# EPR-Bell realism as a part of logic

I. Schmelzer

###### Abstract

We argue that for the proof of Bell’s theorem no assumptions about realism or
free will are necessary. The key formula

$E(AB|a,b)=\int A(a,b,\lambda)B(a,b,\lambda)\rho(\lambda)d\lambda$

follows from the logic of plausible reasoning (the objective Bayesian
interpretation of probability theory) taken alone, without any further
assumptions about realism. The space $\Lambda$, usually interpreted as some
space of “hidden variables”, can be constructed for an arbitrary “field of
discourse” using Stone’s theorem. The rejection of superdeterminism follows
from logical independence – the non-existence of information which suggest a
dependence – of the free decisions of the experimenters from everything else.

To prove the Bell inequality it is, then, sufficient to reduce this to

$E(AB|a,b)=\int A(a,\lambda)B(b,\lambda)\rho(\lambda)d\lambda.$

This follows for space-like separated measurements from Einstein causality
alone. So, the consequence of the violation of Bell’s inequality is that
Einstein causality has been empirically falsified, without any loopholes left.

We consider and reject the idea of weakening the logic of plausible reasoning
could be used to create a loophole. Finally, we discuss what can be used to
replace Einstein causality. While weak (signal) Einstein causality holds, it
follows either from a notion of strong causality or from human inability to
prepare states which violate quantum uncertainty, which makes it uninteresting
for fundamental considerations. A notion of strong causality has either
unacceptable causal loops or requires a hidden preferred frame. Given that a
hidden preferred frame is completely compatible with modern physics, it is
argued that this choice is preferable.

##  I Introduction

While there is a lot of discussion about the assumptions necessary to prove
the Bell inequalities, there is a wide agreement that one needs, beyond
Einstein causality (usually named “locality”111We will not use “locality”
because it is misleading. One can easily imagine theories with a preferred
frame which are local in the common sense meaning, but with a maximal speed of
information transfer greater than the speed of light. Despite being local in
any reasonable sense, they would have to be named “nonlocal” if “local” is
used for Einstein causality.), some other non-trivial physical assumptions.
Variants of proofs of Bell’s inequality may use different assumptions, in
particular “realism” (however specified), “counter-factual definiteness”, or
Reichenbach’s principle of common cause Wiseman . Moreover, one also has to
exclude superdeterminism.

The main thesis of this paper is that there is no need for any physical
assumptions beyond Einstein causality. All what is necessary beyond Einstein
causality is, in fact, already part of logic – of the logic of plausible
reasoning, usually named “objective Bayesian interpretation” of the rules of
probability theory, as defined by Cox Cox  and Jaynes Jaynes . The
straightforward consequence is that, once the violation of Bell’s inequalities
for space-like separated measurements is accepted, Einstein causality has to
be given up, as falsified by these experiments.

Beyond this main result, we discuss possible loopholes and consequences.

There is no way to avoid this conclusion beyond the questioning of logic, even
if not of classical logic, but only in the less well-studied form of the logic
of plausible reasoning. Given that defenders of Einstein causality have been
ready to reject even such fundamental things as realism (as defined by the EPR
criterion of reality) as well as classical causality (in any version
containing Reichenbach’s principle of common cause), one can expect that at
least some physicists will consider even the rejection of logic as a
reasonable alternative. Once there can be, even in principle, no logical
arguments against the rejection of logic, we cannot prove they are wrong, so
that we simply have to accept this possibility as a fact of life. Nonetheless,
we give a short overview of previous attempts to question logic, because, even
if they have failed to question logic itself, they were not completely
worthless but have reached something, namely a better understanding of logic.
Our suggestion is that questioning the logic of plausible reasoning has a
chance to reach similar results too – but that would not change the main
result of this paper. It would leave the mathematics of the proof, and
therefore the necessity to reject Einstein causality, unchanged. What one can
reasonably hope for is a reinterpretation of these mathematics in common
language, avoiding misinterpretations caused by the vagueness of human
language, which we, as human beings, use to describe even precise mathematical
results.

For those who do not want to question the logic of plausible reasoning, the
rejection of Einstein causality is unavoidable. What follows from this? The
straightforward alternative would be the revival of a preferred frame, which
would allow hidden quantum causal information transfer from past into future
in some hidden absolute time coordinate. While this would be in contradiction
with the standard interpretation of SR as well as GR, it would not be in
conflict with observation. Indeed, in SR the Lorentz ether is simply an
interpretation, even the original one. Moreover, viable alternatives with a
preferred frame have been proposed for modern physics too, as for gravity gle
, as for the standard model of particle physics clm . Thus, to accept a hidden
preferred frame is at least not as problematic as usually assumed.

Another alternative would be weakening causality, so that one can nonetheless
preserve Einstein causality, even if only in a weaker sense. Here, we have a
straightforward candidate worth to be discussed, namely weak or signal
causality. Weak or signal Einstein causality could be violated only with a
real transfer of information faster than light. Given the theorem that such
information transfer is not possible using violations of the Bell
inequalities, it seems reasonable to hope that this weaker notion could be
preserved. And we, indeed, will not question this theorem, so, will not
question signal Einstein causality. In this sense, it remains a way to
preserve Einstein causality.

Nonetheless, signal causality has no chance to have any fundamental relevance.
Given the formulas which follow from the logic alone, there appears to be only
one explanation why signal causality appears to be weaker than strong
causality, namely that we are unable to prepare sufficiently localized states.
If we would be able to prepare states with all really existing parameters
fixed, we could send signals, but all what we can prepare are averages,
quantum equilibrium states, and in these states all the information we would
like to send is averaged out. So, it is the most trivial, straightforward
excuse for our inability to send faster than light signals which appears to be
sufficient.

##  II What we need to prove Bell’s inequality

In the large amount of discussion following Bell’s theorem, there has been no
doubt (beyond the fringes) that Bell’s inequality follows from eq. (2) of
Bell’s paper, which can be rewritten as the formula for the expectation value
of the product $AB$ of the measurement results:

$E(AB|a,b)=\int A(a,\lambda)B(b,\lambda)\rho(\lambda)d\lambda.$ (1)

The part where Einstein causality is used can be also easily identified. If we
can derive the formula

$E(AB|a,b)=\int_{\lambda\in\Lambda}A(a,b,\lambda)B(a,b,\lambda)\rho(\lambda)d\lambda,$
(2)

then we can derive (1) using Einstein causality, because Einstein causality
tells us that the measurement result of Alice $A(a,b,\lambda)$ cannot depend
on the decision $b$ of Bob what to measure, resp. for the measurement result
of Bob $B(a,b,\lambda)$ depending on the decision $a$ of Alice. So, the
question what we need beyond Einstein causality to derive Bell’s inequality
reduces to the question what we need to derive (2). Now, (2) is nothing but an
application of the more general formula

$E(o|a)=\int o(a,\lambda)\rho(\lambda)d\lambda,$ (3)

where $a\in A$ denotes all decisions of experimenters, and $o(a,\lambda)$
denotes some observable which is measured. What do we need to derive this
formula?

The straightforward way to derive it, which we will use here, is to obtain a
general formula for the expectation values in Kolmogorovian probability
theory, namely

$E(f)=\int_{\omega\in\Omega}f(\omega)\rho(\omega)d\omega.$ (4)

Then we apply this formula to the general experimental situation where we
distinguish between the parameters $a\in A$ which can be freely chosen by the
experimenters, and all the other parameters $\lambda\in\Lambda$, so that
instead of a general $\omega$ we consider now pairs $\omega=(a,\lambda)\in
A\times\Lambda$. This would give

$E(f)=\int f(a,\lambda)\rho(a,\lambda)dad\lambda.$ (5)

What remains is to get rid of the dependence of $\rho(a,\lambda)$ on $a$. That
means, the free decisions of the experimenters should be independent of
everything what defines the state of the system which is measured. This would
give

$E(f)=\int f(a,\lambda)\rho(a)\rho(\lambda)dad\lambda$ (6)

and as a consequence (3). This step is the rejection of superdeterminism.

So, what we have to do is to obtain formula (4) of Kolmogorovian probability
theory, and to justify the rejection of superdeterminism.

##  III Kolmogorovian probability theory as the logic of plausible reasoning

To justify the application of formulas of probability theory based on logic
alone, we have to switch from the frequency interpretation of probability to
the Bayesian interpretation of probability in its objective variant, as
proposed by Cox Cox  and Jaynes Jaynes . If we interpret (4) as a formula of
the frequency interpretation, we have no chance to derive it out of nothing
but logic. Indeed, in the frequency interpretation any statements about
probabilities are physical hypotheses about some frequencies, which, as any
other empirical hypotheses, may be falsified by observation. Instead, in the
objective Bayesian interpretation, the probabilities are degrees of
plausibility of a statement given all the available information. These degrees
of plausibility have to follow logical rules of consistency. It appears that
these rules of consistency, combined with a few simple conventions, are
equivalent to the rules of probability theory.

Unfortunately, the rules derived by Cox Cox  and Jaynes Jaynes  for the logic
of plausible reasoning do not cover all formulas of Kolmogorovian probability
theory. What Jaynes Jaynes  writes about differences between Kolmogorovian
probability theory and the logic of plausible reasoning is not encouraging:

> Our system of probability, however, differs conceptually from that of
> Kolmogorov in that we do not interpret propositions in terms of sets, but we
> do interpret probability distributions as carriers of incomplete
> information. Jaynes  p.50
>
> The Kolmogorov system of probability (henceforth denoted by KSP) is a game
> played on a sample space $\Omega$ of elementary propositions $\omega_{i}$
> (or ‘events’; it does not matter what we call them at this level of
> abstraction). …For all practical purposes, then, our system will agree with
> KSP if we are applying it in the set-theory context. But in more general
> applications, although we have a field of discourse F and probability
> measure P on F with the same properties, we do not need, and do not always
> have, any set $\Omega$ of elementary propositions into which the elements of
> F can be resolved. Jaynes  p.651ff

Fortunately, it appears that the set $\Omega$ can be constructed, always.
Let’s see. What is considered in the logic of plausible reasoning is a “field
of discourse” $\mathcal{F}$, which is the set of all propositions considered.
It is an abstract Boolean algebra, that means, we can apply the Boolean
operations $\land,\lor,\lnot$ and they follow the rules of classical logic.
Instead, Kolmogorovian probability considers only a special realization of a
Boolean algebra, namely of the algebra of subsets of a set $\Omega$, with the
set-theoretic operations $\cap,\cup,\bar{\quad}$ of intersection, union and
complement.

