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\begin{document}

\draft

\title{Self-Consistent Effective-Medium Approximations with Path Integrals}

\author{Yves-Patrick Pellegrini$^1$ and Marc Barth\'el\'emy$^{1,2}$}

\address{ $^1$ Service de Physique de la Mati\`ere Condens\'ee, \\
		Commissariat \`a l'Energie Atomique,\\
		BP12, 91680 Bruy\`eres-le-Ch\^atel, France.\\
	  $^2$ Center for Polymer Studies and Dept. of Physics,
		Boston University, Boston, MA 02215.}

\date{Last modified: December 9, 1999. Printed: \today\\
To be published in Physical Review E.}

\maketitle

\begin{abstract}
We study effective-medium approximations for linear composite media
by means of a path integral formalism with replicas.
We show how to recover the Bruggeman
and Hori-Yonezawa effective-medium formulas.
Using a replica-coupling ansatz,
these formulas are extended into new ones
which have the same percolation thresholds as that of the Bethe
lattice and Potts model of percolation,
and critical exponents $s=0$
and $t=2$ in any space dimension $d\geq 2$.
Like the Bruggeman and Hori-Yonezawa formulas, the new formulas
are exact to second order in the weak-contrast and dilute limits.
The dimensional range of validity of the four effective-medium
formulas is discussed, and it is argued that the new ones are
of better relevance than the classical ones in dimensions $d=3,4$
for systems obeying the Nodes-Links-Blobs picture, such as
random-resistor networks.
\end{abstract}

\vskip 1cm
\pacs{PACS numbers: 05.10.-a, 05.40.-a, 05.50.+q, 72.70.+m }

%\begin{multicols}{2}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
Among various effective-medium formulas used to model the effective
behavior of random conducting linear composites, the symmetrical
Bruggeman formula \cite{BRUG35,LAND52} is undoubtly the most
popular. Applied to an insulator/conductor binary mixture, it predicts
a percolation-like transition \cite{KIRK71,CLER90,STAU92,SAHI98} for a
volumic fraction of conductor $p_c=1/d$, where $d$ is the space
dimension. The critical exponents are $s=t=1$, and its critical properties
are thoroughly discussed in Ref.\ \cite{CLER90}. This formula can be
interpreted in two ways. On one hand, Milton has shown \cite{MILT85} that it
yields the exact effective conductivity of an {\it ad hoc} ideal medium
built with a particular hierarchical structure. On the other hand, the
Bruggeman formula can be seen as a first (one-body) self-consistent
approximation to general disordered symmetric cell-materials
\cite{MILL69}, to which systematic corrections could be worked
out. However, the Bruggeman approximation is very different from a
mean-field theory of random conducting media. Indeed, an exact
mean-field calculation on the Bethe lattice \cite{STIN73,STRA74} predicts a
percolation threshold $p_c\sim 1/(2d)$ and exponents $s=0$ and
$t=3$. These exponents are exact for $d\geq 6$, as well as the
asymptotic behavior of the threshold when $d\to\infty$ (at least, for
the hypercubic lattice \cite{STAU92} to which a continuum theory
naturally compares \cite{NOTE1}). These values are also obtained in a
more systematic mean-field theory for random resistor networks
\cite{STEP77}.
The remarkable discrepancy between the mean-field results and
Bruggeman's formula indicates the ambiguous status of the Bruggeman
theory. As a matter of fact, in spite of various (mostly perturbative)
investigations \cite{HORI75,BERG81,LUCK91} in order to precise its
theoretical status, the reasons for the peculiar critical behavior of
Bruggeman's formula are not completely cleared up.
More surprisingly, another
self-consistent effective-medium approximation \cite{HORI75,HORI77} due
to Hori and Yonezawa (HY), obtained for the same type of media by means
of a completely different approximation scheme (and later derived by
functional methods \cite{BART93}), exhibits the same exponents $s=t=1$
and a similar threshold behavior $p_c=1-\exp(-1/d)\sim 1/d$.

Apart from phenomelogical variants, and up to our knowledge,
the Bruggeman and HY effective-medium formulas are
the only ones obtained from the equations of
electrostatics in continuous media which are able to describe, at least
qualitatively, the overall features of a percolation transition in any
dimension. One intriguing question concerns the possibility of deriving
alternative effective-medium formulas from a continuum formulation, which
do not lead to the seemingly unavoidable values $s=t=1$ and $p_c\sim
1/d$. As we show in this paper, such a possibility exists. Our starting
point is the path integral approach recently put forward by
Barth\'el\'emy and Orland\cite{BART98}, where the effective-medium
problem is recast in a functional form. The problem reduces to
compute a free-energy: roughly, the logarithm, averaged over the
disorder, of a functional integral of Boltzmann-like weights, over
allowed field configurations (which include boundary conditions). The
average of the logarithm is carried out with the replica method (already
used in Ref.\ \cite{STEP77}). In Ref.\ \cite{BART98}, the authors showed
that the path integral formulation
allows one to easily recover the second-order
weak-disorder expansion of the effective permittivity of nonlinear
composites \cite{BLUM91}.

However, this formulation
has not yet been used to derive self-consistent estimates.
In this paper, we show how this can be done.
After a presentation of the functional approach to the homogeneization
problem, and of the replica method (Sec.\ \ref{pifotp}),
we discuss self-consistent effective-medium
approximations (Sec.\ \ref{posa}). As usual
in such approximations, a background reference medium is introduced under the
form of an ansatz for the energy of the system,
whose parameters are to be
determined self-consistently (Sec.\ \ref{o}).
The new feature here is that the ansatz
contains a replica-coupling term, whose significance is
explained (Sec.\ \ref{rcacota}).
The self-consistency conditions to determine its parameters
are next discussed,
and two types of effective-medium formulas are identified (Sec.\ \ref{sc}):
one in which the replica couplings are cancelled
(hereafter referred to as ``type 1''),
and the other one with non-zero replica couplings (``type 2'').
Two different approximations are then worked out for each type
(Sec.\ \ref{tm}). It is found that
type 1 generates the Bruggeman and HY formulas,
whereas type 2 brings in two new effective-medium formulas which
are ``replica coupling counterparts'' of the previous ones.
They possess exponents $s=0$, $t=2$, and
a threshold $p_c\sim 1/(2d)$ (Sec.\ \ref{nef}).
These new formulas are discussed in Sec.\ \ref{d},
where numerical results are presented
before we conclude in Sec.\ \ref{c}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Path integral formulation of the problem}
\label{pifotp}
The effective properties of a random
conducting medium can be defined with the help of the
total dissipated power in the medium\cite{WILL86,SANC87,PONT92b}.
In terms of the electric field
$E(x)$, the dissipated power $w$ in the system of volume $V$ reads
\begin{equation}
W[E]=\int_V dx\,w_x\bigl(E(x)\bigr),
\end{equation}
where $w_x$ is the local power density.

{\em Hereafter, we take the volume $V$ of the sample equal to one}. In
heterogeneous materials, $w_x$ depends on constitutive
parameters randomly varying from point to point. For linear conducting
media with $j(x)=\sigma(x)E(x)$, where $\sigma$ is the local random
conductivity and $j$ is the electric current, we have
\begin{equation}
\label{linmed}
w_x\bigl(E(x)\bigr)=\sigma(x)E^2(x)/2.
\end{equation}
In the analogous effective
permittivity problem the dissipated power is replaced by the stored
energy $\varepsilon(x)E^2(x)/2$ ($\varepsilon$ is the
permittivity). For this reason, we shall abusively refer to $w_x$ as
the ``energy density'' hereafter. In the nonlinear problem, $w_x(E)$
is a non-quadratic function of $E$.

An alternative to solving Maxwell's equations is to minimize the total
energy $W$ subjected to the two constraints \cite{WILL86,SANC87}: (i) $E=- \nabla\phi$ and
(ii) $\overline{E}=E_0$; here, the bar stands for a spatial average,
and $E_0$ is a constant applied electric field. The
minimum, $W^*\bigl(E_0\bigr)$, is expected to be self-averaging, as
occurs for the free-energy in disordered systems. We can therefore write
\begin{equation}
\label{mini}
W^*(E_0)=\left\langle\mathop{\text{min}}_{\overline{E}
=E_0 \atop{E=-\nabla\phi}} W[E]\right\rangle
\end{equation}
where the brackets $\langle\cdot\rangle$ denote the disorder average.
$W^*(E_0)$ is the energy in a homogeneous medium
characterized by an effective
constitutive law \cite{WILL86}
\begin{equation}
\label{constef}
\left\langle j\right\rangle={\partial W^*(E_0)\over \partial
E_0}=\sigma_{\text{eff}} E_0.
\end{equation}
The second equality defines the effective conductivity of the medium.

The problem thus reduces to computing the average of
the constrained minimum of a functional of the electric field. The
electric field derives from a potential and has a fixed mean value. We
can rewrite the constrained minimum in (\ref{mini}) using a path
integral
\begin{equation}
\mathop{\text{min}}_{\overline{E}=E_0\atop{E=-\nabla\phi}}
W[E]=-\lim_{\beta\rightarrow\infty}
\frac{1}{\beta}\mathop{\text{ln}}
\int{\cal D}E\,{\cal D}\phi\,
\delta(E+\nabla\phi)
\delta(\overline{E}-E_0)
e^{-\beta W[E]}.
\end{equation}
The minimum can be interpreted as the ground state energy
associated to the partition function
\be
\label{ZZ}
Z=\int \tilde{\cal D}E\,
e^{-\beta W[E]},
\end{equation}
where we have used the shorthand notation
\begin{equation}
\tilde{\cal D}E={\cal D}E\,\delta(\overline{E}-E_0)\int {\cal D}\phi\,
\delta(E+\nabla\phi)
\end{equation}
for the constrained functional measure.
We need to compute the average of the logarithm of (\ref{ZZ}).  In order
to proceed, we introduce replicas \cite{EDWA75,MEZA87} and use the identity
$\left\langle\ln Z\right\rangle=
\lim_{n\rightarrow 0}(\left\langle Z^n\right\rangle-1)/n$, hence
\begin{equation}
\label{wstar0}
W^*=-\lim_{\beta\rightarrow\infty}
\lim_{n\rightarrow 0}
\frac{1}{n\beta}
(\left\langle Z^n\right\rangle-1).
\end{equation}
The limits do not commute. The equivalent form
\begin{equation}
\label{wstar}
W^*=-\lim_{\beta\rightarrow\infty}\lim_{n\rightarrow
0}\frac{1}{n\beta}\ln\left\langle Z^n\right\rangle
\end{equation}
can be used as well. The replica method
relies on the fact that one can easily compute
the replicated partition function
$\langle Z^n\rangle$ for $n$ integer, and subsequently take the limit
$n\rightarrow 0$. The main quantity of interest therefore is
\be
\left\langle Z^n\right\rangle=
\int\prod_{\alpha=1}^{n}
\tilde{\cal D}E^{\alpha}\,\left\langle
e^{-\beta \sum_{\alpha=1}^{n} W[E^{\alpha}]}
\right\rangle.
\end{equation}
Denoting the replicated measure by
$\tilde{\cal D}\bigl(E^\alpha\bigr)=\prod_{\alpha=1}^{n}
\tilde{\cal D}E^{\alpha}$, the average $\langle Z^n\rangle$ can be written
in terms of an ``effective Hamiltonian''
\be
\label{zneff}
\langle Z^n\rangle=
\int\tilde{\cal D}\bigl(E^\alpha\bigr)\, e^{-\beta{\cal H}_e},
\ee
with
\begin{equation}
\label{heff}
{\cal H}_e=-{1\over\beta}\ln\left\langle e^{-\beta\sum_{\alpha=1}^{n}
W[E^{\alpha}]}
\right\rangle.
\end{equation}

