Fits to the data
used :
These formulae were obtained from the relation (2.13) between Q and x, by inserting a scaling form for dx-1/df3 suggested by the renormalization group
equations [9]. Using the same strategy as before we could determine :
The fits were made for two parameters
-(c4, till) in
d
=4 and (c3, y) in d
=3
-with be fixed at 1.904 (d
=4) and 1.540 (d
=3). For comparison, perturba-
tive solutions of renormalization group equations predict 9=1/4, whereas high-temperature series yield ;
=7/6. We also display, in figure 5, the fit for Q in d
=3.
Fig. 5.
-Plot of In ( 0 > vs. In ! b in d
=3. Only the best
fit is shown.
.