This leads to the natural question if there are abstract Boolean algebras
which cannot be represented by the algebra of subsets of some space. The
answer is given by Stone’s representation theorem for Boolean algebras Stone .
It tells that every Boolean algebra is isomorphic to an algebra of subsets of
some set $\Omega$, its Stone space. The Stone space of a Boolean algebra is
the set of homomorphisms of the algebra to the two-element Boolean algebra
$\\{0,1\\}$. That means, each $\omega\in\Omega$ defines for each
$A\in\mathcal{F}$ a truth value $A(\omega)\in\\{0,1\\}$. Every proposition
$A\in\mathcal{F}$ is the equivalent to the subset $A\subset\Omega$ of
propositions which are true for $A$, by $\\{\omega\in\Omega|A(\omega)=1\\}$.
These subsets can be used to define a topology as well as the corresponding
Borel algebra, so that every $A$ appears to be measurable.

Let’s note here that the Stone space is a completely natural thing from point
of view of plausible reasoning. An element of the Stone space
$\omega\in\Omega$ fixes a truth value for every proposition, and, as a
homomorphisms, does this in a consistent way, so that if $A$ and $B$ are
assumed to be true, then $A\land B$ also has to be true. So, it is simply what
one could describe as a consistent, conceivable, imaginable possibility,
eventuality, story or scenario. So, the space $\Omega$ is a quite natural
object, constructed in a straightforward way out of what is open and well-
known, the set of consistent scenarios.

This changes the interpretation of the space $\Lambda$, as we know it from
usual discussions of Bell’s theorem, where it considered to be a space of
“hidden parameters”. The space $\Omega\cong A\times\Lambda$ is, instead,
constructed out of elements which are completely well-known – the propositions
of the field of discourse $\mathfrak{F}$ which is considered. Ok, the
particular truth values fixed by the particular $\omega\in\Omega$ may remain
hidden from observation, so in this sense the $\omega$ contains some hidden
parameters. But the meaning of the particular parameter is well-defined and
part of what is considered, and it is only the particular value which remains
hidden. This completely contradicts the usual perceptions about hidden
variables: The construction of the space of “hidden variables” is a trivial,
straightforward operation, given the set of statements under consideration.

Moreover, it also changes the interpretation of the role of random variables –
in particular of all observables – being well-defined functions of $\omega$.
One can naively think that behind this is hidden an assumption about
determinism. It is not – a random variable $f$ may be as fundamentally,
inherently random as one likes, the proposition “$f$ has the value $f_{0}$”
will be a valid proposition of our field of discourse, and therefore it is,
for every $\omega\in\Omega$, either true or false, and in any case well-
defined and fixed for this particular $\omega$.

##  IV Why logical independence is sufficient to ignore superdeterminism

An important difference between the frequency interpretation of probability
and the logic of plausible reasoning is the very different notion of
independence. In the frequency interpretation, independence is a very
nontrivial physical hypothesis – the null hypothesis. It may appear false –
falsified by observation, which shows a correlation. The situation appears
completely different in plausible reasoning. Here, the question is a
completely different one – what would be rational to think, given the
available information? In a situation where we have no information which
indicates any connection, it would be irrational to assume there exists one.
Will $A$ increase or decrease the probability of $B$? Without any information,
we have no reason to prefer one of the two answers. The only rational answer
is the one which makes no difference – that $A$ is irrelevant for $B$, that
both are independent of each other, $P(A\land B)=P(A)P(B)$. This approach, of
course, does not prevent us from errors – we may not have had the information
required to recognize their correlation. But this is not a problem for
plausible reasoning: Once the statistical data which suggest a dependence are
known, they will be applied to modify the probabilities, in a process known as
Bayesian updating. And despite the fact that the initial conclusion about the
independence appeared to be wrong, this does not change the point that it was
the only reasonable one, given the information which was available at this
time.

This difference becomes essential for the consideration of superdeterminism –
the question if the general probability distribution
$\rho(\omega)=\rho(a,\lambda)$ of (5) splits into $\rho(a)\rho(\lambda)$ of
(6). This is nothing but the usual situation – we have the decisions of the
experimenters, and in general, we have no information which would suggest that
it is somehow correlated with anything relevant for the preparation or the
execution of the experiment. And this lack of information is already
sufficient, it is all we need in plausible reasoning to switch from (5) to
(6).

Let’s note that this does not mean that superdeterminism is wrong. What we
consider here is logic – and nothing nontrivial about reality can follow from
pure logic taken alone. And superdeterminism is, even if it is a completely
unreasonable big conspiracy theory, a physical hypothesis. But we can ignore
superdeterminism. Once we have switched to the logic of plausible reasoning,
(6) no longer means that the physical hypothesis of superdeterminism has been
somehow falsified, but that, given the information we actually have, it would
be irrational to assume that superdeterminism is true.

That means, logical independence allows us to obtain formula (6) out of of (5)
without the necessity to reject superdeterminism – we can simply ignore it.
Even if superdeterminism would be true, it would remain rational for us to use
(6) instead of (5).

This completes the first part of the paper, and gives the main result: Formula
(3) appears to be a formula of the logic of plausible reasoning. It does not
require any nontrivial physical hypothesis to justify this formula – pure
logic is sufficient.

##  V Questioning the logic of plausible reasoning

The most direct, straightforward possibility to preserve Einstein causality
would be to question the logic of plausible reasoning. A rejection of logic
sounds, of course, problematic. But that does not mean that there is no
chance. First of all, one does not have to reject logic completely, it could
be sufficient to weaken it. Moreover, there have been attempts, and, even if
they were not really successful, they were also not completely unreasonable.
Of course, a weakening of logic cannot be rejected in general, because we
could not use logic to reject it. All what one can reject would be particular
proposals to weaken logic, so that one could base a rejection on those parts
of logic which remain untouched. Nonetheless, to look at the results of
previous attempts to criticize logic appears interesting, and allows us to
estimate what possible future attempts can give.

###  V.1 Relevance logic

One such criticism against classical logic was “relevance logic” relevance-
logic . Many philosophers, in particular Hugh MacColl MacColl , have claimed
that some formulas of classical logic, like $p\to(q\to p)$, or $\lnot
p\to(p\to q)$, or $(p\to q)\lor(q\to r)$ are counterintuitive if we interpret
$\to$ as representing the concept of implication that we have before we learn
classical logic. Relevance logicians claim that what is unsettling about these
so-called paradoxes is that in each of them the antecedent seems irrelevant to
the consequent. This has failed, not because relevance logic has failed to
make proposals how to define implication more close to the common sense
meaning of the word, but because for all purposes where implication has been
used in logic before, it was sufficient to use the $p\to q$ as simply a sign
for $\lnot(p\land\lnot q)$ without some relevance beyond this. So, what has
changed was not classical logic, but the interpretation of the meaning of the
expression $p\to q$, removing “relevance” from its meaning. Given that a
relevant implication $p\to_{rel}q$ (however defined) also proves $p\to q$,
logic was not even weakened by this reinterpretation.

###  V.2 Intuitionism

Another classical criticism of logic were various variants of “constructive
mathematics” constructive-mathematics , in particular “intuitionism”. They
rejected existence proofs of classical mathematics which have not given
explicit constructions of the objects in question, and proposed different
rules of logic, especially for the meaning of “exists”, so that “there exists”
becomes close to “we can construct”. This criticism was solved by the Gödel-
Gentzen translation Goedel ; Gentzen , which in particular translated a
classical “$(\exists x)A(x)$” into “$\lnot((\forall x)\lnot A(x))$”. As the
result of this translation, every proof valid in classical logic became a
valid proof in intuitionist logic too. So, the answer was quite similar: the
mathematical apparatus of logic remained unchanged, and the criticism has only
succeeded to modify the interpretation of “exists”, distinguishing it from “we
can construct”. Similarly to the case above, classical logic was not weakened
too – an explicit construction can be used to prove existence too.

So, the history of criticism of classical logic shows a clear pattern: The
mathematical apparatus of logic remained unchanged. The only thing which has
changed was the interpretation of the meaning of some mathematical
expressions, away from the stronger common sense meaning toward a weaker
interpretation which corresponds to the formulas of logic. While new
mathematical constructions, which covered the common sense meaning in a more
accurate way, have been proposed, they have not given much, and are now
essentially abandoned. What has survived was the mathematical concept as used
in pure logic, and the modified, weaker interpretation of the logical rule.

###  V.3 Frequentism vs. logic of plausible reasoning

The difference between the frequentist interpretation and the Bayesian
interpretation of probability theory – the logic of plausible reasoning – can
be understood along similar lines. Jaynes Jaynes  sees the Bayesian
interpretation in a classical tradition going back to Laplace, so that
frequentism can be seen as a criticism of this tradition too. The key
advantage of the logic of plausible reasoning was that it can be applied to a
much wider range – namely, it can be used to assign degrees of plausibility to
theories and hypotheses, which cannot be interpreted as frequencies. On the
other hand, similarly to the cases above, where the frequency interpretation
is applicable, the resulting probabilities are the same.