For simplicity, we restrict ourselves to cell materials where the local
properties are statistically uncorrelated from site to site. Volume
integrals may then to be identified with sums over sites (each
pertaining to one cell) according to the correspondence $\int
dx\leftrightarrow v\sum_x$, where $v$ is an infinitesimal cell volume
(which defines the microscopic correlation length of the
problem). Then, ${\cal H}_e$ simplifies to
\begin{equation}
\label{calf}
{\cal H}_e=-{1\over\beta}\int\frac{dx}{v} \ln\left\langle
e^{-\beta v\sum_\alpha w_x\bigl(E^\alpha(x)\bigr)}\right\rangle.
\end{equation}

Note that our discussion in Sec.\ III will be specialized to binary
disorder for which the constitutive parameters can take only two
values (but the proofs are general). That is, we assume that the
local energy density is distributed according to the probability
distribution
\be
P\bigl(w=w_x(E)\bigr)=p\delta \bigl(w-w_1(E)\bigr)+
q\delta \bigl(w-w_2(E)\bigr).
\ee
(where $q=1-p$). With this choice,
\be
\label{znmoyen}
{\cal H}_e=-{1\over\beta}\int\frac{dx}{v} \ln
\left[p e^{-\beta v\sum_{\alpha=1}^n w_1\left(E^{\alpha}(x)\right)}
+qe^{-\beta v \sum_{\alpha=1}^n w_2\left(E^{\alpha}(x)\right)}\right].
\end{equation}
The above formalism applies to any form of the energy density, and in
particular to nonlinear media \cite{BERG92,PONT92a,YU94,PONT97}.
A method for extracting from the path integral the second-order
weak-contrast perturbation expansion of the effective potential
$W^*(E_0)$, for nonlinear media, has been introduced in Ref.\ \cite{BART98}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Principle of self-consistent approximations}
\label{posa}
In this paper, we consider the linear problem only. This section is devoted to self-consistent approximations to $W^*$. We first present
the principle for building such approximations through the
introduction of a trial Hamiltonian. Then, we
discuss the choice of a trial Hamiltonian with replica couplings.
Finally, we explain how to exploit these replica couplings
in order to obtain two kinds of self-consistent formulas.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Overview}
\label{o}
 The common ingredient to the approximations discussed
below is the introduction of a linear comparison medium
described by a trial Hamiltonian ${\cal H}_0$ which is quadratic in the
electric field and non-random, e.g.\ the one-parameter ansatz
\begin{equation}
\label{trybrug}
{\cal H}_0=\frac{\sigma_0}{2}\int dx\,\sum_{\alpha}{E^{\alpha}}^2(x),
\end{equation}
where $\sigma_0>0$ is to be determined by an
appropriate self-consistency condition.
This Hamiltonian is that of a (replicated) homogeneous medium,
but without couplings between replicas. Its meaning and that of
other possible choices with replica couplings are discussed below.

The partition function $\langle Z^n\rangle$ can be rewritten as
\begin{equation}
\left\langle Z^n\right\rangle =
\frac{
\int{\tilde{\cal D}}\bigl(E^{\alpha}\bigr)\, e^{-\beta({\cal
H}_e-{\cal H}_0)}e^{-\beta{\cal H}_0}
}{
\int{\tilde{\cal D}}\bigl(E^{\alpha}\bigr)\,e^{-\beta{\cal H}_0}
}
\int{\tilde{\cal D}}\bigl(E^{\alpha}\bigr)\,e^{-\beta{\cal H}_0}
,
\end{equation}
or, with another notation
\begin{equation}
\label{znmoyen2}
\left\langle Z^n\right\rangle =
\left\langle
e^{-\beta({\cal H}_e-{\cal H}_0)}
\right\rangle_0 Z_0,
\end{equation}
where $Z_0$ is the partition function associated to ${\cal H}_0$, and
$\left\langle\cdot\right\rangle_0$ stands for the functional average with
weights $e^{-\beta{\cal H}_0}/Z_0$. Equ.\ (\ref{wstar}) thus reads
\begin{equation}
\label{wsplit}
W^*=W_0+\Delta W,
\end{equation}
where
\begin{eqnarray}
\label{wo}
W_0(E_0)&=&-\lim_{n\to 0\atop
\beta\to\infty}\frac{1}{n\beta}\ln Z_0,\\
\label{dwo}
\Delta W(E_0)&=&-\lim_{n\to 0\atop \beta\to\infty}\frac{1}{n\beta}
\ln \left\langle e^{-\beta({\cal H}_e-{\cal H}_0)}
\right\rangle_0.
\end{eqnarray}
The quantity $\Delta W(E_0)$ is difficult to compute (an exact evaluation
would lead to the exact result for the effective conductivity),
and we have to resort to approximations.

A natural self-consistency condition for ${\cal H}_0$
is
\begin{equation}
\label{sccond}
\Delta W(E_0)=0,
\end{equation}
which completely determines ${\cal H}_0$ in the case where it
depends on one single parameter, as in (\ref{trybrug}). For more general
choices of ${\cal H}_0$ with several free parameters, (\ref{sccond})
only provides a relation between these parameters, and additional
considerations are in order to determine them all. First of all,
we have to precise the form of the ansatz to be used in our calculations.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Replica couplings and choice of the ansatz}
\label{rcacota}
We deduce here the form of the trial Hamiltonian ${\cal H}_0$ from an
analysis of the effective Hamiltonian ${\cal H}_e$. Eq.\ (\ref{heff})
shows that the effective Hamiltonian is non-random but that the
average over disorder introduced a coupling between different
replicas. The meaning of these couplings is more transparent if we
carry out an expansion of (\ref{heff}) around the average field
$\overline{E}=E_0$ as in the weak-contrast expansion \cite{BART98}.
With $\partial_i=\partial/\partial E_i$ and $\Delta
E^\alpha=E^\alpha-E_0$ we have
\begin{eqnarray}
{\cal H}_e&=&n\langle w_x(E_0)\rangle\nonumber\\
&+&\frac{1}{2}
\left[ \sum_{\alpha}\int dx\,a_{ij}\Delta E^\alpha_i(x)
\Delta E^\alpha_j(x)-\beta\sum_{\alpha,\gamma}
\int dx dy\, c^{(2)}_{ij}(x-y)\Delta E^\alpha_i(x)
\Delta E^\gamma_j(y)\right] +\cdots,
\end{eqnarray}
where
\begin{eqnarray}
\label{aa}
a_{ij}&=&\langle \partial^2_{ij} w_x(E_0)\rangle,\\
\label{bb}
c^{(2)}_{ij}(x-y)&=&\langle\partial_{i} w_x(E_0)\partial_{j} w_y(E_0)
\rangle-\langle \partial_{i} w_x(E_0)\rangle\langle\partial_{j}
w_y(E_0)\rangle.
\end{eqnarray}
The first non-zero replica-coupling term is proportional to $\beta
c^{(2)}$. We thus see that the coupling between replicas acts only
within clusters defined by $n$-point connected correlation functions
$c^{(n)}$, and accounts for the fluctuations of the electric field in
these clusters. The replica coupling would vanish if there were no
disorder at all. In the limit where the size of the region defined by
$c^{(2)}$ shrinks to zero -- which means that the system is observed
at a macroscopic level, we can approximate
\begin{equation}
c^{(2)}_{ij}(x-y)\simeq v c^{(2)}_{ij}(0)\delta(x-y),
\end{equation}
and we recover the expansion
\begin{equation}
\label{expan}
{\cal H}_e=n\langle w_x(E_0)\rangle+\frac{1}{2}\int dx\,\left[
\sum_{\alpha}a_{ij}\Delta E^\alpha_i(x)\Delta E^\alpha_j(x)-v\beta
\sum_{\alpha,\gamma}c^{(2)}_{ij}(0)\Delta E^\alpha_i(x)\Delta
E^\gamma_j(x)\right] +\cdots.
\end{equation}
which could directly be obtained from (\ref{calf}). The presence of
$v$ in front of the replica coupling term is the macroscopic remnant
of a microscopic average having been taken within a two-particle
cluster, of center $x$ and volume $v$. This discussion therefore
enlightens a relation between replica coupling and the electric field
fluctuations within clusters.

Expansion (\ref{expan}) suggests a two-parameter
replica-symmetric ansatz of the form
\begin{equation}
\label{ansatz}
{\cal H}_0=\frac{1}{2} \sum_{\alpha\gamma}\int dx\, M^{\alpha\gamma}
E^\alpha_i  E^\gamma_i,
\end{equation}
where
\begin{equation}
\label{matmok}
M^{\alpha\gamma}=\sigma_0\delta_{\alpha\gamma}-v\beta Q E_0^2.
\end{equation}
The free parameters are $\sigma_0$ and $Q$. Note that $Q$ has the
dimension of a squared conductivity, because it is related to a
quantity relative to two points. For simplicity, the ansatz $M$ is
diagonal in the euclidean vector space. However, we tried calculations
with a tensorial structure reproducing that of $a_{ij}$ and
$c^{(2)}_{ij}$ in Eqs.\ (\ref{aa}), (\ref{bb}); but, apart from a
different normalization for $Q$, no differences showed up in the final
effective-medium theories (as far as linear media are concerned).

An interesting feature of the ansatz (\ref{matmok}) is that, though
being non-random, it embodies underlying disorder through its replica
couplings. In order to understand this point, we compute
$W_0(E_0)$ given by (\ref{wo})
\begin{equation}
\label{wopath}
W_0(E_0)=-\lim_{n\to 0\atop \beta\to\infty}\frac{1}{n\beta}\ln \int
\tilde{\cal D} E\,e^{-\frac{\beta}{2}\int
dx\,\left(\sigma_0\sum_\alpha{E^\alpha}^2-v\beta Q E_0^2
\sum_{\alpha,\gamma}{E^\alpha}\cdot{E^\gamma}\right)}.
\end{equation}
After writing $E=E_0-\nabla \phi$, and going to the Fourier transform
of $\phi$ \cite{NOTE2}, we arrive at
\begin{eqnarray}
W_0(E_0)&=&-\lim_{n\to 0\atop \beta\to\infty}\frac{1}{n\beta}\ln
\left[(\mathop{\text{Det}}M)^{-1/2v}
e^{-\frac{\beta}{2}\sum_{\alpha\gamma}
M_{\alpha\gamma}E_0^2}\right]\nonumber\\
\label{wocomp}
&=&\frac{1}{2}\sigma_0(1-Q/\sigma_0^2) E_0^2.
\end{eqnarray}

Carrying out the derivative of (\ref{wopath})
with respect to $\sigma_0$, we obtain
\begin{equation}
\lim_{n\to 0\atop \beta\to\infty}\frac{1}{n}\left\langle\sum_\alpha
\langle{E^\alpha}^2\rangle\right\rangle_0=E_0^2+ \frac{Q}{\sigma_0^2}
E_0^2,
\end{equation}
where volume averages $\overline{E^2}$ have been replaced by statistical
ones, the microscopic size $v^{1/d}$ being much smaller than
that of the system, $V^{1/d}=1$.
All the replicas are equivalent, and the functional average $\langle\cdot\rangle_0$ selects in the limit $\beta\to\infty$ the real
field in the medium. Hence, setting $\Delta E=E-E_0$, the previous
equation leads to
\begin{equation}
\label{qq}
\frac{\langle\Delta E^2\rangle}{E_0^2}=\frac{Q}{\sigma_0^2}.
\end{equation}
which implies that $Q\geq 0$. When $Q\not=0$, the electric field
fluctuates in the medium, whereas it is uniform when $Q=0$. The ansatz
${\cal H}_0$ therefore represents a medium which is homogeneized
(because it is non-random), but which nonetheless
accounts for field fluctuations.
We thus expect new effective medium approximations
when the replica coupling $Q$ is non zero.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Self-consistency}
\label{sc}
Up to this point the discussion focused on the ansatz itself, without
referring to ${\cal H}_e$. In particular, $\sigma_0$ was treated as a
mere number. We now discuss what happens when self-consistency is
imposed, within some approximation scheme. The medium is made of $N$
phases labelled by $\nu$, of respective conductivities $\sigma_\nu$
and volume concentrations $p_\nu$. The self-consistency relation
$\Delta W(E_0)=0$, which imposes constraints on the ansatz,
determines $Q$ as a function $Q=Q(\sigma_0,\{\sigma_\nu\})$.
Then $W^*=W_0$ and, with (\ref{constef}),
\begin{equation}
\label{sigeff}
\sigma_{\text{eff}}=\sigma_0\left[1-\frac{Q(\sigma_0,\{\sigma_\nu\})}
{\sigma_0^2}\right].
\end{equation}

