books.google.de - This work dealing with percolation theory clustering, criticallity, diffusion, fractals and phase transitions takes a broad approach to the subject, covering basic theory and also specialized fields like disordered systems and renormalization groups....https://books.google.de/books/about/Introduction_To_Percolation_Theory.html?id=v66plleij5QC&utm_source=gb-gplus-shareIntroduction To Percolation Theory
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Dietrich Stauffer, Ammon Aharony. PERCOLATION THEORY Revised Second
Edition lis) Taylor & Francis Introduction to Percolation Theory Introduction to
Theory Percolation Revised Second. DIETRICH STAUFFER AND AMNON
AHARONY.
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Dietrich Stauffer, Ammon Aharony. Introduction to Percolation Theory Introduction
to Theory Percolation Revised Second Edition Dietrich Stauffer and.
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Dietrich Stauffer, Ammon Aharony. A.3 Computerized cluster counting 166
Further reading 178 B Dimension-Dependent Approximations 179 B.1 Upper
critical dimension 179 B.2 Flory approximation 180 B.3 !-Expansion 181 Further
reading ...
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Percolation theory is still the simplest context in which all these tools can be
introduced and explained. Another new element in this edition is the set of
exercises. Many of these were originally research problems, and their solutions
are hidden ...
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This book is an attempt to introduce the reader to a research field which is
already more than forty years old but which ... But in contrast to many other
modern research fronts, percolation theory is a problem which is, in principle,
easy to define.
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I am indebted to J.Kertész for information about his percolation seminar at
Munich Technical University, and his comments and those of D.W. Heermann,
H.J.Herrmann, A.Margolina, B.Mühlschlegel, R.B.Pandey, S.Redner, and M.
Sahimi on a ...
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CHAPTER. 1. Introduction: Forest. Fires,. Fractal. Oil. Fields,. and. Diffusion. 1.1.
WHAT IS PERCOLATION? Imagine a large array of squares as shown in Fig. 1(a)
. We imagine this array to be so large that any effects from its boundaries are ...
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Dietrich Stauffer, Ammon Aharony. Fig. 2. Example for percolation on a 60!50
square lattice, for various p as indicated. Occupied squares are shown as *,
empty squares are ignored. Near the threshold concentration 0·5928 the largest
cluster ...
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Dietrich Stauffer, Ammon Aharony. squares, and N is a very large number, then
pN of these squares are occupied, and the remaining (1!p)N of these squares are
empty. This case of random percolation is what we concentrate on here: Each ...
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Historically, percolation theory goes back to Flory and to Stockmayer who during
World War II used it to describe how small ... the start of percolation theory is
associated with a 1957 publication of Broadbent and Hammersley which
introduced ...
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Why is there a special value of p, which we call the percolation threshold pc ,
where the lifetime seems to diverge? For p near unity, each row can immediately
ignite the trees in Fig. 3. Average termination time for forest fires, as simulated on
a ...
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In the percolation case, if p increases smoothly from zero to unity, then we have
no percolating cluster for p<pc and (at least) one percolating cluster for p>pc .
Thus at p=p c , and only there, something peculiar happens: for the first time a
path ...
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This brings up many questions concerning dynamics on the percolation clusters,
that we shall discuss below. The simplest example, concerning diffusion, is briefly
introduced in the next section. The reader should be warned, however, that ...
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Dietrich Stauffer, Ammon Aharony. and j different, the scalar product can be +1 or
!1 with equal probability since we assumed that the motion is completely random.
Moreover, in half of the cases the scalar product is zero since di and dj are ...
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For regular lattices, and in the homogeneous regime describing the largest
percolation cluster on large length scales above pc ... COMING ATTRACTIONS
This introduction should have given you an impression of modern percolation
theory.
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Out of the 21 articles there, our introduction is related in particular to that of
Hammersley on the origins of percolation theory, of Jouhier et al. on gelation of
macromolecules, and of Mitescu and Roussenq on ant diffusion. For a more
recent ...