###  V.4 Why this makes sense

In all three cases, the result can be understood as being quite rational. Once
the formulas of logic can be applied always and everywhere, it would not be
reasonable to reject them. If there appears to be a contradiction between the
common sense meaning of the phrase used to describe the logical rule and the
formulas, the adequate reaction is, first, to find a more accurate description
of the logical rule in common language, and, on the other hand, to find a new
formalism which adequately describes the common sense meaning used before for
the logical rule. Then, everybody is free to use what is more important for
him. Given that the logical rule has a much wider range of applicability –
everywhere – one can reasonably expect that the logical rule will be more
widely used.

So, history and rationality seem to lead to similar conclusions: The formulas
which are part of logic will survive. But it is possible that they have to be
reinterpreted, and this reinterpretation has to be done in a careful way. One
has to distinguish the purely logical notion – usually weak, because
applicable without any restriction – from the common sense notion, which
covers usually only archetypal applications instead of the most general one.
The archetypal, common sense applications nonetheless remain valid
applications, so that logic is not weakened by such a reinterpretation.

###  V.5 The situation with causality

Regarding causality, the situation seems quite similar. With (3) as a logical
formula, we have a sort of weak, purely logical notion of causal influence. It
is defined by the function $o(a,\lambda)$, which always exists. This is
different from the classical, common sense meaning of causality, which is
considered to be a sort of physical hypothesis, which is part of some theories
but not of others.

The other thing we have observed in all three examples above, namely that if
the stronger common sense notion is applicable, the weaker logical one is also
valid, holds here too. In a theory with common sense causality, a dependence
of the function $o(a,\lambda)$ on $a$ would be interpreted as a causal
influence.

To summarize, there is no reason to hope that questioning the logic of
plausible reasoning may allow to save Einstein causality. All one can hope for
is a metaphysical reinterpretation of the meaning of causality.

##  VI The example of quantum theory

It makes sense to consider this weaker notion of logical causality for quantum
theory. At least in the minimal interpretation of quantum theory it is not a
theory which makes statements about causality at all. The de Broglie-Bohm
interpretation (which is essentially a different theory, which gives the
quantum predictions only in a subset of quantum equilibrium states) has also
been named “causal interpretation”, with the obvious hint that all other
interpretations are not causal.

To consider the example of quantum theory is useful also because of other
counterarguments. So it has been explicitly doubted that Kolmogorovian
probability theory is sufficient to handle quantum theory, for example, by
Khrennikov Khrennikov :

> Roughly speaking for any fixed experimental context a Kolmogorov space can
> be used. Probabilistic data collected in a fixed experiment in quantum (as
> well as classical) physics can be described by the conventional measure-
> theoretic model, but not data collected for a few incompatible observables.
> Khrennikov  p. vii

Then, the Stone space we have used here is, in a general situation, a quite
complex object. The problem is that to construct a particular
$\omega\in\Omega$ in a situation where we have an infinite space of
propositions $\mathcal{F}$ would require a similarly infinite amount of
arbitrary choices. As a consequence, to prove it we have to use the axiom of
choice222Technically, a slightly weaker assumption, the Boolean prime ideal
theorem, is sufficient, but this would not change the point., which usually
suggests that the construction is a quite artificial one.

Fortunately, the axiom of choice is not as problematic as it is presented in
popular culture. It is known today that this axiom does not lead to
contradictions. If set theory without it is free of contradiction, it remains
free of contradictions if we add the axiom of choice. Then, for the
interesting applications this does not matter at all. In fact, whenever we
consider a particular conceivable scenario, we try to fix only the relevant
properties. This requires only a finite number of choices. And if we later
have to consider some other property, which we have not fixed yet, we can fix
it and continue. Stone’s theorem is useful only because it gives us certainty
that even in the purely theoretical case of an infinite number of
propositions, which could force us into the necessity of an infinite number of
such choices, would not lead us into contradictions.

Nonetheless, given the importance of quantum theory, it seems useful to given
an explicit and usable construction of this space for quantum theory.

###  VI.1 Explicit construction for quantum theory

For every particular quantum observation, as defined by an operator $\hat{F}$,
quantum theory already defines a particular Kolmogorovian space with a
probability distribution on it. The space is defined by the possible
measurement results of the observable, say, $f$, together with the quantum
state $\hat{\rho}$. The corresponding conceivable possibility would be the
proposition “the measurement of $F$ for the quantum state
$\hat{\rho}\in\mathfrak{S}$ would give the result $f\in F$”. This gives us a
space $\Lambda_{F}$ consisting of pairs $(\hat{\rho},f)\in\mathfrak{S}\times
F\cong\Lambda_{F}$, with probability distribution $\rho(f)df$ defined by the
eigenstates $f\in F$ of the operator $\hat{F}$ as
$\langle\psi_{f}|\hat{\rho}|\psi_{f}\rangle$.

Now let’s consider an arbitrary set of incompatible measurements
$\hat{F}_{a},a\in A$, and combine them into a single big experiment, where we
make some random choice $a\in A$ what to measure. For simplicity, we can
restrict ourselves to a classical random device deciding about the $a\in A$.
Let’s denote the measurement results for different measurements as $f_{a}\in
F_{a}$. What defines in this case the space $\Omega$?

Given that it is conceivable that $\hat{F}_{a}$ is measured, we have to
include the conceivable measurement results $f_{a}\in F_{a}$ for all $a\in A$.
This gives us the product space $F_{A}\cong\prod_{a\in A}F_{a}$. The prepared
state $\hat{\rho}\in\mathfrak{S}$ will be part too, which gives our resulting
space $\Lambda\cong\mathfrak{S}\times F_{A}$. The measurement which is
actually made is, of course, also relevant, so that we obtain the space
$\Omega\cong A\times\Lambda$ used in (3).

Let’s define now the probability distribution on $\Omega$. For given
$(a,\hat{\rho})$, quantum theory defines the probability distribution on the
space $F_{a}$ of measurement results $\rho_{a}(f_{a},\hat{\rho})$. On the
product space, we can use

$\rho(\lambda)=\rho(f_{A},\hat{\rho})=\prod_{a\in
A}\rho_{a}(f_{a},\hat{\rho})\text{ on }\Lambda\cong\prod_{a\in
A}F_{a}\times\mathfrak{S},$ (7)

which is a probability distribution of type $\rho(\lambda)d\lambda$ on
$\Lambda$ we need. To show that this probability distribution is viable, all
one needs is that it gives, for every fixed $a\in A$, the correct distribution
on the space of the actual measurement results $F_{a}$, which is trivial by
construction:

$\begin{split}E(o|a)&=\int o(f_{a})\rho(f_{A},\hat{\rho})df_{A}\\\ &=\int
o(f_{a})\rho(f_{a})df_{a}\prod_{a^{\prime}\neq
a}\rho_{a^{\prime}}(f_{a^{\prime}})df_{a^{\prime}}\\\ &=\int
o(f_{a})\rho(f_{a})df_{a}.\end{split}$ (8)

Given that it gives the correct predictions for all experiments allowed in
quantum theory, there would be no need for any further justification. If this
would have been a guess, so what? A positivist has certainly no reason to
object.

Nonetheless, this probability distribution can be derived from the principles
of plausible reasoning. All we need to obtain (7) is, beyond the information
provided by quantum theory about the probability distributions
$\rho_{a}(f_{a},\hat{\rho})$ of particular experiments, a basic principle of
plausible reasoning, the principle of logical independence: If we have no
information which indicates some dependence between two variables, then we
have to assume that they are independent.

We have to apply it two times. The first time when we have made the split
$\rho(\omega)=\rho(a,\lambda)\to\rho(a)\rho(\lambda)$, assuming the
independence of the free decisions of the experimenters what to measure, as
discussed above. The second time when we have made the split of
$\rho(f_{A},\hat{\rho})$ into the product of the $\rho_{a}(f_{a},\hat{\rho})$
in (7). In both cases, we have, indeed, no information provided by quantum
theory that there is a connection between the entities assumed as independent.

###  VI.2 Details about causal dependencies would have to follow from hidden
variable theories

While the independence seems unproblematic for the free will decisions, the
independence assumption made in (7) seems less plausible. Here one can
reasonably expect that the future will give us additional information which
will force us to reject the independence. What type of information could this
be? We can give here a quite definitive answer, namely that it has to be a
hidden variable theory. It has to provide some information which is not
provided by quantum theory, that means, it is hidden information. What else is
a hidden variable theory if not a theory which provides information not
available in quantum theory?

Let’s note here that the construction (7) in itself is not unknown. Kochen and
Specker KochenSpecker  have given a variant of this construction, naming it
“somewhat trivial” (which, indeed, it is) and “artificial” (it is not, but
considering it as a “hidden variable theory” is indeed artificial, given that
it does not contain any hidden information). The purpose for giving this
construction was to “to point out the insufficiency of condition (1)
[essentially our (4)] as a test for the adequacy of the solution of the
[hidden variable] problem”. There is a reason to agree. That we can fulfill
(4) without adding any information to quantum theory means that fulfilling it
does not “solve” any hidden variable problem. What it does is something
different, namely to put an end on attacks on formula (4). One can hardly
argue that this formula is inadequate and can no longer be applied if it can
be obtained even for quantum theory as it is, based on a trivial construction.

Kochen and Specker continued by proposing an additional condition for what
would be an adequate solution of the hidden variable problem, today named non-
contextuality, only to show that it cannot be fulfilled. While it is certainly
important to find out properties which have to be violated by hidden variable
theories, this is not a point of interest here. Our point is the opposite: The
probability distribution (7) may appear inadequate, because the independence
assumption is too strong. This happens always if we have more information,
information which indicates a dependence. No problem – but what we can say is
that another theory, which provides such information, will be a hidden
variable theory in the most straightforward meaning of the word, simply
because it provides – it has to provide – information which is hidden in
quantum theory.