Suppose now that $\sigma_0=\sigma_0(\{\sigma_\nu\})$ is determined
by an additional condition (to be precised below). Using the exact
formula \cite{BERG78}  (cf.\ Appendix \ref{qfotf})
\begin{equation}
\label{berg}
\frac{\langle\Delta E^2\rangle}{E_0^2}=\sum_\nu \frac{\partial
\sigma_{\text{eff}}}{\partial \sigma_\nu} -1,
\end{equation}
the fluctuations of the electric field deduced from (\ref{sigeff})
can be written
\begin{equation}
\label{qberg}
\frac{\langle\Delta
E^2\rangle}{E_0^2}=\sigma_{\text{eff}}'(\sigma_0)\left(\sum_\nu
\frac{\partial \sigma_0}{\partial
\sigma_\nu}-1\right)+\frac{Q}{\sigma_0^2}-\frac{1}{\sigma_0}
\left[\frac{\partial Q}{\partial
\sigma_0}+\sum_\nu\left(\frac{\partial Q}{\partial
\sigma_\nu}\right)_{\sigma_0}\right]
\end{equation}
where the last derivative is performed at constant $\sigma_0$.  This
expression distinguishes between different contributions to the field
fluctuations: (i) the first term represents fluctuations coming from
the ``macroscopic'' background effective medium $\sigma_0$; (ii) the
second one is that already found in (\ref{qq}), and would be the only
one if $Q$ were independent from $\sigma_0$, and if
$\sigma_0$ were equal to $\langle\sigma\rangle$,
 the trivial value corresponding
to a non-fluctuating reference medium for a multiphase composite, cf.\
Appendix \ref{qfotf}; (iii) and finally a third
term comes from the dependence of $Q$ on $\sigma_0$ and
$\sigma_\nu$. Both last terms are, according to the interpretation of
replica coupling developped in the previous section, of
``microscopic'' origin.

We now turn to the determination of $\sigma_0(\{\sigma_\nu\})$. A
first obvious self-consistency condition for $\sigma_0$ is $Q\equiv 0$,
so that $\sigma_{\text{eff}}=\sigma_0$. The effective-medium formulas obtained
this way are referred to as ``type 1'' hereafter. As is shown
below, to this type pertain the Bruggeman and HY formulas.

``Type 2'' effective-medium formulas are obtained by taking $\sigma_0$
as the solution of $\sigma_{\text{eff}}'(\sigma_0)=0$, and by using
this value in $\sigma_{\text{eff}}$. According to (\ref{qberg}), this
procedure makes the effective-medium insensitive to the fluctuations
generated in the reference medium $\sigma_0$, so that relevant fluctuations
only come from $Q$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Two approximations}
\label{tm}
In this section, the ideas introduced above are used within two
different approximations to $\Delta W(E_0)$, based on the ansatz
(\ref{ansatz}), (\ref{matmok}). For each approximation to $\Delta W$,
``type 1'' and ``type 2'' formulas are obtained.  Herafter,
$q=Q/\sigma_0^2$, so that
\begin{equation}
\label{sigefq}
\sigma_{\text{eff}}=\sigma_0(1-q).
\end{equation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{One-impurity approximation}
\label{oia}

We first consider a ``one-impurity'' (or ``local'') calculation. The
Bruggeman formula emerges as the ``type 1'' effective-medium formula
in this approximation, which is not suprising since it can be seen as
a one-site (self-consistent) theory \cite{HORI75,LUCK91}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Approximation scheme}
The approximation for $\Delta W(E_0)$ [Eq.\ (\ref{dwo})], detailed in
Appendix \ref{appb}, is a one-impurity approximation where interactions
between different points are ignored. Let us denote by $w_0$ the
trial Hamiltonian density, which depends on all the replicas,
defined from (\ref{ansatz}), (\ref{matmok}) by
\begin{equation}
\label{hdens}
{\cal H}_0\equiv\int dx\, w_0[E(x)].
\end{equation}
Here and in Appendix \ref{appb}, the notation $[\cdot]$ indicates
a dependence with respect to all the replicas.
Setting
\begin{equation}
\label{dwx}
\Delta w_x[E(x)]=\sum_\alpha w_x\bigl(E^\alpha(x)\bigr)-w_0[E(x)],
\end{equation}
the one-impurity approximation results in
\begin{equation}
\label{oneimp}
\left\langle e^{-\beta({\cal H}_e-{\cal H}_0)}\right\rangle_0\simeq
1+\frac{1}{v}\left\langle \left\langle e^{-\beta v \Delta w_x[E(x)]}\right\rangle_0-1\right\rangle.
\end{equation}
Because of statistical translation invariance, the final result is
independant of the point $x$. The right-hand side can be computed
exactly for any potential $w_x$ in the limit $\beta\to\infty$ using
a saddle-point method. Setting $\Delta\sigma=\sigma-\sigma_0$
and
\begin{equation}
\mu=\left(1+\frac{\Delta\sigma}{d\sigma_0}\right)^{-1},
\end{equation}
we arrive at
\begin{equation}
\label{dwoi}
\Delta W(E_0)=\frac{1}{2}\langle\Delta\sigma \mu\rangle E_0^2
+\frac{q}{2}\langle \sigma\mu\rangle E_0^2.
\end{equation}
The condition $\Delta W(E_0)=0$ yields
\begin{equation}
\label{qbrug}
q=-\frac{\langle\Delta\sigma \mu\rangle}{\langle \sigma\mu\rangle}.
\end{equation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Type 1 formula: Bruggeman's}
Letting $q\equiv 0$ amounts to imposing $\langle\Delta\sigma
\mu\rangle=0$, which is nothing but the Bruggeman equation
\begin{equation}
\left\langle\frac{\sigma-\sigma_0}{\sigma+(d-1)\sigma_0}\right\rangle=0.
\end{equation}
The Bruggeman equation can also be written $\langle\mu\rangle=1$, or $\sigma_0=\langle\sigma\mu\rangle$ if $d\not= 1$, or $\sigma_0=\langle\sigma\mu\rangle/\langle\mu\rangle$.
The last expression is suitable for computing $\sigma_0$ iteratively
(starting, e.g., from $\sigma_0=\langle\sigma\rangle$) in any
dimension. The Bruggeman conductivity $\sigma_{\text{eff}}
=\sigma_0$ possesses a percolation threshold $p_c=1/d$, and critical
exponents $s=t=1$ \cite{CLER90}.

The fluctuations computed from (\ref{berg}) read
\begin{equation}
\frac{\langle\Delta E^2\rangle}{E_0^2}=
\frac{\sigma_0\langle\mu^2\rangle}
{\langle\sigma\mu^2\rangle}.
\end{equation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Type 2 formula}
We now let $q\neq 0$ and given by (\ref{qbrug}) and
$\sigma_{\text{eff}}(\sigma_0)=\sigma_0\bigl(1-q(\sigma_0)\bigr)$. The
equation $\sigma_{\text{eff}}'(\sigma_0)=0$ reads
\begin{equation}
\label{eqsigo}
\sigma_0=\frac{\langle\sigma\mu\rangle}{\langle\mu\rangle}
\left(1+\frac{\langle\mu\rangle\langle\sigma^2\mu^2\rangle
-\langle\sigma\mu^2\rangle\langle\sigma\mu\rangle}
{2d\langle\sigma\mu\rangle^2}\right).
\end{equation}
Like Bruggeman's, this equation is easily solved by iterations starting
from $\sigma_0=\langle\sigma\rangle$. The iterations then always
converge to the physical solution, which we denote by
$\sigma_0^*$. The effective conductivity thus is
$\sigma_{\text{eff}}=\sigma_0^*\bigl(1-q(\sigma_0^*)\bigr)$.  To study
its critical behavior, we consider a binary mixture, where
$\sigma=\sigma_1$ with probability $(1-p)$, and $\sigma=\sigma_2$ with
probability $p$. In the conductor/superconductor limit where
$\sigma_2\to\infty$ we find, setting $p_c=1/(2d-1)$,
\begin{equation}
\label{sols0}
\sigma_0^*=\frac{\sigma_1}{p(d-1)}\left(\sqrt{\frac{1-p}{1-p/p_c}}
-1\right)\qquad (p<p_c)
\end{equation}
and
\begin{equation}
\label{solseff}
\sigma_{\text{eff}}=2\sigma_1\frac{\left[1-dp
-\sqrt{(1-p)(1-p/p_c)}\right]}{p^2(d-1)^2}\qquad  (p<p_c).
\end{equation}
The critical concentration $p_c$ can be interpreted as a percolation
threshold, and is the same as that obtained in the mean-field model on
a Bethe lattice\cite{STEP77} with connectivity $z=2d$. Since
$\sigma_{\text{eff}}=2(2d-1)\sigma_1/(d-1)\sim(p_c-p)^0$ for
$p\lesssim p_c$, the superconductivity exponent is $s=0$. Note however
that $\sigma_0$ displays a square-root cusp at $p=p_c$.  The critical
behavior for $p>p_c$ is obtained by exmining the insulator/conductor
mixture where $\sigma_2$ is finite and $\sigma_1=0$.  Then
\begin{eqnarray}
\sigma_0^*&=&\sigma_2\frac{p/p_c-1}{2(d-1)},\\
\sigma_{\text{eff}}&=&\sigma_2\frac{(p/p_c-1)^2}{4(d-1)^2 p}
\qquad  (p>p_c).
\end{eqnarray}
Since $\sigma_{\text{eff}}\sim(p-p_c)^2$ for $p\gtrsim p_c$,
the conductivity exponent is $t=2$.

For the special case of $d=1$, $\mu=\sigma_0/\sigma$ so that
(\ref{eqsigo}) reduces to $\sigma_0=\langle1/\sigma\rangle^{-1}$,
and $q=0$. Therefore, $\sigma_{\text{eff}}=\langle1/\sigma\rangle^{-1}$,
which is the exact result. Like Bruggeman's, the new formula
is also exact to second order in the
contrast, in any dimension
\begin{equation}
\label{wcl}
\sigma_{\text{eff}}=\langle\sigma\rangle\left[1-\frac{\langle
\sigma^2\rangle-\langle\sigma\rangle^2}{d\langle\sigma\rangle^2}
+\cdots\right];
\end{equation}
and in the dilute limit where (e.g.) $p_2\ll 1$
\begin{equation}
\label{dl}
\sigma_{\text{eff}}=\sigma_1\left[1+d\frac{\sigma_2-\sigma_1}
{\sigma_2+(d-1)\sigma_1}+\cdots\right].
\end{equation}
In the discussion (Section \ref{d}), it is argued that because
of its exponents $s\not= t$, and because it is less trivial than
the Bruggeman formula (especially in the insulator/conductor case
where it does not reduce to a straight line), this formula may
constitute an easy-to-handle alternative to the latter in dimensions
$d\geq 3$. Graphical comparisons between different effective-medium
formulas are discussed in Sec.\ \ref{d}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Cumulant series approximation}
\label{asa}

In this section, we show how to recover by means of a cumulant
approximation to $\Delta W$ the effective-medium formula of HY,
together with its ``type 2'' counterpart.