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Dietrich Stauffer, Ammon Aharony. Physica D., 38, 1–398 (1989). [Conference,
Mandelbrot's birthday]. PhysicaA., 191, 1–577 (1992). [Conference on Fractals
and Disordered Systems]. First percolation theory for polymer gelation Flory, P.J.,
...
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Therefore, different percolation lattices will contain clusters of different sizes and
shapes. In order to discuss their ... Series expansions and Monte Carlo
simulations are used to check these theoretical predictions. 2.1. THE TRUTH
ABOUT ...
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Dietrich Stauffer, Ammon Aharony. Fig. 6. Definition of triangular, honeycomb and
cubic lattices. For the triangular lattice, every intersection of the lines in (a) is a
lattice site; for the honeycomb lattice, the centres of the triangles in (a) form the ...
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It took about two decades from the first numerical estimates in 1960 for square
bond percolation, over non-rigorous arguments that ... The purpose of the present
book is to offer an introduction to percolation theory, not a comprehensive review.
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Dietrich Stauffer, Ammon Aharony. This normalized cluster number is crucial for
many of our later discussions in two or three dimensions. It equals the probability,
in an infinite chain, of an arbitrary site being the left hand end of the cluster.
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(5) The mean cluster size diverges if we approach the percolation threshold. We
will obtain similar results later in more than one dimension. This divergence is
very plausible, for if there is an infinite cluster present above the percolation ...
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Have we thus solved the percolation cluster problem exactly in d dimensions,
leaving the evaluations of mean cluster size and correlation length as an
exercise analogous to the one-dimensional case? Unfortunately, our
straightforward, exact, ...
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They are listed here without proof since according to our present knowledge
these animals, even if domesticated by exact solutions, do not help us in an exact
solution for percolation clusters at the threshold. The perimeter t, averaged over
all ...
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Percolation theory started with the exact solution on this somewhat artificial
structure. in d dimensions. We see that in the limit d¡# the surface becomes
proportional to the volume; this is also true if we look at squares, cubes, '
hypercubes' etc.
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When we now talk about percolation in the Bethe lattice, we therefore always
have in mind the behaviour in the interior of the Bethe lattice, and not the effects
due to the surface, which are also important. Let us now find the percolation ...
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Figure 11 displays this result, which goes back to Flory (1941), for it is in polymer
chemistry that the first percolation theory was developed by studying bond
percolation on this Bethe lattice. (As we saw above, for this special case the
difference ...
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The total cluster size is zero if the origin is empty and (1+3T) if the origin is
occupied; therefore the mean size is (17) We have thus derived exact formulae
for the mean cluster size S below the percolation threshold (Eq. (17)) and the
strength P ...
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Critical phenomena also occur for thermal phase transitions; the Bethe lattice
approximation for percolation theory then is somewhat analogous to the
molecular field approximation for magnetism, or the van der Waals equation for
fluids.
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The calculation which follows now uses tricks which occur again and again in the
scaling theory of percolation clusters. We assume p to be only slightly smaller
than pc: (21) where we used the above result that c vanishes quadratically in p(
pc ...
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Moreover, near the 2 percolation threshold we no longer require c to vanish as (p
(p c) but instead allow a more general power law: (26) Here * is another free
exponent, not necessarily equal to 1/2 as in the Bethe lattice solution given by Eq
.
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Instead a critical exponent # is introduced which describes how the strength of
the infinite network goes to zero at the percolation threshold. Secondly, let us
calculate how the mean cluster size S diverges at the threshold. As for Eq. (21),
we ...
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Dietrich Stauffer, Ammon Aharony. problem already in the evaluation of the first
moment; to get the strength P of the infinite network we had to subtract from the
sum its value at p=p c , and then replace the sum by Some caution is necessary if
a ...
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These relationships, known as scaling laws, have been used since the 1960s for
thermal phase transitions and in the 1970s were extended to percolation theory.
By going through the above formalism the reader will have a better feeling for ...
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Nature made it simple for us: f(z) has only one maximum, and not many, for usual
percolation problems. We call that value of f(z) at this maximum fmax, and the
negative value of z at this maximum is called z . Thus, max (35a) For a fixed
cluster ...