A more detailed consideration of the question of hidden variable in quantum
theory shows that quantum theory itself contains some elements which are not
fixed by quantum theory, but depend on choices of the theoretician who
describes a particular situation. Then, if we compare different choices of the
theoretician, it appears that one description contains information which is
hidden in the other one.

One such theoretical decision is the classical-quantum split in the Copenhagen
interpretation. Imagine one and the same experiment with different splits
between the classical and the quantum part. Then, the description with the
larger classical part contains information about the classical trajectory in
the intermediate region, information which is hidden if the quantum part is
larger. One split puts the mechanism which possibly kills the cat into the
quantum part, one into the classical. Then, the second split contains the
trajectory of the mechanism, which is hidden in the other split.

Another situation where quantum theory differently interpreted appears to be a
hidden variable theory is the possibility to describe the two measurements in
Bell tests in different ways – as a common measurement, or as two separate
experiments, with two possible orderings in time. There exists either a
dependence of $A(a,b,\lambda)$ on $b$ or of $B(a,b,\lambda)$ on $a$, as a
consequence that Bell’s inequality is violated. But if Alice measures first,
then Bob’s decision what to measure cannot influence the measurement result of
Alice, so that $A(a,b,\lambda)=A(a,\lambda)$. Similarly, if Bob measures
first, $B(a,b,\lambda)=B(b,\lambda)$.

These considerations already provide example of useful applications of the
formalism. Once one recognizes that the space $\Lambda$ is a logical
triviality, which does not indicate any nontrivial information about hidden
variables, one can much easier identify those elements where hidden variables
appear.

##  VII Consequences: What remains from Einstein causality?

Let’s consider now the consequences of the main result. Einstein causality –
or, more accurate, any interpretation of Einstein causality which requires
that $A(a,b,\lambda),B(a,b,\lambda)$ which appear in (2) have the form
$A(a,\lambda),B(b,\lambda)$ – is sufficient to derive the Bell inequalities,
and therefore has to be rejected. What could be a replacement? One radical
solution would be, in principle, to give up causality completely. But, as we
will see, there is no necessity for this.

###  VII.1 Weakening Einstein causality: Signal causality

One straightforward possibility, which seems quite popular even now, is to
give up the usual, strong version of Einstein causality, and to go back to a
weaker notion of causality – weak or signal causality. With weak causality, it
is not sufficient for a causal connection that some function $o(a,\lambda)$
depends on some variable $a$, but, first, we care only about observables
$o(a,\lambda)$, only if they depend on free decisions of experimenters $a$,
and we require not only that the function $o(a,\lambda)$ itself depends on
$a$, but if the expectation value $E(o|a)=\int
o(a,\lambda)\rho(\lambda)d\lambda$ depends on $a$.

While such a restriction, with its restriction to observable effects, is quite
plausible from point of view of positivism, positivism is no longer a
reasonable candidate for a philosophy of science. In Popper’s critical
rationalism, scientific theories are hypotheses, thus, can and will contain
hypotheses which are not immediately open to empirical tests. They have to
make testable empirical predictions, the more the better, but if some
hypothesis about unobservable elements of reality leads, however implicit, to
some additional testable predictions, this is sufficient to justify the
hypothesis.

Such a weak notion of causality makes, of course, sense as a useful
information about restrictions for human devices, preventing hopeless affords
similar to perpetuum mobile construction. But as some fundamental principle it
seems useless. The point is that now we know that for every theory there will
exist some description in terms of $o(a,\lambda)$ and $E(o|a)=\int
o(a,\lambda)\rho(\lambda)d\lambda$, so that the information that $E(o|a)$ does
not depend on $a$ either follows from $o(a,\lambda)$ not depending on $a$,
which would mean strong causality, or it is a consequence of the impossibility
to fix in our preparations sufficiently precise distributions
$\rho(\lambda)d\lambda$, thus, it would be only a side effect of restricted
human abilities with no fundamental importance.

So, even if one accepts weak causality as an interesting information, one
would not consider it as something fundamental. It is derived from the
fundamental, strong notion of causality defined by the functions
$o(a,\lambda)$ together with restrictions of our ability to prepare initial
states.

###  VII.2 Classical causality in the Lorentz ether

If one wants to preserve for a fundamental causality at least the non-
existence of causal loops (as necessary to avoid the grandfather paradox), one
has no choice but to accept a hidden preferred frame.

Indeed, assume we can, for any pair of events, observe violations of the Bell
inequalities. Let’s fix the event $e_{a}$ when Alice makes the measurement and
try to identify an equal time event on Bob’s world-line $e_{b}(t)$. We assume
that we have moments $t_{-}$ long before the experiment and $t_{+}$ long after
so that $e_{b}(t_{-})\to e_{a}\to e_{b}(t_{+})$. Then all we have to do is to
use Dedekind cuts to construct a sequence $(t^{n}_{-},t^{n}_{+})$ with
$e_{b}(t^{n}_{-})\to e_{a}\to e_{b}(t^{n}_{+})$ and
$|t^{n}_{+}-t^{n}_{-}|=2^{-n}|t_{+}-t_{-}|$. We observe a violation of the
Bell inequalities for $t^{i+1}=\frac{1}{2}(t^{i}_{+}+t^{i}_{-})$. Based on
strong causality, we conclude that either $e_{a}\to e_{b}(t^{i+1})$ or
$e_{b}(t^{i+1})\to e_{a}$, so that we can put $t^{i+1}$ either in place of
$t^{i+1}_{+}$ or $t^{i+1}_{-}$, reusing the other $t^{i}_{\pm}$ as
$t^{i+1}_{\pm}$. This gives $e_{b}(t^{i+1}_{-})\to e_{a}\to
e_{b}(t^{i+1}_{+})$. This sequence gives a limit $t_{*}$ with the property
that for every $\varepsilon>0$ we have $e_{b}(t_{*}-\varepsilon)\to e_{a}\to
e_{b}(t_{*}+\varepsilon)$. The $e_{b}(t_{*})$ defines the moment contemporary
to $e_{a}$, so that we obtain in this way a global contemporaneity.

Given that we can only observe a violation of the Bell inequalities, but
cannot find out what was the direction of causal influence which has caused
it, the resulting preferred foliation remains hidden from observation. But is
a hidden preferred frame compatible with modern physics?

In the domain of special relativity, this would require nothing but a return
to the Lorentz interpretation, where a preferred rest frame of the Lorentz
ether exists, and relativistic effects are distortions of clocks and rulers
caused by velocity against a static homogeneous ether. A generalization of the
Lorentz ether into the domain of relativistic gravity which gives, in a
natural limit, the Einstein equations of GR in harmonic coordinates, is also
known gle . This generalized Lorentz ether is no longer static and
homogeneous, and its density and stress tensor also influences clocks and
rulers. There exists also a proposal for an interpretation of the standard
model of particle physics compatible with the Lorentz ether clm . This model
gives the SM gauge group and the three generations of fermions, together with
some additional massive scalar fields. So, compatibility of the Lorentz ether
with modern physics is unproblematic. The classical notion of causality can
also survive in quantum theory, as, for example, in realistic and causal
interpretations of quantum theory like the de Broglie-Bohm interpretation.
Moreover, given that in the Lorentz ether causality is connected with the
fixed Newtonian background of absolute space and time, to preserve causality
during quantization of the Lorentz ether is unproblematic too – the background
remains fixed, so that causality remains fixed too, and is not endangered by
uncertainty of the metric.

##  VIII Conclusion

We have shown that a basic formula in the proof of Bell’s inequality, (2), is
a formula following from the logic of plausible reasoning applied to the
situation of the experiment considered in Bell’s paper, without any further
physical assumptions like realism or causality. As a consequence, all what is
necessary to prove Bell’s theorem is Einstein causality. That means, given
that the Bell inequalities are violated for space-like separated pairs of
measurements, Einstein causality has been empirically falsified.

None of the loopholes usually discussed – rejection of realism and causality,
or superdeterminism – are left. The only possibility to preserve Einstein
causality would be to reject the logic of plausible reasoning. We have
considered earlier attempts to criticize classical logic, and came to the
conclusion that all one can hope for is a reinterpretation of the meaning of
the formulas of logic, away from the common sense meaning toward a weaker,
more general, purely logical interpretation. Such a reinterpretation of the
meaning of causality can be found here too.

We have illustrated this considering quantum theory, where we have identified
a rather weak notion of causality. We have found that any strengthening of
this weak notion would require some hidden variables.

Last but not least, we have considered the possibilities to weaken Einstein
causality. One possibility would be a notion of weak Einstein causality based
on signal causality. Such a notion of causality cannot have any fundamental
importance, because the impossibility to send signals faster than light is
only a consequence of uncertainty relations preventing the preparation of
sufficiently certain states. If one wants to have a fundamental notion of
causality without causal loops, one has to accept the introduction of a hidden
preferred frame. Fortunately, nothing in modern science is in conflict with
such a hidden preferred frame, theories based on such a preferred frame exist
as for relativistic gravity as for the standard model of particle physics.