\subsubsection{Approximation scheme}
We consider the first-order cumulant approximation
\begin{equation}
\label{cumul}
\left\langle
e^{-\beta({\cal H}_e-{\cal H}_0)}
\right\rangle_0\simeq
e^{-\beta\left\langle{\cal H}_e-{\cal H}_0\right\rangle_0}.
\end{equation}

We have then
\begin{equation}
\label{dwsa}
\Delta W(E_0)\simeq\lim_{n\to 0\atop \beta\to\infty}\frac{1}{n}
\left\langle
{\cal H}_e-{\cal H}_0\right\rangle_0.
\end{equation}
As is shown in Appendix C, the calculations here involve an expansion
in a series of the cumulants of the distribution of $\sigma$, whose
significance has been discussed at length in the original paper by
HY\cite{HORI77}. After some algebra, we obtain (cf.\ Appendix C)
\begin{equation}
\label{fmfodeu}
\Delta W(E_0)
=-{\sigma_0\over 2}\Bigl\{\left[1+d h_0
\bigl(1/(d\sigma_0)\bigr)\right]+d q h_0
\bigl(1/(d\sigma_0)\bigr)\Bigr\}E_0^2,
\end{equation}
where
\begin{equation}
h_0(z)=\int_0^{\infty}du\, e^{-u}\ln\langle e^{-u\sigma z}\rangle.
\end{equation}
The family of functions $h_k$ is defined in Appendix C.
The self-consistency $\Delta W(E_0)=0$ now yields
\begin{equation}
\label{q}
q=-\left[1+\frac{1}{d h_0
\bigl(1/(d\sigma_0)\bigr)}\right].
\end{equation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Type 1 formula: the Hori-Yonezawa formula}
\label{thfr}

We first consider the case with no couplings between replicas,
i.e. $q=0$. The HY formula for $\sigma_0$ reads
\begin{equation}
\label{hy}
h_0 \bigl(1/(d\sigma_0)\bigr)=-{1\over d},
\end{equation}
and the effective conductivity is $\sigma_{\text{eff}}=\sigma_0$.
It can be shown that $\sigma_{\text{eff}}$ displays a percolation
threshold
$p_c=1-\exp(-1/d)$, and exponents $s=t=1$\cite{HORI77}.
Applying (\ref{berg})  and using (\ref{deriv}), the fluctuations read
\begin{equation}
\frac{\langle\Delta E^2\rangle}{E_0^2}=
-{2 +d h_1\bigl(1/(d\sigma_0)\bigr)\over 1+d h_1\bigl(1/(d\sigma_0)\bigr)}.
\end{equation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Type 2 formula}
\label{nef}

We now consider $q\neq 0$ and determined as a function of $\sigma_0$
by (\ref{q}). Then
\begin{equation}
\label{phiosol}
\sigma_{\text{eff}}(\sigma_0)=\sigma_0
\left[2+{1\over dh_0
\bigl(1/(d\sigma_0)\bigr)}
\right].
\end{equation}
The equation for $\sigma_0$ is $\sigma_{\text{eff}}'(\sigma_0)=0$;
that is, with (\ref{deriv})
\begin{equation}
\label{deriveq}
2+{1\over d}{h_1\bigl(1/(d\sigma_0)\bigr)\over
h_0^2\bigl(1/(d\sigma_0)\bigr)}=0.
\end{equation}
In order to study the critical behavior of $\sigma_{\text{eff}}$,
we consider again a binary mixture where $\sigma=\sigma_1$ with
probability $(1-p)$, and $\sigma=\sigma_2$ with probability
$p$. In the conductor/superconductor case where
$\sigma_2\to\infty$, we have
$h_0\bigl(1/(d\sigma_0))\bigr)=\ln(1-p)-\sigma_1/(d\sigma_0)$ and
a similar equation for $h_1$, so that (\ref{deriveq}) reduces to
a second-degree polynomial equation. Its physical solution reads
\begin{equation}
\label{solinf}
\sigma_0^*={2\sigma_1\over\sqrt{1+2d\ln(1-p)}
\left[1+\sqrt{1+2d\ln(1-p)}\right]}.
\end{equation}
It is defined for $p$ less than a critical value
\begin{equation}
p_c=1-e^{-1/(2d)}.
\end{equation}
This percolation threshold is the same as the one obtained in the Potts
model at the mean-field level, and in the
mean-field theory
of Ref.\ \cite{STEP77}. Reporting (\ref{solinf}) into (\ref{phiosol}),
we arrive at
\begin{equation}
\sigma_{\text{eff}}={4\sigma_1\over
\left[1+\sqrt{1+2d\ln(1-p)}\right]^2}\qquad(p<p_c).\\
\end{equation}
Since $\sigma_{\text{eff}}\propto (p_c-p)^0$
for $p\lesssim p_c$, the superconductivity exponent is $s=0$.

In the opposite insulator/conductor case, where $\sigma_1=0$
and $\sigma_2$ is finite, the solution for $p>p_c$ can only be found perturbatively around the percolation threshold. Expanding the
logarithm in $h_0$ and $h_1$ as
\begin{equation}
\ln\left[(1-p)+p
e^{-u\sigma_2/(d\sigma_0)}\right]=\ln(1-p)+\sum_{l\geq
1}{(-1)^{l-1}\over l}\left({p\over 1-p}\right)^l
e^{-lu\sigma_2/(d\sigma_0)},
\end{equation}
and defining
\begin{equation}
A(x)=\sum_{l\geq 1} {(-1)^{l-1}\over l^2} x^l=\int_0^x {dt\over t}
\ln(1+t),
\end{equation}
we find that
\begin{equation}
\sigma_0^*={\sigma_2\over 4 d^2 A\bigl(p_c/(1-p_c)\bigr)}(p-p_c)+O
\left((p-p_c)^2\right),
\end{equation}
and that
\begin{equation}
\sigma_{\text{eff}}={\sigma_2\over
4d^2 A\bigl(p_c/(1-p_c)\bigr)}(p-p_c)^2+O\left((p-p_c)^3\right)
\qquad (p\gtrsim p_c),
\end{equation}
where the conductivity exponent is $t=2$. Hence, as in the previous
``one-impurity'' approximation, the replica-coupling ansatz yields
critical exponents $s=0$, $t=2$, and an asymptotic dependence of the
threshold $p_c\sim 1/(2d)$ when $2d\gg 1$. One can easily check that
this new ``type 2'' effective-medium formula is exact to second order
in the weak-contrast limit and in the dilute limit.

After a few manipulations, we now obtain with (\ref{berg}) and (\ref{q})
\begin{equation}
{\left\langle \Delta E^2\right\rangle\over \left\langle E\right\rangle^2}
=2q\left(1+\frac{q}{2}\right).
\end{equation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Discussion}
\label{d}

We plot in Fig. \ref{fig1} (resp.\ \ref{fig2}) the ``type 2'' scaled
conductivities $\sigma_{\text{eff}}/\sigma_1$ versus $p$, the volume
fraction of material 2, for a dielectric ratio $\sigma_2/\sigma_1=10$
(resp.\ $\sigma_2/\sigma_1=1000$). We also show the Hashin-Shtrikman
(HS) bounds \cite{HASH62}, the Hori-Yonezawa formula which comes from
a cumulant series (CS) approximation, and the one-impurity (OI)
Bruggeman formula.  The dimension is $d=2$. Figs.\ \ref{fig3} and
\ref{fig4} display similar plots for $d=3$.

For moderate contrast (Figs.\ 1 and 3), we observe that all four
self-consistent formulas lie close to each other. This is a
consequence of the fact that they are exact to second order in the
contrast. Also, for any contrast, the slopes at $p=0$ and $p=1$ are
all identical, which is a consequence of the fact that they are exact
to second order in the dilute limit $p\to 0$ (the expression near
$p=1$ is obtained by replacing $p$ by $p-1$ and by interchanging
$\sigma_1$ and $\sigma_2$). We also observe that the HS bounds are
satisfied in each case considered. However, the formulas obtained via
the cumulant series summation, i.e.\ both the HY formula and its
``type 2'' counterpart, do not reduce to the exact result
$\sigma_{\text{eff}}=\langle 1/\sigma\rangle^{-1}$ in dimension 1 (not
shown). This exact result is also the common value of the HS bounds
for $d=1$. Hence, formulas derived from the cumulant series
approximation do not obey the HS bounds in dimension $d=1$. On the
other hand, both the Bruggeman formula and its ``type 2'' counterpart
do reduce to the exact result when $d=1$, and can be seen to always
obey the HS bounds whatever $d$ is. The one-impurity approximation
scheme therefore appears to be of better physical relevance for all
dimensions, than the cumulant series approximation.

We now discuss the critical behavior. First of all, the percolation
thresholds found in the``type 2'' formulas are
$p_c=1-\exp\bigl(-1/(2d)\bigr)$ (cumulant series) and $p_c=1/(2d-1)$
(one-impurity). These thresholds are the percolation thresholds of the
Potts model, and that of the Bethe lattice model, respectively. Both
thresholds decrease as $1/(2d)$ when $d\to\infty$, which is the exact
asymptotics. The Bethe and Potts models are mean-field models, where
emphasis is put on fluctuations in the couplings between a given site
and its neighbours. On the contrary, in effective-medium theories,
interactions between impurities are taked into account through the
self-consistent background medium. Such interactions are more
important for low dimensions. Effective-medium theories therefore
overestimate interactions in high dimensions, whereas mean-field
models are expected to underestimate them in low dimensions. Above the
upper critical dimension where mean-field models are accurate,
interactions between impurities become irrelevant. According to this
discussion, our ``type 2'' formulas appear as hybrids between
mean-field and usual effective-medium theories, and are expected to be
mostly relevant in dimensions intermediate between $d=1$ and the upper
critical dimension $d=6$.  Indeed, the condition
$\sigma'_{\text{eff}}(\sigma_0)=0$ minimizes the influence of the
background medium and, according to the interpretation developped in
Sec.\ \ref{rcacota}, replica coupling has to do with couplings
between neighboring points. The reason for which the introduction of a
replica-coupling ansatz yields the exact thresholds of mean-field
theories will have to be clarified in the future.  In Fig.\
\ref{fig5}, we plot the quadratic fluctuations $\left\langle\Delta
E^2\right\rangle/\left\langle E\right\rangle^2$ as a function of
$p$. The fluctuations in the ``type 2'' estimates are greatly reduced
compared to those of the Bruggeman and HY formula. This is consistent
with the fact that the influence of the background is reduced.

``Type 1'' formulas give exponents $s=t=1$, while for ``type 2''
formulas they are $s=0$, $t=2$. Mean-field theories yield $s=0$, $t=3$
which are the exact values for $d\geq 6$.  It is interesting to
compare these values to exact bounds deduced from the
Nodes-Links-Blobs (NLB) model, in all dimensions. The NLB model is
currently accepted as a good one for the backbone structure of real
random resistor networks\cite{STAU92}. The bounds read
\begin{mathletters}
\label{expbounds}
\begin{eqnarray}
&&t\ge 1+(d-2)\nu,\\
&&s\ge 1+(2-d)\nu,
\end{eqnarray}
\end{mathletters}
where $\nu>0$ is the correlation length exponent: $\xi\propto
|p-p_c|^{-\nu}$. They hold for $2\leq d\leq 6$, whereas for $d>6$ the
right-hand sides in (\ref{expbounds}) are fixed to their $d=6$
values. These bounds follow, e.g., from comparing the lower and upper
exact bounds obtained in Ref.\ \cite{WRIG86} for the noise exponent
$\kappa$ in weakly nonlinear networks, within the NLB scheme. They
are satisfied by simulation results \cite{WRIG86}.
Using the usual
effective-medium values $s=t=1$ in (\ref{expbounds}) implies the
absurd value $\nu=0$, save for $d=2$ where a finite value of $\nu$ is
allowed. Though information about the correlation length $\xi$ (and
therefore about $\nu$) is not included in the Bruggeman nor in the HY
formulas, the above bounds show that, as long as they are meant to
model percolating systems obeying the NLB picture, these formulas are
truly adequate only in dimension $d=2$ -- and $d=1$ where the Bruggeman
formula is exact. As to ``type 2'' formulas, we insert the
values $s=0$, $t=2$ into (\ref{expbounds}) and deduce that
$\nu=1/(d-2)$, a reasonable expression for $d\geq 3$ only. If we
furthermore insist on having $\nu\geq 1/2$ as in real systems, these
heuristic arguments restrict the range of validity of the new formulas
to $d=3,4$. Note, moreover, that only in dimension $d=2$ are the
exponents equal: $s=t$, because of self-duality \cite{STRA77}.  A
formula with unequal exponents therefore is expected to be essentially
relevant to dimensions $\geq 3$.