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Dietrich Stauffer, Ammon Aharony. polynomials and plotting the ratio versus
Equation (33) then asserts that for different s the results all lie on the same curve f
=f(z). (Some people call this effect 'data collapsing'.) Of course, in reality they do
not ...
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For percolation, usually these series expansions give the most accurate
estimates for the exponents. The determination of the threshold is often less
accurate than that by the Monte Carlo method described in Chapter 4.
Sometimes the best ...
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Dietrich Stauffer, Ammon Aharony. 2 2 same counter if and only if the resulting x
+y is smaller than unity. After, say, one million such pairs of random numbers
have been used, our counter will be at about 106'/ 4. The pair (x, y) gives a point
...
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Instead of giving high-quality data again as Having tested the validity of scaling
theory right at the percolation threshold, we now reviewed, for example, in
Stauffer (1979) (see Further Reading, Chapter 1), we now take poor data which
the ...
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Dietrich Stauffer, Ammon Aharony. Our computer output in Appendix A also gives
the size 'INF' of the largest cluster and the second moment 'CHI' of the cluster size
distribution Table 2. Percolation exponents for d= 2, 3, 4, 5, 6—e and in the ...
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There are also interesting effects in the cluster numbers far away from the
percolation threshold. It may be possible that some of their aspects are then
approximated reasonably well by the Bethe lattice solution, or by effective
medium theories.
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Again this latter result is not rigorously proven for all p above pc but widely
believed to be valid in that whole range for simple percolation provided .
Numerical data, for example from the exact cluster numbers at intermediate s,
confirm that Eq.
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Dietrich Stauffer, Ammon Aharony. (concentration p) percolate through the lattice
but also the empty sites (con-centration (1!p)). Nearly every occupied site is part
of the infinite network of occupied sites, and nearly every empty site belongs to ...
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1/2, analogous to numerous mean-field theories for thermal phase transitions.
See also the discussion on cluster structure at high dimensions, towards the end
of Section 5.3. We have seen in Chapter 2 that many quantities diverge at the ...
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Thus the finite clusters at the percolation threshold are fractals in the sense that
their fractal dimension D is smaller than their ... Percolation theory supports equal
rights for clusters above and below pc; if above p c they have the freedom to ...
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Dietrich Stauffer, Ammon Aharony. As we have mentioned several times, many of
the results quoted as power laws are only asymptotic, i.e. they are valid only for
very small (p(p c) or for very large s. This also applies to Eq. (48), which should ...
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In later chapters we relate this scaling behaviour to the role of " as the only
relevant length, and we explain it using the modern theory of the renormalization
group. As we saw, Eq. (52) contains three different asymptotic fractal dimensions,
...
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Dietrich Stauffer, Ammon Aharony. to the mean cluster size and similar properties
. From Eqs. (48) and (49), the size of these clusters is This is exactly the cluster
size that dominated the moments of the mass distribution, see Eq. (31). In fact ...
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Dietrich Stauffer, Ammon Aharony. and the rest do not. Thus the answer to
whether an infinite cluster is present is simply a 'perhaps'. Instead we may look at
the largest cluster in the finite system (without periodic boundary conditions).
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Monte Carlo data for the size of the largest cluster at the site percolation
threshold p=p c =1/2 of the triangular lattice, as a function of the linear dimension
L of the lattice. The slope of this log-log plot for large sizes gives the fractal
dimension .
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Dietrich Stauffer, Ammon Aharony. At p(p c =0·035, the double-logarithmic plot
shows a constant slope of !0·1 for L<10, implying that in this range and . For L>10
, the curve has zero slope, indicating a constant density, i.e. a homogeneous ...
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Ignoring this history, we will concentrate on real-space renormalization of
percolation (sometimes also called position space renormalization). This seems
the simplest way to introduce renormalization ideas into percolation theory (
Reynolds et ...
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Dietrich Stauffer, Ammon Aharony. line or plane with the bottom line or plane.) In
an infinite system, we have (=1 above and (=0 below pc . The quantity d(/dp gives
the probability (divided by the small interval dp), that the lattice starts to ...
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If we also define an effective percolation threshold for finite lattices as that point
where the curve P(p) has an inflexion point (maximum of dP/dp), this L-
dependent threshold approaches the true p c as L !1/& , as Eq. (56) tells us
immediately.