## References

  * (1) Cox, R. T. (1946), Probability, frequency, and reasonable expectation, Am. J. Phys. 14, 1–13.
  * (2) Bell, J.S. ( 1980). Against Measurement, Physics World, Aug. 1980, 33-40.
  * (3) Bell, J.S. (1964). On the Einstein-Podolsky-Rosen paradox, Physics 1, 195-200, also in Bell , 14-21.
  * (4) Bell, J.S. (1987). Speakable and unspeakable in quantum mechanics, Cambridge University Press, Cambridge.
  * (5) Bridges, D., Palmgren, E. (2016). Constructive Mathematics, in Zalta, E.N. (ed.), The Stanford Encyclopedia of Philosophy (Winter 2016 Edition).
  * (6) Gödel, K. (1933), Zur intuitionistischen Arithmetik und Zahlentheorie, Ergebnisse eines mathematischen Kolloquiums, v. 4, pp. 34–38 (German).
  * (7) G. Gentzen (1936), ”Die Widerspruchfreiheit der reinen Zahlentheorie”, Mathematische Annalen, v. 112, pp. 493–565 (German).
  * (8) Jaynes, E. T. (2003). Probability Theory: The Logic of Science, Cambridge University Press.
  * (9) Khrennikov, A. (2009). Interpretations of Probability, Walter de Gruyter, Berlin.
  * (10) Kochen, S., Specker, E.P. (1967). The Problem of Hidden Variables in Quantum Mechanics, J. Math. Mech. 17(1), 59-87.
  * (11) MacColl, H., (1908), ‘If’ and ‘imply’, Mind, 17: 151–152, 453–455.
  * (12) Mares, E. (2014), Relevance Logic, in Zalta, E.N. (ed.), The Stanford Encyclopedia of Philosophy (Spring 2014 Edition).
  * (13) Schmelzer, I. (2009). A Condensed Matter Interpretation of SM Fermions and Gauge Fields, Found. Phys. 39(1), 73-107, arXiv:0908.0591.
  * (14) Schmelzer, I. (2012). A generalization of the Lorentz ether to gravity with general-relativistic limit, Advances in Applied Clifford Algebras 22(1), 203-242, DOI 10.1007/s00006-011-0303-7, arXiv:gr-qc/0205035.
  * (15) Wiseman, H.M. (2014). The Two Bell’s Theorems of John Bell, J. Phys. A: Math. Theor. 47, 424001, arxiv:1402.0351.
  * (16) Stone, M.H. (1936) The Theory of Representations of Boolean Algebras, Trans. Am. Math. Soc. 40, 37-111.


# Physical properties of a generalized model of multilayer adsorption of
dimers

G Palacios  CRCN-NE/CNEN, 50740-545, Recife, PE, Brazil
<EMAIL_ADDRESS>Sumanta Kundu  Department of Physics and Astronomy,
University of Padova, Via Marzolo 8, I-35131 Padova, Italy  INFN, Sezione di
Padova, Via Marzolo 8, I-35131 Padova, Italy  L A P Santos  CRCN-NE/CNEN,
50740-545, Recife, PE, Brazil  SCIENTS, 53635-015, Igarassu, Brazil  M A F
Gomes  Departamento de Física, Universidade Federal de Pernambuco, 50670-901,
Recife, PE, Brazil

###### Abstract

This article investigates the physical and statistical properties of two-
dimensional structures obtained with a dimer packing model, which may be of
great interest in other scientific and technological areas besides theoretical
physics. The study is based on extensive computer simulations in order to
examine the effect of the orientational anisotropy of dimers deposited
following a ballistic deposition process, leading to multilayer networks of
complex morphologies dominated by branched fractal architectures. The
geometric characteristics of the bulk and contours of these systems, and the
transport properties, particularly the electrical conductivity were
investigated by varying the orientational anisotropy of the dimers. The
results provide information about the fundamental mechanisms underlying
formation and behavior of such types of amorphous and disordered matter that
are of paramount importance both to physics as well as to environmental and
material sciences.

##  I Introduction

Disordered and amorphous structures appear as objects of study in several
theoretical, experimental and applied domains in the realms of condensed
matter physics, and in geological, chemical and biological physics, as well as
in technological processes.

A widely studied model that allows obtaining amorphous structures is the
ballistic deposition model, that has several versions [1, 2, 3, 4, 5, 6].
However, all share the following general rules: the particles (usually
monomers) are added one at a time to a growing cluster or aggregate using
initial randomly selected positions and directions. These models lead to
clusters with complex porous structures with compelling properties which have
sometimes been described in terms of fractal dimensions. However, it has
already been demonstrated [3, 7, 8, 9] that the internal structure of
ballistic aggregates is uniform at all scales, except those of short length.

In ballistic deposition models the interface or active zone evolves with time,
changing its roughness while being subjected to random noise [10]. The
roughness width of the interface, which is defined as the standard deviation
of the height, increases initially fast and reaches a plateau as the number of
deposited particles becomes large (with respect to the horizontal length of
the grid) [2, 10]. Furthermore, it has already been shown that the active zone
is not a fractal [2]. Many of these processes of ballistic deposition are
believed to share the property of at least approximate scale invariance in
both space and time. Scale invariance is indicated by self-affine or self-
similar interface profiles.

Porous media are often used experimentally in the purification process of
gases and liquids, and for the separation of various compounds. The porous
media can also be used in the storage of gases such as hydrogen or the
production of activated carbon [11, 12, 13, 14]. In these systems the
measurement of the electrical conductivity makes it possible to determine the
concentration of filtered agents [15, 16]. It has been demonstrated that
natural porous media such as rock and sediment reservoirs, generally have
complex internal structures whose fractal geometry influences the transport
properties[17]. For example, although it is known that conductivity is
strongly dependent on the amount of minerals dispersed in water, in a water-
saturated porous medium there is an influence of porosity. It is known that
the conductivity in these systems involves power-law dependencies encapsulated
in the Archie’s law [18]. Karst network structures (dendritic structures,
morphologically very similar to the structures studied in this work) cover 20
% of the earth’s surface. Therefore, understanding them is crucial for many
practical purposes such as the problems of contamination in the environment as
well as the assessment of soil stability [19].

In a previous work, we studied the transport properties in amorphous
structures resulting from the application of a version of the ballistic
deposition model, which is the unrestricted multilayer adsorption of dimers
[20]. In this case, the dimers are dropped vertically and are deposited on the
top of a randomly selected column with growing structure formation and thus,
in turn, may prevent from accessing the lower layers at the selected column
for the future deposits, generating pores in the structure. Several properties
of this model were examined when all the deposited dimers are horizontally
oriented [20]. Here we propose a more realistic version, in which the dimers
have the chance to deposit oriented also in the vertical direction and their
occurrence probability can be controlled.

The structure of this paper is the following: in Section II we give the
details of our simulation; in Section III, the main results are presented and
a discussion is made of the principal aspects, including the percolation
transition (III.1), the fractal and diffusion on the percolation clusters
obtained (III.2), the electrical conductivity (III.3) and the fractal
properties of the free interface (III.4). In Section IV we summarize our main
findings and present some future perspectives.

##  II Model and simulation details

We consider a variant of the ballistic deposition model for the formation of a
growing structure on a one dimensional lattice of size $L$. Dimers are dropped
one by one onto the randomly selected lattice sites from a far apart distance
from the growing structure along the vertical direction. Specifically, a dimer
follows a vertical trajectory from its release point and lands on the top of
the growing surface where it first encounters a previously deposited dimer.
Note that the dimers are non-sticky in nature and therefore, during its
vertical motion even if a dimer finds a previously deposited dimer at the side
of the nearest neighbor columns, it continues its motion until it hits a dimer
beneath it and it cannot go down. This dynamics leads to a complex porous
structure formation due to the orientational anisotropy of the incoming
dimers.

We implement the dynamics in the following way: at each instant of time $t$,
by selecting the orientation of a dimer randomly with probability $p_{0}$ and
$(1-p_{0})$ for horizontal and vertical, respectively, we drop the dimer from
a randomly selected position $x_{0}$ onto the lattice. For a horizontal
(vertical) dimer, we choose the location of the one end of the dimer to be at
the site:

$x_{0}\in\begin{cases}[1,L-1],&\text{if horizontal}\\\ [1,L],&\text{if
vertical}\end{cases}$ (1)

then, the dimer is placed at the sites:

$\begin{cases}([x_{0},x_{0}+1],h_{max}),&\text{if horizontal}\\\
(x_{0},[h_{max},h_{max}+1]),&\text{if vertical}\end{cases}$ (2)

where $h_{max}$ is the maximum height of the growing structure at the
corresponding columns. Summarizing, the dynamics considered here always allows
adhesion on top of two different dimers or on top of a single dimer, blocking
the incoming dimers that arrive from the top to the bottom on the growing
surface due to the presence of overhangs (screening effects [21]). At any
stage, the maximum height of the entire growing structure along the vertical
direction is denoted by $H_{max}$. Fig. 1 shows typical configurations of the
growing structure for different values of $p_{0}$.

Figure 1: Typical multilayer packing configurations of the model studied in
the present article for $L=H_{max}=512$, and for selection probabilities
$p_{0}=0.1,0.3,0.5,0.9$ from (a) to (d), respectively. The dimers are colored
according to their time of deposition.

These images suggest that the multilayer packing structure of dimers results
in dendritic structures with a complex pore structure, perhaps of a fractal
nature. Their morphologies are basically dominated by the contribution of the
complex geometry of the internal region of the structure and of the boundary,
known as active region (to be examined in subsection III.4). Both have very
interesting fractal properties.

It should be noted that even if the morphology of the formation might be
different due to the orientational anisotropy of the dimers, the roughness
exponent associated with the growing front is the same as the ordinary
Ballistic deposition model.

In this work, the focus of the analysis is based on the physical properties of
the percolation cluster. To identify the percolation cluster, we combine the
deposition algorithm discussed above with the Hoshen–Kopelman algorithm [22].
This allows to label the dimers at each deposition step. To determine when the
percolation cluster is created it was used the following trick: the first and
last columns of the deposition grid are assigned two different labels. During
the deposition process, clusters of interconnected dimers are created with
different labels. When the labels of the first and the last column of the
deposition grid are the same, it means that the percolation cluster has
emerged. We have also used the Burning algorithm [23] to calculate the
percolation threshold.