We also quote theoretical bounds for $t$ due to Golden, valid for
hierarchical NLB models: $1\leq t\leq 2$ for $d=2,3$ and $2\leq t\leq
3$ for $d\geq 4$ \cite{GOLD90}. The above analysis is consistent with
these bounds, and can be summarized as a set of prescriptions for
using the ``best'' available effective medium theories, as far as a
non-conflicting critical behaviour is concerned: for $d=1$,
Bruggeman's formula, or its ``type 2'' counterpart are exact; for
$d=2$, the Bruggeman or HY formulas are adequate; for $d=3,4$, ``type
2'' formulas are applicable; finally, for $d\geq 5$ mean-field
theories would be the most relevant.  Estimates or exact values for
the exponents are \cite{STAU92}: $(\nu,s,t)=(4/3,1.3,1.3)_{d=2}$,
$(0.88,0.73,2.00)_{d=3}$, $(0.68,0.4,2.4)_{d=4}$,
$(0.57,0.1,2.7)_{d=5}$, $(1/2,0,3)_{d\geq 6}$. These values support
our prescriptions.

An interesting observation is that actually both ``type 1''
and ``type 2'' formulas can be given by a variational formulation as
\begin{eqnarray}
\label{min}
\sigma_{\text{eff}}^{\text{type 1}}&=& \min_{\sigma_0\geq 0\atop 0\leq
q(\sigma_0)\leq 1}\sigma_{\text{eff}}(\sigma_0),\\
\label{max}
\sigma_{\text{eff}}^{\text{type 2}}&=&
\max_{\sigma_0\geq 0\atop 0\leq q(\sigma_0)\leq 1}
\sigma_{\text{eff}}(\sigma_0),
\end{eqnarray}
provided that an unphysical solution $\sigma_{\text{eff}}=0$ is
discarded in the minimization (\ref{min}). Indeed, at least in the
framework of the two different models introduced in Sec.\ \ref{tm},
the curves for $q(\sigma_0)$ and $\sigma_{\text{eff}}(\sigma_0)$ are
found to have the form shown in Fig.\ 6. The infimum (\ref{min})
occurs at $q=0$, whereas the solution $\sigma_0^*$ to the equation
$\sigma_{\text{eff}}'(\sigma_0)=0$ corresponds to a maximum of
$\sigma_{\text{eff}}$. Both types of theories can therefore be
interpreted as extremal theories in the framework of self-consistent
models built on the replica-coupling ansatz.  The physical meaning of
this interpretation is still not clear. However, ``type 2''
formulas should not been disregarded as unphysical because of their
showing up as maximal ones: the minimization principle states that the
dissipated power is minimized with respect to the electric field; but
there is no reason why an extremization with respect to arbitrary
variational parameters should not lead to a maximum of the dissipated
power.  Eqs.\ (\ref{min}), (\ref{max}) explain why for a given
approximation, one always has $\sigma_{\text{eff}}^{\text{type 1}}
\leq\sigma_{\text{eff}}^{\text{type 2}}$ in Figs.\ 1-4.

We now consider some points that were not explicitly treated in the
paper. First, we presented the formalism in terms of the electric
field $E$, from which we obtained a conductivity
$\sigma_{\text{eff}}$. The electric current $j$ (or the induction
$D$), could be used instead \cite{BART98}. In such a formulation, the
random constitutive parameter is the resistivity $\rho(x)=1/\sigma(x)$
and the constraints are $\nabla\cdot j=0$ and $\overline{j}=j_0$. One
then computes an effective resistivity $\rho_{\text{eff}}$. Both
formulations are equivalent, but a given approximation scheme in
general leads to different results for $\sigma_{\text{eff}}$ and
$\tilde\sigma_{\text{eff}}=1/\rho_{\text{eff}}$.  Preliminary
investigations of ``type 2'' formulas have been led in this
case. These will be presented elsewhere.  Finally, we discuss the
natural question about the possibility of replica-symmetry breaking
\cite{PARI84}. Replica symmetry breaking introduces more free
parameters in the ansatz, and the final extremization has to be
carried out with respect to several variables. There is no frustration
in this problem, and we therefore expect the replica symmetric
solution to be the only one. In order to test this, we tried a one-step
symmetry-breaking solution and did indeed not find any new solution.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}
\label{c}

We presented a functional approach to the calculation of
effective-medium properties of random media. We showed how to recover
the Bruggeman and Hori-Yonezawa formulas by using specific
approximation schemes to the basic functional integral.  We also
discussed the introduction of a replica-coupling parameter in a
gaussian Ansatz, from which new effective-medium formulas were
obtained. These formulas appear to be more adequate in $d=3$ compared
to the standard ones by Bruggeman and HY. Because it yields a
sensible result in all dimensions, and fulfills all the constraints
required to deserve the label of a ``good'' effective-medium theory,
the ``type 2'' counterpart of the Bruggeman formula offers an
interesting alternative to the latter. Indeed, it has a percolation
threshold equal to $p_c=1/5$ in three dimensions. This is closer to
values observed in real materials, compared to the $p_c=1/3$ of the
Bruggeman formula which often constitutes an overestimation.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\acknowledgements

We gratefully acknowledge H.~Orland for stimulating discussions. One
of us (MB) wants to thank H.E.~Stanley for his hospitality at the CPS
and the DGA for financial support.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\appendix
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Quadratic fluctuations of the field}
\label{qfotf}
For completeness, we give here the demonstration of Eq.\ (\ref{berg})
\cite{BERG78}. Volume averages are identified to statistical ones.
Because $W^*=(1/2)\langle \sigma E^2\rangle =(1/2)\sum_\nu p_\nu
\sigma_\nu \langle E^2\rangle_\nu$, where $\langle \cdot \rangle_\nu$
denotes an average on the phase $\nu$ on which the conductivity
$\sigma_\nu$ is constant, the effective conductivity reads
\begin{equation}
\sigma_{\text{eff}}=\sum_\nu p_\nu \sigma_\nu \frac{\langle E^2\rangle_\nu}{E_0^2}.
\end{equation}
On the other hand, $\sigma_{\text{eff}}$ has to be an homogeneous
function of degree one of the $\sigma_\nu$, whence
\begin{equation}
\sigma_{\text{eff}}=\sum_\nu \sigma_\nu \frac{\partial \sigma_{\text{eff}}}{\partial \sigma_\nu}.
\end{equation}
Comparing both equations yields the values of the
$\langle E^2\rangle_\nu$ and consequently that of $\langle E^2\rangle
=\sum_\nu p_\nu \langle E^2\rangle_\nu$. Equ.\ (\ref{berg}) follows.
We note that if $\sigma_{\text{eff}}=\langle\sigma\rangle$ (an exact
upper bound for the effective conductivity), then
$\langle \Delta E^2\rangle=0$. Therefore, $\sigma_{\text{eff}}=\langle\sigma\rangle$ defines a trivial model
of a medium which is a composite, but from which field fluctations
are nonetheless absent.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Calculations in the ``one-impurity'' approximation}
\label{appb}
The approximation which leads to (\ref{oneimp}) is built as follows. We first expand the exponential
\begin{equation}
\label{ser}
\left\langle e^{-\beta({\cal H}_e-{\cal H}_0)}\right\rangle_0=\sum_{k\geq 0}
{(-\beta)^k\over k!} \left\langle({\cal H}_e-{\cal H}_0)^k\right\rangle_0.
\end{equation}
Since ${\cal H}_0$ is non-random, using the hamiltonian
density $w_0$ defined in (\ref{hdens}) and $\Delta w_x[E(x)]$ defined by (\ref{dwx}) we can rewrite  the difference ${\cal H}_e-{\cal H}_0$ as
\begin{equation}
{\cal H}_e-{\cal H}_0=\int dx\, \Delta H(x),
\end{equation}
where
\begin{equation}
\label{dh}
\Delta H(x)=-\frac{1}{\beta v}\ln\left\langle e^{-\beta v \Delta w_x[E(x)]}\right\rangle.
\end{equation}
The one-impurity approximation consists in writing ($k\geq 1$)
\begin{mathletters}
\begin{eqnarray}
\label{intint}
({\cal H}_e-{\cal H}_0)^k
&=&
\int dx_1\ldots dx_k\,\Delta H(x_1)
\ldots
\Delta H(x_k)\\
&\simeq& v^{k-1}\int dy\,
\Delta H(y)^k.
\end{eqnarray}
\end{mathletters}
The last expression only retains contributions from identical points
in Eq.\ (\ref{intint}). Summing back the series in (\ref{ser}), and using (\ref{dh}) yields
\begin{eqnarray}
\label{ebracko}
&&\left\langle e^{-\beta({\cal H}_e-{\cal H}_0)}\right\rangle_0
\simeq
1+\int {dy\over v}
\left\langle {\cal I}(y)-1\right\rangle,\\
\label{caliy}
&&{\cal I}(y)=\frac{\int \tilde{\cal D}(E^{\alpha})\,
e^{-\beta\int dx\, \left\{w_0[E(x)]
+v\,\Delta w_{\bf x}[E(x)]
\delta(x-y)\right\}}}
{\int \tilde{\cal D}(E^{\alpha})\,e^{-\beta {\cal H}_0}},
\end{eqnarray}
where a one-impurity-type integral is involved. Since the fundamental
size of the theory ($\sim v^{1/d}$) is much smaller than the volume
$V=1$ of the system, and since the latter is statistically
translation-invariant, the outer integral over $y$ is redundant with
the disorder average, and can be dropped. We therefore arrive at
(\ref{oneimp}).