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The similarity idea as a foundation of thermal critical phenomena and scaling
goes back to the 1960s (see the review of Kadanoff et al., 1967) and leads to
Wilson's first renormalization theory. In real-space renormalization, we replace a
cell of ...
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Dietrich Stauffer, Ammon Aharony. being occupied if every original site is
occupied with probability p. The super-site is occupied if a spanning cluster exists
. In our triangle this is the case if either all three sites are occupied (probability p3
), or if ...
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Dietrich Stauffer, Ammon Aharony. The two edges AD and CF will thus be
connected with probability (64) where the different terms correspond to
connecting configurations with 5, 4, 3 and 2 bonds. It is easy to check that Thus
and . Again, the ...
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It shows a simulation of site percolation on the triangular lattice at p c =1/2. The
sites on the largest cluster, which connect between the boundaries of the finite
sample, are emphasized by showing the bonds which connect them. Figure 22(b)
...
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Similar ideas were introduced into the theory of critical phenomena in the mid-
1960s by Kadanoff, and led Wilson to the introduction of the detailed
renormalization group. The details of p'(p) are, however, necessary for obtaining
amplitudes, ...
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Dietrich Stauffer, Ammon Aharony. becomes steeper as b increases, this
intersection p* approches pc , and Eq. (57) implies that p*(pc=const·b!1/&. The
slope *=d(/dp at p* diverges as b1/& (see Sec. 4.2), and thus the difference may
have a ...
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Dietrich Stauffer, Ammon Aharony. How can we in practice determine the
asymptotic exponent 1/, from our widths &(b) for finite cell sizes b? We plot the
ratio log (1/&)/log b versus 1/log b and look for a smooth curve fitting these data
and ...
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Dietrich Stauffer, Ammon Aharony. We denote the conductivity exponent by $: (72
) for p¡pc. Often this exponent is also called t but here we need t as the symbol for
time (and earlier we used it for the perimeter.) Perhaps we should leave it as ...
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Dietrich Stauffer, Ammon Aharony. calculate only conductivities of such strips, but
with Cray-computer speed. In two dimensions, its result #/.=0·9745±0·0015 was
found by extrapolating conductivities in strips of width L and confirmed with ...
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Dietrich Stauffer, Ammon Aharony. All the above arguments led to the 'links-
nodes-blobs' picture (Stanley, 1977). This picture is summarized schematically in
Fig. 28: nodes at distance " are connected by generalized links, which contain
both ...
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Dietrich Stauffer, Ammon Aharony. (79) with in d=2,3. We shall describe below
several physical applications of the minimal path. One obvious application
concerns the forest fires mentioned in Chapter 1: if it takes a unit time to transfer
the fire ...
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Dietrich Stauffer, Ammon Aharony. and y(")=D SC . When q¡0, every term in the
sum which arises from Ib&0 will contribute 1; hence (neglecting the few non-
current-carrying bonds on the backbone) and y(0)=DB. Since (Ib/I)2q is a ...
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Dietrich Stauffer, Ammon Aharony. Fig. 29. (a) Initial stages of the build-up of the
Sierpinski carpet. Empty squares are shadowed. At each step of the iteration, the
linear dimension L is enlarged by a factor 3 and the mass by a factor 8, since ...
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Dietrich Stauffer, Ammon Aharony. widths. However, when such a bond occurs
inside a blob, there almost always exist other bonds in parallel to it, with much
larger conductances. Therefore, the total resistance is dominated by the
narrowest ...
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Dietrich Stauffer, Ammon Aharony. FURTHER READING First experiments on
random resistor networks Last, B.J. and Thouless, D.J., Phys. Rev. Lett., 27, 1719
(1971). Monte Carlo measurements of conductivity Derrida, B., Zabolitzky, J.G., ...
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Dietrich Stauffer, Ammon Aharony. function r(z) for z(" thus must vary as in order
that t cancels out: Equation (105) now requires this exponent k/x to equal .
Equating these two expressions for k we get or (108a) (108b) from which follows.