##  III Results and discussion

###  III.1 Percolation

The focus of this subsection is to study the percolation properties of the
growing structure formed due to the particle deposition. Note that the entire
growth process can be classified into three stages in time (see Fig. 2): (i)
Initial stage: increase in the number of isolated clusters of dimers connected
through their nearest neighbors and their size growth with time, (ii)
Intermediate stage: merging of growing clusters of different sizes, and (iii)
Final stage: size growth of only one single cluster. At this stage there may
also be other clusters, but their growth is stopped, as a new incoming dimer
cannot penetrate deep interior of the formed structure due to the screening
effect.

Figure 2: Snapshots of the growing structure on a lattice of size $L=612$ for
$p_{0}=0.7$ at different instants of time $t$ = 4408 (a), 7466 (b), 10330 (c),
and 12343 (d). The corresponding values of $H_{max}$=40, 60, 80, and 98,
respectively. Different colors represent different clusters.

When the $H_{max}$ of entire structure is sufficiently high, there exists a
spanning path from the left to the right of the system through the largest
cluster at the final stage, whereas such a spanning cluster is absent at the
initial stage. At the intermediate stage of the cluster merging process, as
$H_{max}$ gradually increases, a percolation transition occurs when such a
spanning cluster first appears between left and right boundaries of the system
at a critical value of $H_{max}=H_{c}(L)$. Figs. 3(a-c) display the left to
right spanning probability $P_{p}$ of the system as a function of $H_{max}$ on
a lattice of linear size $L$ ranging from 128 to 8192 for three different
values of $p_{0}$. The curves clearly depend both on $L$ and $p_{0}$.

Figure 3: Percolation probability $P_{p}$ as function of the maximum height
$H_{max}$ of the multilayer growing structure for $p_{0}$ = 0.3 (a), 0.5 (b),
0.7 (c). (d) The plot of $P_{p}$ vs. $H_{max}$ for different values of $p_{0}$
using $L=1024$ shows clearly its dependence on $p_{0}$. The results are based
on averages over $10^{6}$ (for first two smallest systems) to 13750 (for
largest system) independent samples. For $L=1024$, we consider (at least)
$2.2\times 10^{5}$ samples.

Notice that, for a specific value of $L$, the sharp rise of the curve for
$P_{p}$ shifts to the higher value of $H_{max}$ with increasing the value of
$p_{0}$ (see Fig. 3(d)). It indicates that the presence of vertical dimers
promotes the percolation. Qualitatively, such a behavior can be understood in
the following way: for small $p_{0}$ the interior of the percolation cluster
is more homogeneous and all the deposited dimers belong to the cluster. If
$p_{0}$ is increased, at the base of the packing structure isolated clusters
are formed that will never help the growing structure to establish a spanning
path (see Fig. 4). This implies that more time (or more incoming dimers) is
required for the structure to grow to obtain a spanning path. Correspondingly,
the height of the percolating cluster at the percolation transition increases.

Figure 4: The growing structure on a lattice of size $L=512$ at the
percolation threshold for $p_{0}$=0.2 (a), 0.5 (b), and 1.0 (c). The critical
height at which percolation transition occurs increases with increasing the
value of $p_{0}$. The blue and red colors represent the percolating cluster
and all other isolated clusters, respectively.

Numerically, the precise value of the percolation threshold $H_{c}(L)$ for a
given value of $L$ and $p_{0}$ is determined using the bisection method [25,
24]. We start with a pair of values of deposition time $t^{\text{hi}}$ and
$t^{\text{low}}$ such that a spanning cluster exists at $t=t^{\text{hi}}$, but
not at $t=t^{\text{low}}$. This interval is then successively bisected and
checked if there exists a spanning cluster using the Burning algorithm [23]
until $t^{\text{hi}}-t^{\text{low}}=1$. At this stage, the corresponding
height of the entire structure at $t=t^{\text{hi}}$ defines the critical
height for a given run. Note that, we initially stored the sequence of dimer
deposition on the lattice sites up to $t=t^{\text{hi}}$ and used this sequence
during the iterative process. By repeating the entire procedure for a large
number of independent runs and averaging the corresponding critical height
values the percolation threshold $H_{c}(L)$ is obtained.

Figure 5: Size-scaling of (a) $H_{c}(L)$, and (b) $S_{max}(L)$ for
$p_{0}=0.3,0.5,$ and $0.7$. The solid line represent the fit using Eq. 3. The
number of samples used to find out these results are indicated in Fig. 3.

The dependence of $H_{c}(L)$ on $L$ is exhibited in Fig. 5. It appears that
the data points are consistent with the following functional form:

$H_{c}(L)=AL^{\nu}+c,$ (3)

where $c$ and $A$ are the fitting constants and $\nu>0$. We find that the
exponent value $\nu$ has a slight dependence on $p_{0}$ values:
$\nu(p_{0}=0.3)=0.158\pm 0.004$, $\nu(p_{0}=0.5)=0.130\pm 0.004$ and
$\nu(p_{0}=0.7)=0.108\pm 0.003$.

In Fig. 5 (b) the finite-size scaling of the size of the percolation
(spanning) cluster is presented. In this case we fit the data using a power-
law similar to Eq. 3 but with $c=0$. It was verify that for sufficiently large
$L$, $S_{max}(L)$ scales as: $S_{max}(L)\sim L^{\alpha}$ with
$\alpha(p_{0}=0.3)=1.268\pm 0.006$, $\alpha(p_{0}=0.5)=1.258\pm 0.007$ and
$\alpha(p_{0}=0.7)=1.250\pm 0.007$. Note that irrespective of the value of
$p_{0}$ we obtain nearly same exponent value $\alpha$ associated with the
variation of $S_{max}(L)$.

###  III.2 Fractal and diffusion dimension on the percolation cluster

This subsection will be dedicate to the study of the fractal and diffusion
properties associated to the percolation clusters generated with our model. It
should be noted that in this case we cannot use the habitual way of
calculating the fractal dimension in which the scale exponent of the mass (or
area) of the percolation cluster with $L$ is exactly the fractal dimension.
Thus, there is an anisotropic system where the height $H_{c}$ scales non-
trivially with $L$. That is why one should use some alternative method. To
obtain their corresponding fractal dimensions a method based on the density
autocorrelation function was chosen; the averaged density of occupied sites
within neighborhoods of radius $r$ centered in all points belonging to the
cluster is firstly calculated. Afterwards, a log-log plot of density against
distance $r$ then yields a straight line with slope $D_{f}-d$, where $d$ is
the dimensionality of the space (2 in this case). For each value of $p_{0}$
the fractal dimension for each $L=128;256;512$ and $1024$ is obtained as
averages on $16,000;2,800;400$ and $100$ replicas respectively. The diffusion
exponent $D_{w}$ [26] of the percolation cluster was also calculated. For
this, random walks are generated starting at a random occupied site of the
percolation cluster, allowing to walk
$t=100;200;500;1,000;5,000;10,000;100,000$ steps and determining in each case
the average of the square of the distance between the starting and ending
point of the walk $\langle r^{2}\rangle$. The average $\langle r^{2}\rangle$
for each value of $t$ is taken with different number of replicas, i.e.,
$N_{rep}=73,000;44,000;20,000;10,000;2,000;1,000;100$ respectively. For a
fractal $D_{w}$ is defined by the scaling $\langle r^{2}\rangle\sim
t^{2/D_{w}}$ which reduces to the known result $\langle r^{2}\rangle\sim t$
for brownian walkers ($D_{w}=2$).

The behaviour of the fractal dimension $D_{f}$ as function of $L$, for
different values of the probability $p_{0}$ and the corresponding diffusion
exponent $D_{w}$ are shown, respectively, in the left and right columns of
Fig. 6.

Figure 6: In the left column we show the fractal dimension $D_{f}$ of the
percolation cluster for different values of the probability $p_{0}$ and for
different sizes $L$ of the substrate. In the right column we show the
diffusion exponent. The solid line in all plots represent the fit using Eq. 4.
In all cases the error bar (variance) is much smaller than the size of the
symbols. See text for detail.

Asymptotic analysis $D_{f}(L\to\infty)$ and $D_{w}(L\to\infty)$ was done for
all curves of the Fig. 6. $D(L)$ for the different values of $p_{0}$ was
fitted with following exponencial model:

$D(L)=ae^{-bL}+c,$ (4)

The result of the fits is shown in Fig. 6 as a solid line in each case. Table
1 shows the values of the fitted asymptotic coefficients $c=D(L\to\infty)$
valid for the thermodynamic limit, for each value of $p_{0}$.

$p_{0}$ |  $D_{f}$ |  $D_{w}$
---|---|---
0.2 |  $1.806\pm 0.037$ |  $2.912\pm 0.099$
0.4 |  $1.838\pm 0.027$ |  $2.939\pm 0.124$
0.6 |  $1.852\pm 0.010$ |  $2.783\pm 0.137$
0.8 |  $1.877\pm 0.024$ |  $2.638\pm 0.065$
1.0 |  $1.888\pm 0.024$ |  $2.649\pm 0.050$
Table 1: Values of fractal dimension ($D_{f}$) and diffusion exponent
($D_{w}$).

One can note that $D_{w}$ is close to the classical percolation $D_{w}=2.8784$
[27] for $p_{0}$ small, just where $D_{f}$ is more distant from the classical
percolation, whose value is 91/48, which in turn approaches the $D_{f}$ values
from this paper for $p_{0}$ close to the unit. Curiously, Ben-Avraham and
Havlin [28, 29] found $D_{w}=2.7\pm 0.1$ for classical 2d percolation, close
to our values $D_{w}=2.638$ for $p_{0}=0.8$ and $D_{w}=2.649$ for $p_{0}=1$.