When $\beta\to\infty$, the functional ${\cal I}(y)$ can be computed
exactly. Let us briefly indicate how to do it. We first introduce the
notation $\vec{h}$ for vectors of dimension $nd$, and components
$h^\alpha_i$, with $\alpha=1,\ldots,n$, $i=1,\ldots,d$.  Hence,
$\Delta w_y[E(y)]\equiv \Delta w_y\bigl(\vec{E}(y)\bigr)$. The next
step is to use the formal identity
\begin{equation}
e^{-\beta v\Delta w_y(\vec{E})}=\int {d\vec{h}\,d\vec{h}'\over
(2\pi)^{nd}}e^{-i\vec{h}\cdot \vec{h}'} e^{-\beta v\Delta
w_y\bigl(-i\frac{\partial}{\partial \vec{h}'}\bigr)} e^{i
\vec{h}'\cdot \vec{E}}
\end{equation}
to write the numerator of (\ref{caliy}), which we denote hereafter by
${\cal J}(y)$, as
\begin{equation}
{\cal J}(y)=\int {d\vec{h}\, d\vec{h}'\over
(2\pi)^{nd}}e^{-i\vec{h}\cdot \vec{h}'} e^{-\beta v \Delta
w_y\bigl(-i\frac{\partial}{\partial\vec{h}'}\bigr)}\int\tilde{\cal
D}E\,e^{-(\beta/2)\int dx\, \vec{E}(x)\cdot \tilde{M} \cdot
\vec{E}(x)+i\vec{h}'\cdot \vec{E}(y)},
\end{equation}
where $\tilde{M}$ is the matrix defined from the replica-coupling
matrix $M$ in $w_0$ by $\tilde{M}^{\alpha\gamma}_{ij}\equiv
M^{\alpha\gamma}\delta_{ij}$. After an integration over the fields $E$
and $\phi$ implied in the measure $\tilde{\cal D}E$ (which can be
easily done using the Fourier components of $\phi$, and with $y=0$
since (\ref{oneimp}) is independent of $y$), $\cal J$ reads, up to
inessential factors \cite{NOTE2}:
\begin{equation}
{\cal J}(y)=e^{-(\beta/2)\sum_{\alpha\gamma}M^{\alpha\gamma}E_0^2}
\int {d\vec{h}\, d\vec{h}'\over (2\pi)^{nd}}e^{-i\vec{h}\cdot \vec{h}'}
e^{-v\beta\Delta w_y\bigl(-i\frac{\partial}{\partial \vec{h}'}\bigr)}
e^{i \vec{h}'\cdot \vec{E}_0-\vec{h}'\cdot \tilde{M}^{-1}
\cdot \vec{h}'/(2\beta v d)}.
\end{equation}
Formally expanding $\exp(-v\beta\Delta w_y)$ in powers of
$-i\partial/\partial \vec{h}'$, and carrying out successive
integrations by parts over $\vec{h}'$ yields
\begin{equation}
{\cal J}(y)=\left[\mathop{\text{Det}}(M)\left(\frac{v\beta
d}{2\pi}\right)^n\right]^{d/2}
e^{-(\beta/2)\sum_{\alpha\gamma}M^{\alpha\gamma}E_0^2}\int d\vec{h}\,
e^{-(v \beta d/2)(\vec{E}_0-\vec{h})\cdot
\tilde{M}\cdot(\vec{E}_0-\vec{h})-v\beta\Delta w_y(\vec{h})},
\end{equation}
where the determinant is evaluated in replica space. Finally,
${\cal I}(y)={\cal J}(y)/{\cal J}(y;\Delta w_y=0)$:
\begin{equation}
\label{iy}
{\cal I}(y)=\left[\mathop{\text{Det}}(M)\left(\frac{v\beta
d}{2\pi}\right)^n\right]^{d/2}\int \prod_\alpha dh^\alpha \,
e^{-v\beta\left\{{d\over 2}
\sum_{\alpha\gamma}M^{\alpha\gamma}(E_0-h^\alpha)_i
(E_0-h^\gamma)_i+\Delta w_y[h]\right\}}.
\end{equation}
For any $\Delta w_y$, this integral over the replicated vector field
$h$ can be computed exactly using a saddle-point method \cite{NEGE87}
in the limit $\beta\to\infty$, as announced. This allows for a
possible extension of the theory to nonlinear media in the
``one-impurity'' approximation. Here, for the linear problem at hand,
Eq.\ (\ref{iy}) is a simple gaussian integral. Setting
$\Delta\sigma=\sigma-\sigma_0$ and
\begin{equation}
\mu=\left(1+\frac{\Delta\sigma}{d\sigma_0}\right)^{-1},
\end{equation}
we obtain
\begin{equation}
\ln{\cal I}(y)=\left[-\frac{\beta v}{2}\Delta\sigma\,\mu E_0^2
+\frac{d}{2}(\ln \mu-\beta v q \sigma\mu E_0^2/d)\right]n+O(n^2),
\end{equation}
from which follows (\ref{dwoi}).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Calculation in the ``cumulant series'' approximation}
\label{co}
We have to compute $\left\langle {\cal H}_e\right\rangle_0$ and
$\left\langle {\cal H}_0\right\rangle_0$ in Equ.\ (\ref{dwsa}).
The calculation of $\left\langle {\cal H}_0\right\rangle_0$ is easy
with the methods already employed, and yields
\begin{equation}
\lim_{n\to 0}\left\langle
{\cal H}_0\right\rangle_0/n =\frac{1}{2}\sigma_0 E_0^2.
\end{equation}
As to $\left\langle {\cal H}_e\right\rangle_0$,
we first expand ${\cal H}_e$ (Eq.\ (\ref{calf})) in the cumulants
$C_k$ of the disorder averages of $\sigma(x)$, according to their
definition by the generating funtion ($X$ is a generic expansion
variable)
\begin{equation}
\ln\left<e^{X\sigma}\right>=\sum_{k\ge 1} {X^k\over k!}C_k(\sigma).
\end{equation}
We therefore have:
\begin{equation}
\label{fexpand}
{\cal H}_e=-{1\over \beta}\sum_{\bf x}\sum_{k\ge 1}{1\over k!}
\left[-{\beta v\over 2}\sum_\alpha {E^\alpha}({\bf x})^2\right]^k
C_k(\sigma).
\end{equation}
We deduce that
\begin{equation}
\label{fob}
{1\over n}\left\langle{\cal H}_e\right\rangle_0=-{1\over V\beta}
\sum_{\bf x}\sum_{k\ge
1}{(-\beta v)^k\over k!}  C_k(\sigma)\,{\cal C}_k(E^2/2),
\end{equation}
where
\begin{equation}
\label{calck}
{\cal C}_k(E^2/2)={1\over n}
\left\langle \left[\sum_\alpha {E^\alpha(x)}^2/2\right]^k
\right\rangle_0
\end{equation}
(because of statistical homogeneity, these coefficients do not depend
on the position variable $x$). It is convenient to introduce the
following generating function ${\cal Z}(X)$ in order to compute the ${\cal C}_k$:
\begin{eqnarray}
{\cal Z}(X)&=&\sum_{k\geq 1} {(-X)^k\over k!}{\cal C}_k(E^2/2)
\nonumber\\
&=&{1\over n} \left\{\left\langle\exp\left[-{1\over 2}X
\sum_\alpha{E^\alpha}^2(x)\right]
\right\rangle_0-1\right\}
\nonumber\\
&=&{1\over n}\ln\left\langle\exp
\left[-{1\over 2}X \sum_\alpha{E^\alpha}^2(x)\right]
\right\rangle_0+O(n).
\end{eqnarray}
Setting $A^{\alpha\gamma}=\delta_{\alpha\gamma}+(X/v\beta d) [M^{-1}]^{\alpha\gamma}$, we obtain \cite{NOTE2}
\begin{equation}
\label{zx}
{\cal Z}(X)=-{1\over 2n} \left\{d\mathop{\text{Tr}}
\mathop{\text{Ln}} A+X E_0^2
\sum_{\alpha\gamma} [A^{-1}]^{\alpha\gamma}\right\}+O(n)
\end{equation}
(the trace and the logarithm act in the replica space).

Expanding (\ref{zx}) in powers of $X$ then allows for the identification
\begin{equation}
{\cal C}_k(E^2/2)={k!\over 2}\left({1\over v\beta d}\right)^k \frac{d}{n}\left[{1\over k}\mathop{\text{Tr}} (M^{-k})+{v\beta d} E_0^2 \sum_{\alpha\gamma} [M^{1-k}]^{\alpha\gamma}\right]+O(n).
\end{equation}
Use of this expression in (\ref{fob}) cancels the convergence factor
$k!$: we reintroduce it by inserting the identity
\begin{equation}
{1\over m!}\int_0^\infty du\, e^{-u} u^m =1,
\end{equation}
applied to $m=k-1$ and $m=k$ in the resulting cumulant series. This
permits its Borel summation, which brings in the functions $h_m(x)$
defined by (\ref{hm}). This results in
\begin{equation}
\label{fmfo}
{1\over n}
\langle{\cal H}_e\rangle_0
=-{1\over n}
\left[
{d\over 2}E_0^2
\sum_{\alpha\gamma}\left[ M
h_0(M^{-1}/d)\right]^{\alpha\gamma}
+{1\over 2 v \beta}\mathop{\text{Tr}}\, h_{-1}(M^{-1}/d)
\right]+O(n),
\end{equation}
where we defined the family of functions
\begin{equation}
\label{hm}
h_m(z)=\int_0^\infty du\,u^m e^{-u} \ln\left\langle e^{-u\sigma
z}\right\rangle\qquad(m>-2).
\end{equation}
Note for further use that
\begin{equation}
\label{deriv}
h_m'(z)={1\over z}\left[h_{m+1}(z)-(k+1)h_{m}(z)\right].
\end{equation}

For $M$ given by (\ref{matmok}), the differents terms in (\ref{fmfo}) are
\begin{eqnarray}
&&\lim_{n\to 0}{1\over n} \sum_{\alpha\gamma} \left[M
h_0(M^{-1}/d)\right]^{\alpha\gamma}=\sigma_0 h_0
\bigl(1/(d\sigma_0)\bigr),
\nonumber\\
&&\lim_{n\to 0}{1\over n}\mathop{\text{Tr}}
h_{-1}(M^{-1}/d)=dh_{-1}\bigl(1/(d\sigma_0)\bigr)+v\beta\, d\sigma_0 q\, h_0 \bigl(1/(d\sigma_0)\bigr)E_0^2.
\end{eqnarray}
which leads to Eq.\ (\ref{fmfodeu}).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{references}
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{\bf 24}, 636 (1935).

\bibitem{LAND52} R.~Landauer, J.~Appl.~Phys. {\bf 23}, 779 (1952);
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Inhomogeneous Media}, edited by J.~C.~Garland and D.~B.~Tanner.
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J.~M.~Luck, Adv.\ Phys. {\bf 39}, 191 (1990).

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{\em Introduction to Percolation Theory} (Taylor \& Francis, London, 1992).

\bibitem{SAHI98} M.\ Sahimi, Phys.\ Rep. {\bf 306}, 213 (1998),
and references therein.

\bibitem{MILT85} G.\ W.\ Milton, Commun.\ Math.\ Phys. {\bf 99}, 463
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\bibitem{STIN73} R.~B.~Stinchcombe, J.\ Phys.\ C {\bf 6}, L1 (1973).

\bibitem{STRA74} J.~P.~Straley, J.\ Phys.\ C {\bf 9}, 783 (1976).

\bibitem{NOTE1} This is because the key ingredient of the continuum
theory, namely the Green function $\sum_k k_i k_j/k^2$ (computed at $x=0$
in the calculations of this paper, cf.\ \cite{NOTE2}),
has the same low-momentum structure as its counterpart on an
hypercubic lattice (see, e.g.\ Ref.\ \cite{LUCK91}).

\bibitem{STEP77} M.~J.~Stephen, Phys.\ Rev.\ B {\bf 17}, 4444 (1977).

\bibitem{HORI75} M.~Hori and F.~Yonezawa, J.\ Math.\ Phys.  {\bf 16},
352 (1975).

\bibitem{BERG81} D.~J.~Bergman and Y.~Kantor, J.\ Phys.\ C {\bf 14},
3365 (1981).

\bibitem{LUCK91} J.~M.~Luck, Phys.\ Rev.\ B {\bf 43}, 3933 (1991).

\bibitem{HORI77} M.~Hori, J. Math. Phys. {\bf 18}, 487 (1977).

\bibitem{BART93} M.~Barth\'el\'emy and H.~Orland, J.\ Phys.\ I
(France) {\bf 3} 2171 (1993).

\bibitem{BART98} M.~Barth\'el\'emy and H.~Orland, Eur.\ Phys.\ J.\ B
{\bf 6} 537 (1998), cond-mat/9806302.

\bibitem{BLUM91} R.~Blumenfeld and D.~J.~Bergman, Phys.\ Rev.\ B {\bf
44}, 7378 (1991); D.~J.~Bergman and R.~Blumenfeld, Phys.\ Rev.\ B {\bf
54}, 7378 (1996).