Page 128
The diffusivity in the whole lattice and the diffusivity in the infinite network or very
large cluster are thus related by (110) an equality not restricted to the region very
close to the percolation threshold. Now we see that our characteristic time ...
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Dietrich Stauffer, Ammon Aharony. law for the strength of the infinite cluster (
exponent #) and the mean cluster size (exponent $) (Eq. (53)) now reads in d
dimensions. For d=2, the numerically determined exponents are very close to .
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Dietrich Stauffer, Ammon Aharony. This value of applies when we start with a
given specific realization of the cluster, and we consider all the random walks
which start at a fixed origin on that cluster. Such walks, and their corresponding ...
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Dietrich Stauffer, Ammon Aharony. waves, which are spread over the whole
sample, a perturbation at the origin turns out to decay with distance, i.e. to remain
localized. Such excitations are called fractons. If the perturbation is periodic, with
...
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Dietrich Stauffer, Ammon Aharony. In this section we limit ourselves to the
number of sites on the accessible external perimeter in two dimensions. As we
shall see, there exist several ways to identify such a perimeter. We start with the
hull, ...
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Dietrich Stauffer, Ammon Aharony. If the size of the diffusing particle is larger than
2a, then it will also not be able to enter into the gate between h and j in Fig. 33(a).
This will then also exclude sites 20 and i from the external perimeter. It is thus ...
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The percolation clusters discussed so far represented the randomness of the
medium, e.g. the pore structure of a given rock. In the last two sections of this
chapter and in the next chapter we describe dynamic algorithms, under which the
...
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Dietrich Stauffer, Ammon Aharony. (130) Indeed, Sapoval et al. confirmed this
relation numerically, with . They also noted that the above procedure, of finding
the average location of the hull, is a very accurate method to determine p c in two
...
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Dietrich Stauffer, Ammon Aharony. Probability distributions, superlocalization, etc
. Aharony, A. and Harris, A.B., Physica A, 163, 38 (1990). Bunde, A., Havlin, S.
and Roman, H.E., Phys. Rev. A, 42, 6274 (1990). Fractons Alexander, S. and ...
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Criminals are said always to return to the site of their crime, and thus we now
return to one of the original motivations for percolation research, the droplet
description for thermal critical phenomena. We summarize what percolation
theory for ...
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Dietrich Stauffer, Ammon Aharony. each other. A pair of spins then has the '
exchange' energy (J, if it is parallel, and +J if it is antiparallel. The total energy E
in a field H is then, with spins Si=±1: Here the first sum runs only over nearest ...
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Thus, if at low temperatures the concentration p of spins is varied to approach the
percolation threshold, then with the percolation exponents #, $ and not the
undiluted Ising exponents. Thus, in this case we find an exact correspondence of
the ...
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Dietrich Stauffer, Ammon Aharony. Also in a vapour which is undersaturated, i.e.
which is in complete and not in metastable equilibrium, tiny droplets can form and
decay again. If no such droplets occur at all, then we have the classical ideal ...
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DROPLET DEFINITION FOR ISING MODEL IN ZERO FIELD For clarity we now
describe as 'clusters' the groups of neighbouring occupied sites familiar from
percolation theory, and as 'droplets' the properly defined elementary excitations ...
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Dietrich Stauffer, Ammon Aharony. comes, from the Kertész line which we had
hoped to have buried for good, but which is resurfacing again. Along this line the
droplet numbers ns no longer become critical, due There is also a new and ...
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Figure 38 shows schematically this 'phase diagram'. Only the ferromagnetic
region has a spontaneous magnetization. While for T 0 we find, as a function of p(
pc , percolation exponents for m 0 and 1, at p=1 we find, as a function of T c (T,
Ising ...
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Dietrich Stauffer, Ammon Aharony. magnetic moments are three-component
vectors, Si, which are free to rotate in space. The Magnetic systems are often
described by the Heisenberg model, in which the local 'exchange' energy of two ...
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Our first aim was to show that percolation is an active field of research. ... In short,
theoretical physics in general and percolation theory in particular is a human
enterprise and not the fixed body of knowledge which it often appears to be when
...