###  III.3 Electrical conductivity

The Frank and Lobb algorithm [30] was used for finding the conductivity
between the left and right borders of the two-dimensional structure that
results from the dimer deposition process. Therefore the same procedure
described in [20] was used. The calculations of the conductivity $\sigma$ were
performed for each time until the multilayer reaches a given height using the
same methodology applied in [20]. For each given value of $H_{max}$, the
computer experiments were repeated 1200 times.

In Fig. 7 we show the behavior of the mean conductivity as a function of
$H_{max}$ for different values of $L$ and for two values of $p_{0}$.

Figure 7: Mean conductivity as a function of $H_{max}$ for
$L=100;300;500;1,000$ and for $p_{0}=0.2$ (a) and $p_{0}=0.8$ (b). Inset:
Scaling analysis of conductivity $\sigma_{+(-)}$ after (before) the
percolation transition. In the main plot and in the inset, the size of the
symbols is larger than the corresponding error bars.

As is known, the percolation phase transition occurs when a material undergoes
a transition from an insulating state to a conducting state. In general exists
a power law scaling relation between the electrical conductivity $\sigma$ and
the linear size $L$ of the structure near the percolation threshold through
two different scaling relations for $\sigma_{-}$ (conductivity before the
percolation transition) and $\sigma_{+}$ (conductivity after the percolation
transition) [31, 32],i.e.,

$\sigma_{\pm}\sim L^{-\gamma_{\pm}}.$ (5)

In Table 2 we show the values of the scaling exponents $\gamma_{-}$ and
$\gamma_{+}$ for the same values of $p_{0}$ used in Table 1.

$p_{0}$ |  $\gamma_{-}$ |  $\gamma_{+}$
---|---|---
0.2 |  $1.010\pm 0.003$ |  $1.069\pm 0.019$
0.4 |  $1.012\pm 0.003$ |  $1.073\pm 0.018$
0.6 |  $1.016\pm 0.004$ |  $1.091\pm 0.026$
0.8 |  $1.022\pm 0.005$ |  $1.107\pm 0.031$
1.0 |  $1.017\pm 0.013$ |  $1.129\pm 0.116$
Table 2: Scaling exponents of $\sigma_{-}$ and $\sigma_{+}$.

The first interesting result is that for $H_{max}$ below the percolation
transition the system presents a certain universality, which is reflected in a
constant value (equal to 1) of the scale exponents of $\sigma_{-}$, regardless
of the value of $p_{0}$. This value for the scaling exponent implies that the
system belongs to the universality class of percolation in two dimensions [33,
34]. The case of scaling exponents for $\sigma_{+}$ is more interesting. The
results shown in the third column of the Table 2 suggest that after the
percolation point, the behaviour of the conductivity depends critically of
$p_{0}$. At a given $H_{max}$, the conductivity can drop more faster after the
percolation threshold if horizontal alignment of dimers increases, i.e.,
horizontal alignment induces decrease of the conductivity at given conditions,
as expected.

Electrical conduction in amorphous materials is not simply described by the
diffusion of particles within the paradigm of classical (weakly) disordered
systems. The knowledge of disordered systems has benefited in recent decades
from percolation theory and the advent of fractals [28, 29].

In a uniform euclidean system, the mean square displacement $\langle
r(t)^{2}\rangle$ of a random walker is proportional to time $t$, for any
number of spatial dimensions. However, for disordered systems, this linear
relationship is not valid in general, and has to be replaced by:

$\langle r^{2}(t)\rangle\sim t^{2/D_{w}}.$ (6)

In the last expression, as previously defined in subsection III.2 $D_{w}>2$ is
the diffusion exponent which assumes a value different from the classical
brownian exponent $D_{w}=2$. This delay in transport is caused by the
peculiarities in the spatial distribution of scattering centres in the
disordered structure [35]. Regarding to geometric and structural aspects, the
electrical resistance $R$ of a material depends both on the size and the space
dimensionality $d$ and on the topology of the system, then the traditional
expression for the electrical resistance $R=\rho L/A$ (where $\rho$ is the
electrical resistivity of the material, $L$ is the length or distance between
the electrodes and $A$ is the cross section area) implies for a
$d$-dimensional euclidean system (whose area $A$ is proportional to $L^{d-1}$)
that:

$R\sim L^{2-d},$ (7)

that is, from equation (7) follows that for homogeneous square samples the
electrical resistance does not depend on its size.

From the microscopic point of view, the electrical resistance of a system can
be written in general terms as:

$R\sim N/M,$ (8)

where $N$ is the is the average number of scattering centers in the system of
length $L$, and $M$ is the number of possible scattering centers [36]. The
denominator $M$ is proportional to the number of atoms in the system, $L^{d}$
, whereas the numerator $N$ is proportional to the average time required for
the electron to go through the distance between the ends of the system. From
equation (6) it is expected that $N$ is proportional to $t\sim\langle
r^{2}(t)\rangle^{d_{w}/2}=(L^{2})^{d_{w}/2}$. For a non-uniform fractal
system, one can further express equation (8) in terms of $L$ by assuming that
the resistor has a fractal dimension $D$ to obtain
$R\sim\frac{N}{M}\sim\frac{L^{d_{w}}}{L^{D}}\sim L^{d_{w}-D}$ [29], or
equivalently, for the electrical conductance, $C$:

$C\sim L^{D-d_{w}}.$ (9)

Then, in a two-dimensional medium and with diffusion exponent $d_{w}=2$ the
electrical conductance scales as $C\sim L^{\alpha}$, where $\alpha=D-d_{w}=0$.
For media with fractal dimension and with different diffusion exponent it is
expected that $\alpha\neq 0$. Thus, for instance, for a percolation cluster in
two dimensions, $D=91/48=1.8958$ and $d_{w}=2.8784$ [27], and $C\sim
L^{D-d_{w}}=L^{1.8958-2.8784}=L^{-0.9826}$.

Figure 8: Scaling of the conductivity of percolation cluster for different
values of the probability $p_{0}=0.2;0.4;0.6;0.8;1$. The solid lines represent
the fit to power law model $\sigma\sim L^{-\gamma_{0}(p_{0})}$ whose exponents
$\gamma_{0}$ are inserted in the figure.

In order to check the validity of this last result the scaling behavior of the
conductivity of the percolation cluster was calculated. The results are shown
in Fig. 8 with the solid line representing the fit to the power law model
$\sigma_{pc}(p_{0})\sim L^{-\gamma_{0}(p_{0})}$. In the inset of the Fig. 8
one can observe that at the critical point of percolation, the conductivity
presents a universal behavior with the scaling exponent assuming the value
$\gamma_{0}(p_{0})=\gamma_{0}=0.754\pm 0.004$ independent of $p_{0}$.

For completeness, Table 3 brings a comparison between the exponent
$\gamma_{0}$ and the value of the difference $D_{f}-D_{w}$ for the values of
$p_{0}$ shown in the Table 2.

$p_{0}$ |  $\gamma_{0}$ |  $D_{f}-D_{w}$
---|---|---
0.2 |  $-0.755\pm 0.002$ |  $-1.10\pm 0.11$
0.4 |  $-0.753\pm 0.003$ |  $-1.10\pm 0.10$
0.6 |  $-0.755\pm 0.003$ |  $-0.93\pm 0.11$
0.8 |  $-0.754\pm 0.002$ |  $-0.75\pm 0.07$
1.0 |  $-0.756\pm 0.002$ |  $-0.76\pm 0.07$
Table 3: Comparison between the exponent $\gamma_{0}$ and the values of the
difference $D_{f}-D_{w}$.

It is very interesting that, according to the results, the eq. (9) is valid
only for $p_{0}$ close to $1$ exactly where the $D_{f}$ values are close to
the values obtained for the case of classical percolation (see Table 2). At
this point, it is important to mention that there are no works in the
literature where the transport properties are studied in the way it was done
in this work, so it is difficult to compare with results obtained by other
authors.

###  III.4 Fractal properties of the interface

In the context of growth models, within the proposed multilayer deposition
model falls, a matter of great interest is the study of the interface profile,
that is, the curve that separates the multilayered structure from the
environment that surrounds it. The fundamental motivation of the presented
study is that the great majority of the physical processes of interest take
place at the interface. Fig. 9 shows three typical examples of boundary
profiles for different values of $p_{0}$.

Figure 9: Typical upper boundaries or active regions for $p_{0}=0$ (a),
$p_{0}=0.3$ (b), and $p_{0}=1$ (c). For these simulations $L=512$ and
$H_{max}=512$.

In Fig. 10 (a) we show the dependence of the fractal dimension $D$ of the
active region. For the determination of the fractal dimension the one-
dimensional box counting method was applied. It consists of measuring the
perimeter of the curve with ”rulers” of size $\epsilon$ and writing down the
number $N$ of possible rulers of this size that completely cover the curve. A
log-log plot of $N$ against inverse of $\epsilon$ then yields a straight line
with slope $D$. This figure shows that for sufficiently large values of
$p_{0}$ the fractal dimension of the upper boundary decreases in a linear
manner.

Figure 10: (a) The fractal dimension $D$ of the upper boundary as function of
the probability $p_{0}$. (b) Asymptotic analysis of $D$. The inset presents
the behavior of the parameter $c=D(L\to\infty)$ in Eq. 4, giving the value of
the fractal dimension of the upper boundary in the thermodynamic limit as a
function of $p_{0}$. In all cases the error bar (variance) is much smaller
than the size of the symbols. See text for detail.

Asymptotic analysis $D(L\to\infty)$ based on the same model adopted in
subsection III.2 (equation (4)) is shown in the Fig. 10 (b). The inset
exhibits the fine details of $c=D(L\to\infty)$ for $p_{0}\geq 0.2$.