\bibitem{WILL86} J.~R.~Willis, in {\it Homogeneization and Effective Moduli
of Materials and Media}, edited by J.~L.~Ericksen {\it et al.}
(Springer-Verlag, New York, 1896), p.\ 247; D.~R.~S.~Talbot and J.~R.~Willis,
IMA J.\ Appl.\ Math. {\bf 39}, 215 (1987).

\bibitem{SANC87} E.\ Sanchez-Palencia and A.\ Zaoui eds.\,
{\em Homogeneization Techniques for Composite Media. Proceedings, Udine,
Italy 1985}, Lecture notes in Physics {\bf 272} (Springer-Verlag, Berlin,
1987), and references therein.

\bibitem{PONT92b} P.~Ponte Casta\~neda, SIAM J.\ Appl.\ Math. {\bf
52}, 1321 (1992).

\bibitem{EDWA75} S.~F.~Edwards and P.~W.~Anderson, J.\ Phys.\ F {\bf
5} 965 (1975).

\bibitem{MEZA87} M.~M\'ezard, G.~Parisi, and M.~Virasoro, {\it
Spin Glass Theory and Beyond} (World Scientific, Singapore, 1987).

\bibitem{BERG92} D.~J.~Bergman and D.~Stroud, Solid State Phys.  {\bf
46}, 147 (1992).

\bibitem{PONT92a} P.~Ponte Casta\~neda, G.~DeBotton, and G.~Li, Phys.\
Rev.\ B {\bf 46}, 4387 (1992).

\bibitem{YU94} K.~W.~Yu, Y.~C.~Chu, and E.~M.~Y.~Chan, Phys.\ Rev.\ B
{\bf 50}, 4387 (1994).

\bibitem{PONT97} P.~Ponte Casta\~neda and M.\ Kailasam, Proc.\ R.\
Soc.\ London A {\bf 453}, 793 (1997).

\bibitem{NOTE2} Integration is performed over the independant scalars
$\phi_k$ and $\phi_{-k}$ with $1/(2v)$ such couples of Fourier modes
involved. Use is made in the calculations of the
equality $1/v=\sum_k 1\simeq\int dk /(2\pi)^d$, with an
upper cut-off on the $k$ implied. In particular, one appeals
to the formula $\sum_k k_i k_j/k^2=\delta_{ij}/(dv)$.

\bibitem{BERG78} D.~J.~Bergman, Phys.\ Rep. {\bf 43}, 377 (1978).

\bibitem{HASH62} Z.~Hashin and S.~Shtrikman, J.\ Appl.\ Phys.
{\bf 33}, 3125 (1962).

\bibitem{WRIG86} D.\ C.\ Wright, D.\ J.\ Bergman, and Y.\ Kantor,
Phys.\ Rev.\ B {\bf 33}, 396 (1986), and references therein.

\bibitem{STRA77} J.\ P.\ Straley, Phys.\ rev.\ B {\bf 15}, 5733
(1977).

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\bibitem{PARI84} G.~Parisi, in {\it Les Houches, Session XXXIX , 1982 --
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Statistique / Recent Advances in Field Theory and Statistical Mechanics},
edited by  J.-B.\ Zuber and R.~Stora (North Holland, Amsterdam, 1984),
pp.\ 473--523.

\bibitem{NEGE87} J.~W.~Negele and H.~Orland,
{\it Quantum Many-Particle Systems} (Addison-Wesley, New York, 1987).
\end{references}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%  FIGURES
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\twocolumn
%
\begin{figure}
\narrowtext
\vspace*{0.0cm}
\centerline{
\epsfysize=0.9\columnwidth{\rotate[r]{\epsfbox{fig1_c.eps}}}}
\vspace*{0.3cm}
\caption{Rescaled effective conductivities in dimension $d=2$
for a binary medium, versus the volume concentation $p$ of
component 2. The conductivity ratio is $\sigma_2/\sigma_1=10$.
Highest and lowest solid curves: Hashin-Shtrikman bounds; Br.:
the Bruggeman formula (``type 1'', one-impurity
approximation -- OI); HY: the Hori-Yonezawa formula (``type 1'',
cumulant series approximation -- CS); both ``type 2'' curves
are the new formulas, within OI and CS approximations.
}
\label{fig1}
\end{figure}

\begin{figure}
\narrowtext
\vspace*{0.0cm}
\centerline{
\epsfysize=0.9\columnwidth{\rotate[r]{\epsfbox{fig2_c.eps}}}}
\vspace*{0.3cm}
\caption{Rescaled effective conductivities in dimension $d=2$
for a binary medium, versus the volume concentation $p$ of
component 2. The conductivity ratio is $\sigma_2/\sigma_1=1000$.
Same plots as in Fig.\ \ref{fig1}.
}
\label{fig2}
\end{figure}

\begin{figure}
\narrowtext
\vspace*{0.0cm}
\centerline{
\epsfysize=0.9\columnwidth{\rotate[r]{\epsfbox{fig3_c.eps}}}}
\vspace*{0.3cm}
\caption{Rescaled effective conductivities in dimension $d=3$
for a binary medium, versus the volume concentation $p$ of
component 2. The conductivity ratio is $\sigma_2/\sigma_1=10$.
Same plots as in Fig.\ \ref{fig1}.\\
\\
\\
\\
}
\label{fig3}
\end{figure}

\begin{figure}
\narrowtext
\vspace*{0.0cm}
\centerline{
\epsfysize=0.9\columnwidth{\rotate[r]{\epsfbox{fig4_c.eps}}}}
\vspace*{0.3cm}
\caption{Rescaled effective conductivities in dimension $d=3$
for a binary medium, versus the volume concentation $p$ of
component 2.The conductivity ratio is $\sigma_2/\sigma_1=1000$.
Same plots as in Fig.\ \ref{fig1}.
}
\label{fig4}
\end{figure}

\begin{figure}
\narrowtext
\vspace*{0.0cm}
\centerline{
\epsfysize=0.9\columnwidth{\rotate[r]{\epsfbox{fig5_c.eps}}}}
\vspace*{0.3cm}
\caption{Relative quadratic fluctuations of the field in dimension
$d=3$ for a binary medium, versus the volume concentation $p$ of
component 2. The conductivities are $\sigma_1=1$, $\sigma_2=1000$.
}
\label{fig5}
\end{figure}

\begin{figure}
\narrowtext
\vspace*{0.0cm}
\centerline{
\epsfysize=0.9\columnwidth{\rotate[r]{\epsfbox{fig6_b.eps}}}}
\vspace*{0.3cm}
\caption{Rescaled effective conductivity
$\sigma_{\text{eff}}(\sigma_0)/\sigma_1$, and reduced
replica coupling parameter $q(\sigma_0)$ vs.\ $\sigma_0$ in
dimension $d=3$ for a binary medium. In this exemple computed
from Eqs.\ (\ref{sigefq}), (\ref{qbrug}), the conductivity ratio is
$\sigma_2/\sigma_1=100$, and the volume fraction $p$ of
component 2 is $p=0.18$. The ``type 1'' effective conductivity is
obtained when $q=0$, whereas the ``type 2'' effective conductivity
corresponds to the maximum of the curve $\sigma_{\text{eff}}(\sigma_0)$.
}
\label{fig6}
\end{figure}

\end{document}


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\graphicspath{{img/}}
\title{Lightning Network imbalance measure and proactive channel rebalancing algorithm}
%\title{Imbalance measure and proactive channel rebalancing algorithm for the Lightning Network}
\begin{document}

\author{\IEEEauthorblockN{1\textsuperscript{st} Rene Pickhardt}
\IEEEauthorblockA{\textit{Department of Computer Science} \\
\textit{Norwegian University of Science and Technology}\\
Gj{\o}vik, Norway \\
rene.m.pickhardt@ntnu.no}
\and
\IEEEauthorblockN{2\textsuperscript{nd} Mariusz Nowostawski}
\IEEEauthorblockA{\textit{Department of Computer Science} \\
\textit{Norwegian University of Science and Technology}\\
Gj{\o}vik, Norway \\
mariusz.nowostawski@ntnu.no}
}

\maketitle
\begin{abstract}
Making a payment in a privacy-aware payment channel network can be achieved by trying out several payment paths until one succeeds.
With a large network, such as the Lightning Network, a completion of a single payment can take up to several minutes.
We introduce a network imbalance measure and formulate the optimization problem of improving the balance of the network as a sequence of rebalancing operations of the funds within the channels along circular paths within the network.
As the funds and balances of channels are not globally known, we introduce a greedy heuristic
that improves every node's local balance despite the uncertainty.
In an empirical simulation on a snapshot of the Lightning Network we demonstrate that the imbalance distribution of the network has a Kolmogorov-Smirnoff distance of $0.74$ in comparison to the imbalance distribution after the heuristic is applied.
We further show that the success rate of a single unit payment increases from $11.2\%$ on the imbalanced network to $98.3\%$ in the balanced network.
Similarly, the median possible payment size across all pairs of participants increases from $0$ to $0.5$ mBTC for initial routing attempts on the cheapest possible path.
Executing $4$ different strategies for selecting rebalancing cycles lead to similar results
indicating that a collaborative approach within the friend of a friend network might be preferable from a practical point of view.\footnote{A full version of this text - including a discussion about fee free rebalancing - is available at: \url{https://arxiv.org/abs/1912.09555}}

\end{abstract}
\begin{IEEEkeywords}
imbalance, rebalancing, optimization, Bitcoin, Lightning Network, payment channel networks, path finding, routing, liquidity, flow control, congestion control, game theory, uncertainty, simulations, collaborative problem solving, privacy
\end{IEEEkeywords}

%==========================================================================
\section{Introduction}
The Lightning Network has been introduced in order to mitigate the scaling issues of blockchain technologies such as Bitcoin\cite{poon2016bitcoin} by creating a network of payment channels.
In order to protect users privacy while routing a payment through several payment channels privacy a payment network can use a source-based onion routing scheme, like the Sphinx Mix format~\cite{danezis2009sphinx}.
While the Lightning Network, for example, shares the capacity of public channels with its participants through its gossip protocol, the local split of the capacity of the channels into the balance of its participants is not shared with the rest of the network for two reasons:
(1) this would compromise the privacy as one could collect all changes of the channel balances and reconstruct the flow of payments;
(2) propagating this information would essentially mean that every node in the network is made aware of every payment which would have the same poor scaling properties as blockchain technology and other broadcast networks.
The decision to use source-based routing together with the unknown channel balances of the network results in a challenge for finding a path of payment channels so that all channels along the path have enough liquidity to be able to forward an attempted payment.
Currently, this challenge is met by probing paths until one succeeds.
It has been demonstrated that probing for paths can take more than 3 minutes for more than $5\%$ of the attempted payments~\cite{decker2019lnconf}. This leads to a poor payment latency and poor user experience.
Imagine a grocery store in which for every $20^{th}$ customer the cash register would have to wait three minutes until the payment was received.

This study examins the consequences of nodes proactively and collaboratively distributing their funds evenly across their channels.
We introduce the notation of an imbalanced network.
The imbalance is measured as the average of the imbalance scores of its nodes.
The node's imbalance of payment channels is defined as the Gini coefficient of a node's channel balance coefficients which are the relative amount of funds a node owns in a channel in comparison to the capacity of that channel.
Nodes can reduce their imbalance either by initiating a rebalancing their own channels themselves, or, by collaboratively rebalancing the channels with the help of their channel partners by conducting circular payments.

We formulate an optimization problem of finding a sequence of rebalancing operations that minimize the network's imbalance.
As the information necessary to solve the optimization problem is not publicly known in privacy-aware payment channel networks, we provide a greedy heuristic for its participants to find a minimum for the problem.
In an empirical study, we show that the greedy heuristic will lead to a success rate of $98.3\%$ for a small payment between two arbitrary selected nodes.
Also the median possible payment size between all pairs of nodes increases from $0$ in the imbalanced network to $0.5$ mBTC in the best balanced networks that we found.