As a closing to this section, we consider a few comments to be pertinent: (i)
It has not escaped our notice that these measured values of $D$, exhibited in
the inset of Fig. 10 (b), denoting the fractal dimension of the surface of the
aggregates, in the thermodynamic limit, in a very wide interval of $p_{0}$ are
equal to the fractal dimension of the important structure of the two-
dimensional self-avoiding walks, within typical uncertainties of less than 1%.
(ii) An equally intriguing result, is that in the opposite direction, i.e. in
the low-$p_{0}$ region, the Fig. 10(a) indicates that $D$ for the top surface,
in the limit $p_{0}=0$, converges close to the value found for the backbone of
the ordinary percolation problem ($d_{B}=1.62\pm 0.02$) [37]. (iii) These
findings (i) and (ii) concerning $D$ in two different domains of $p_{0}$
appear to be related, because the backbone of the percolation cluster consists
of all the sites visited by all possible self-avoiding walks from the
injection site(s) to the exit site(s) [38].

##  IV Conclusion

We numerically investigated the morphology of a growing multilayer structure
built on a one-dimensional substrate that represents a model of disordered and
amorphous matters involving ballistic deposition of dimers. The occurrence
probability of dimers with two possible orientations, horizontal and vertical,
are selected at random with probability $p_{0}$ and $(1-p_{0})$, respectively.
The most interesting characteristic of the multilayer packing structures of
dimers studied here is the dendritic morphology as depicted in Figs. 1, 2 and
4, with a complex pore structure that is dominated by the contribution of the
complex geometry of both the internal and the boundary parts of the system. In
this work, a great deal of effort is dedicated to deeply analyze the physical
properties of the percolation cluster, its critical behavior, nontrivial
scaling laws, critical exponents in the thermodynamic limit (Sections III.1
and III.2, Fig. 5, Table 1) and the corresponding emergent electrical
conductivity (Sec. III.3, Figs. 7 and 8, Tables 2 and 3). The fractal aspects
of the bulk and perimeter (Sec. III.4, Figs. 9 and 10) of the structure has
also been examined.

As a future problem, one may consider a modified version of the deposition
process by introducing defects into the system, switching on stickiness of the
dimers or in general $k$-mers, and also, bringing up the temperature effects
that might influence the growth process itself, leading to describe a more
realistic dynamics for a range of surface growth phenomena. Moreover, the
present study can help researchers from other areas to quantify and control
different levels of environmental concern linked to the accumulation of
pollutants in trees, as well as in other natural and man-made structures, in
places where contamination can reach different levels.

## Acknowledgement

G Palacios thanks a fellowship from Conselho Nacional de Desenvolvimento
Científico e Tecnológico (CNPq) Process: 381191/2022-2. SK is supported by the
research grant ‘ORLA$\\_$BIRD2020$\\_$01’ of the University of Padova. L A P
Santos acknowledges CNPq for the grant 305017/2021-7. M A F Gomes acknowledges
the financial support from the Brazilian Agency CAPES PROEX
23038.003069/2022-87, no. 0041/2022.

## References

  * [1] Vold, M. A numerical approach to the problem of sediment volume. Journal Of Colloid Science. 14, 168-174 (1959)
  * [2] Meakin, P., Ramanlal, P., Sander, L. & Ball, R. Ballistic deposition on surfaces. Physical Review A. 34, 5091 (1986)
  * [3] Family, F. & Vicsek, T. Scaling of the active zone in the Eden process on percolation networks and the ballistic deposition model. Journal Of Physics A: Mathematical And General. 18, L75 (1985)
  * [4] Meakin, P. Simple models for particle aggregation, deposition and segregation. MRS Online Proceedings Library (OPL). 180 (1990)
  * [5] Yu, J. & Amar, J. Scaling behavior of the surface in ballistic deposition. Physical Review E. 65, 060601 (2002)
  * [6] Comets, F., Dalmau, J. & Saglietti, S. Scaling limit of the heavy-tailed ballistic deposition model with p -sticking. ArXiv Preprint ArXiv:2203.06133. (2022)
  * [7] Meakin, P. Accretion processes with linear particle trajectories. Journal Of Colloid And Interface Science. 105, 240-246 (1985)
  * [8] Bensimon, D., Shraiman, B. & Liang, S. On the ballistic model of aggregation. Physics Letters A. 102, 238-240 (1984)
  * [9] Ball, R. & Witten, T. Causality bound on the density of aggregates. Physical Review A. 29, 2966 (1984)
  * [10] Kardar, M., Parisi, G. & Zhang, Y. Dynamic scaling of growing interfaces. Physical Review Letters. 56, 889 (1986)
  * [11] Blankenship, L., Balahmar, N. & Mokaya, R. Oxygen-rich microporous carbons with exceptional hydrogen storage capacity. Nature Communications. 8, 1-12 (2017)
  * [12] Dincer, I. & Acar, C. Review and evaluation of hydrogen production methods for better sustainability. International Journal Of Hydrogen Energy. 40, 11094-11111 (2015)
  * [13] Heinemann, N., Alcalde, J., Miocic, J., Hangx, S., Kallmeyer, J., Ostertag-Henning, C., Hassanpouryouzband, A., Thaysen, E., Strobel, G., Schmidt-Hattenberger, C. & Others Enabling large-scale hydrogen storage in porous media–the scientific challenges. Energy & Environmental Science. 14, 853-864 (2021)
  * [14] Morris, R. & Wheatley, P. Gas storage in nanoporous materials. Angewandte Chemie International Edition. 47, 4966-4981 (2008)
  * [15] Meng, H., Shi, Q., Liu, T., Liu, F. & Chen, P. The percolation properties of electrical conductivity and permeability for fractal porous media. Energies. 12, 1085 (2019)
  * [16] Printsypar, G., Bruna, M. & Griffiths, I. The influence of porous-medium microstructure on filtration. Journal Of Fluid Mechanics. 861 pp. 484-516 (2019)
  * [17] Wei, W., Cai, J., Hu, X. & Han, Q. An electrical conductivity model for fractal porous media. Geophysical Research Letters. 42, 4833-4840 (2015)
  * [18] Wang, H. & Revil, A. Surface conduction model for fractal porous media. Geophysical Research Letters. 47, e2020GL087553 (2020)
  * [19] Hendrick, M. & Renard, P. Fractal dimension, walk dimension and conductivity exponent of karst networks around Tulum. Frontiers In Physics. 4 pp. 27 (2016)
  * [20] Palacios, G., Santos, L. & Gomes, M. Transport properties in multilayer adsorption of dimers. Physical Review E. 106, 034120 (2022)
  * [21] Bartelt, M. & Privman, V. Kinetics of irreversible multilayer adsorption: One-dimensional models. The Journal Of Chemical Physics. 93, 6820-6823 (1990)
  * [22] Hoshen, J. & Kopelman, R. Percolation and cluster distribution. I. Cluster multiple labeling technique and critical concentration algorithm. Physical Review B. 14, 3438 (1976)
  * [23] Herrmann, H., Hong, D. & Stanley, H. Backbone and elastic backbone of percolation clusters obtained by the new method of ’burning’. Journal Of Physics A: Mathematical And General. 17, L261 (1984)
  * [24] Kundu, S., Araújo, N. A. M. & Manna, S. S. Jamming and percolation properties of random sequential adsorption with relaxation. Phys. Rev. E. 98, 062118 (2018), https://link.aps.org/doi/10.1103/PhysRevE.98.062118
  * [25] Kundu, S., Datta, A. & Manna, S. S. Double transition in a model of oscillating percolation. Phys. Rev. E. 96, 032126 (2017), https://link.aps.org/doi/10.1103/PhysRevE.96.032126
  * [26] Havlin, S. & Ben-Avraham, D. Diffusion in disordered media. Advances In Physics. 36, 695-798 (1987)
  * [27] Grassberger, P. Conductivity exponent and backbone dimension in 2-d percolation. Physica A: Statistical Mechanics And Its Applications. 262, 251-263 (1999)
  * [28] Ben-Avraham, D. & Havlin, S. Diffusion on percolation clusters at criticality. Journal Of Physics A: Mathematical And General. 15, L691 (1982)
  * [29] Ben-Avraham, D. & Havlin, S. Diffusion and reactions in fractals and disordered systems. (Cambridge University Press, 2000)
  * [30] Frank, D. & Lobb, C. Highly efficient algorithm for percolative transport studies in two dimensions. Physical Review B. 37, 302 (1988)
  * [31] Cherkasova, V., Tarasevich, Y., Lebovka, N. & Vygornitskii, N. Percolation of aligned dimers on a square lattice. The European Physical Journal B. 74 pp. 205-209 (2010)
  * [32] Tarasevich, Y., Laptev, V., Goltseva, V. & Lebovka, N. Influence of defects on the effective electrical conductivity of a monolayer produced by random sequential adsorption of linear k-mers onto a square lattice. Physica A: Statistical Mechanics And Its Applications. 477 pp. 195-203 (2017)
  * [33] Jang, H. & Yu, U. Universality class of the percolation in two-dimensional lattices with distortion. Physica A: Statistical Mechanics And Its Applications. 527 pp. 121139 (2019)
  * [34] Bunde, A. & Havlin, S. Fractals and disordered systems. (Springer Science & Business Media,2012)
  * [35] Gefen, Y., Meir, Y., Mandelbrot, B. & Aharony, A. Geometric implementation of hypercubic lattices with noninteger dimensionality by use of low lacunarity fractal lattices. Physical Review Letters. 50, 145 (1983)
  * [36] Gomes, M., Hora, R. & Brito, V. Electrical resistance of complex two-dimensional structures of loops. Journal Of Physics D: Applied Physics. 44, 255401 (2011)
  * [37] Herrmann, H. & Stanley, H. Building Blocks of Percolation Clusters: Volatile Fractals. Phys. Rev. Lett.. 53, 1121-1124 (1984,9)
  * [38] Feder, J. Fractals. (Springer Science & Business Media,2013)