Looking at different strategies of finding rebalancing cycles we suggest, for the sake of speed, that nodes share with their neighbors on which local channels they would like to have inbound or outbound capacity.
This information could easily be probed anyway and will not to worsen the privacy of the nodes involved in rebalancing.

%==========================================================================
\section{Related Work}
\label{sec:relatedWork}

While the Lightning Network white paper~\cite{poon2016bitcoin} does not discuss path finding and states routing as an easy problem it is generally recognized that pathfinding on the Lightning Network is a difficult problem~\cite{piatkivskyi2018split, prihodko2016flare, bagaria2019boomerang, pickhardt2019pathfinding, grunspan2018ant, sivaraman2018routing}.
There is already research conducted in the field of rebalancing channels~\cite{khalil2017revive},which was about the cryptographic protocols used to make sure that participants can enforce the rebalancing that was agreed upon.
There are rebalancing operations for existing lightning implementations: c-lightning\footnote{\url{https://github.com/lightningd/plugins/tree/master/rebalance}} and for lnd\footnote{\url{https://github.com/bitromortac/lndmanage}}.
In particular, the idea of just in time rebalancing while fulfilling routing requests~\cite{pickhardt2019jit} has been implemented as JIT-routing for c-lightning\footnote{\url{https://github.com/lightningd/plugins/pull/66}}.

% ===========================================================================
\section{Formalization and Assumptions}
\label{sec:formalization}

Let $N=(V,E,c)$ be a payment channel network with a finite set of nodes.
The payment channels are the edges in the network such that $E\subset V\times V$.
Additionally, we have a publicly known capacity function $c: E\longrightarrow \mathbb{N}$ that assigns a capacity to every edge of the network.
For every edge $e=(u,v)$ we denote $e_u:=(e,u)$ as the first participant of the channel and $e_v=(e,v)$ as the second participant.
Naturally, the capacity of every channel $e=(u,v)$ is privately split into the local balances with the balance function $b: E\times V\longrightarrow\mathbb{N}$ such that $b(e_u)+b(e_v)\stackrel{!}{=}c(e)$.

We define the channel balance coefficient for $u$ on the channel $e=(u,v)$ as  $\zeta_{(u,v)} = \frac{b(e_u)}{c(e)}$.
This is just the relative amount of funds that the participant $u$ has in the channel $e$.
In general, for an imbalanced chanel we have $\zeta_{(u,v)} \neq \zeta_{(v,u)}$.

The neighbor function $n : V \longrightarrow 2^{E}$ assigns every node a set of all the channels it is part of.
We call $U:=n(u)$ the neighborhood of node $u$.

The total funds of a participant $u$ are denoted as $\tau_u:=\displaystyle{\sum_{e\in U}b(e_u)}$.
The value of $\tau_u$ is constant while no payments are made and no channels are being opened and closed.
The balance function $b$ can vary.
The total capacity of a participant $u$ is denoted as $\kappa_u:=\displaystyle{\sum_{e\in U}c(e)}$.
Using the last two definitions let us define the node balance coefficient for a participant $u$ as $\nu_u = \frac{\tau_u}{\kappa_u}$.

We call a node $u$ {\bf balanced} if its channel balance coefficients $\zeta_{(u,v_1)},\dots,\zeta_{(u,v_d)}$ have the same value.\footnote{This value does not have to take a value of $0.5$, but it certainly is a reasonable goal.}
Consequently, we consider a node {\bf unbalanced}, if its local channel balance coefficients are unequal.
Statistically, inequality of a distribution can be measured with the Gini coefficient.
Thus, for a node $u$ with channel balance coefficients $\zeta_{(u,v_1)},\dots,\zeta_{(u,v_d)}$ we define $G_u = \frac{\displaystyle{\sum_{i\in U} \sum_{j \in U}} | \zeta_i - \zeta_j |}{2 \displaystyle{\sum_{i \in U} \sum_{j \in U} \zeta_j}}$.
If $G_u = 0$ this means that the channel balance coefficients are equal.
This value will be exactly the same as the node's balance coefficient $\nu_u$.
In contrast, if $G_u = 1$ the channel balance coefficients are distributed in the most unequal way.

Finally, $G$ denotes the imbalance of the network $G = \displaystyle{\frac{1}{|V|}\sum_{v\in V}G_v}$. It is the mean of the imbalance values of all nodes in the network.
A perfectly balanced network would be achieved if $G$ takes the value of $0$ whereas the balance is poor if the value of $G$ is close to 1.
Our goal is to find a balance function $b$ which minimizes $G$ given a privacy aware payment channel network with initial distribution of funds $\tau_{u_1},\dots,\tau_{u_n}$.
The constraint to this optimization problem is that the total funds $\tau_u$ are fixed for every node $u \in V$ and any choice of the balance function $b$.
As we lack knowledge about the global network state, we cannot apply standard optimization techniques such as gradient descent, conjugate gradient methods or simulated annealing.
We suggest to use a greedy heuristic in which every participant executes some operations to improve its own balance which others support if it also improves their balance.

\subsection{Description of the rebalancing algorithm}
\label{sec:Greedy Rebalancing Heuristic}

Participants use the local knowledge and make local adjustments trying to solve the optimization problem of finding $b$ such that $G$ is minimized.
\begin{enumerate}
\item A node $u$ computes its node balance coefficient $\nu_u$.
\item $u$ then computes its channel balance coefficients $\zeta_{(u,v_1)},\dots,\zeta_{(u,v_d)}$.
\item All channels $e=(u,v_i)$ for which the channel balance coefficient is higher
than its node balance coefficient, are selected as $C = \{(u,v_i) | \zeta_{(u,v_i)} - \nu_u\ > 0\}$.\footnote{
  Note, that we do not need to take absolute values as $u$ will only be able to initiate a rebalancing operation by sending money which means decreasing its channel balance coefficient of $\zeta_{(u,v_i)}$ towards $\nu_u$.}.
\item From the candidate set $C$ a random channel $e=(u,v)$ is selected.
\item Now the node searches for a circular payment to itself along $e=(u,v)$ by choosing a path $p = [v,x_1,\dots,x_n,u]$. The amount of that payment should decrease the value of $\zeta_{(u,v)}$ to that of $\nu_u$ and can be computed as $a = c(e)\cdot (\zeta_{(u,v)}-\nu_u)$. The end of the circle should be a channel $(x_n,u)$ for which the channel balance coefficient $\zeta_{(u,x_n)}$ is smaller than the node balance coefficient $\nu_u$.\footnote{In one of the experiments we weaken this strong criteria and show empirically that it makes sense to drop it for the benefit of easier path finding at the cost of small oscillations of the algorithm.}
\item The node conducts the payment if all the nodes on the path $p$ agree to participate. It could happen that some nodes will only participate with a value smaller than $a$. As this is already progress $u$ will accept the suggested amount instead of being stubborn.
\item Repeat all steps as long as the local balance coefficients are not even enough and as long as paths are found.
\end{enumerate}

\section{Experimental Setup}
\label{sec:setup}

We evaluated $4$ different strategies for finding rebalancing cycles. \texttt{cycles4} tests all cycles of length $4$ or smaller. In the same way \texttt{cycle5} tests all cycles of length $5$ or smaller. \texttt{foaf} tests most cycles of the friend of a friend network.\footnote{This approximation is achieved by using \texttt{cycle4} as a cycle base as all cycles of length 4 and smaller certainly are part of the friend of a friend network.}
\texttt{mpp} is short for multi-path payments and uses the same heuristic as \texttt{foaf} but will only take a $20^{th}$ of the maximum possible amount for rebalancing operation since it is supposed to split the rebalancing over various cycles.
Our approximation for \texttt{foaf} and \texttt{mpp} explains why the final results for these two strategies are not entirely monotonic but oscillating a bit.

We conducted our experiment on a snapshot of the public Lightning Network from October 2019.
As channels are currently almost always opened by one side\footnote{Dual funded channels are not part of the protocol and no implementation is merged to any of the standard nodes} we randomly guessed who opened the channel by a coin flip and allocated the entire channel capacity to that node.
As a result the histogram of all node balance coefficients had a mode of $0.5$ which is plausible with this random process\footnote{We refer again to the long Version of this paper for more details.}.

After we randomly chose the founder of each channel we compute the largest strongly connected component to remove nodes which had the funds allocated in a way that would not allow any rebalancing.
The strongly connected component consists of $2707$ nodes and $24161$ edges.
The diameter of the strongly connected component had a value of 10.
Almost $40\%$ of shortest paths are longer than the cycles that our strategies have tested.
This demonstrates that the single rebalancing operations which we conducted were not covering the entire network but only taking place within local parts of the network.
This is particularly true for the rebalancing in the friend of a friend network.

In order to conduct the experiments we simulated the proposed algorithm and strategies in the following way.
For each channel we computed up to $5000$ rebalancing cycles following the defined strategy.
The number $5000$ was chosen to be as large as possible so that we were able to conduct the simulation on the $16$ GB of main memory of our machine.

% ==========================================================
\section{Results}
\label{sec:results}

\begin{figure}
 \centering
 \includegraphics[width=6cm]{code/vs/fig/imba_vs_steps.png}
 \caption{Comparing how the imbalance score of the network behaves with the amount of successfull rebalancing operations. As the rebalancing amounts are much smaller with multi path payments the x-axis has a logarithmic scale.}
 \label{fig:imbalancehovertime}
\end{figure}

\cref{fig:imbalancehovertime} confirms that the imbalance of the network is decreasing over time when running our simulation.
This means that the network is becoming more balanced.
We see that most improvement is happening quite fast with $10$k to $100$k rebalancing operations being necessary.
This is not more than $37$ rebalancing operations per node.
Taking into account that the average node degree is $9$ this means that on average each node needs to successfully execute about $4$ rebalancing operations per channel.
This seams plausible in the sense that every channel gets rebalanced at least once.\footnote{Note, that due to the collaborative behaviour of nodes a node might also get its channels rebalanced in the rebalancing attempts of other nodes.}
The multi-path rebalancing needs more operations which makes sense as we only rebalanced for a $20^{th}$ of the possible amount in the multi-path case since we wanted to have multiple other cycles to rebalance that particular channel.
While \texttt{cycle4} improves the balance more quickly, it does not reach the minimum imbalance as well as the other strategies.
In particular, the algorithm converges to a local minimum rather quickly.
Our experimental runs resulted in states that are not identical when we shuffled the order of cycles that we probed, however, the differences were negligible.
We guess that the local minimum might be close to global minimum of the optimization problem.

\begin{figure}
 \centering
 \includegraphics[width=6cm]{code/vs/fig/imba_vs_success_rates.png}
 \caption{Every time the imbalance score reached a new low to a two digit decimal we computed the probability for a random payment of the smallest possible amount to succeed on the cheapest path.}
 \label{fig:imba_vs_success}
\end{figure}

In~\cref{fig:imba_vs_success} we see that a lower imbalance score $G$ yields a higher success rate of payments as one would expect.
The success rate is measured by attempting a payment of $1$ Satoshi between all pairs of nodes and counting the relative amount of payment attempts for which the cheapest path from the source to the destination was successful.
We computed the success rate of every time the balance score hits a new score rounded to two decimals.
It is the first experiment that indicates that the defined imbalance score was well chosen.
Again the multi-path payment strategy sticks out as it achieves a rather high success rate while the network is still rather imbalanced.
A success rate of $80\%$ is reached for an imbalance score of $0.43$.
Comparing this to~\cref{fig:imbalancehovertime} we see that about $250$k rebalance operations are necessary to achieve this success rate.
Even the worst performing algorithm achieves this success rate with far less than $100$k rebalancing operations and a much better overall imbalance score.

This effect, while expected as multi-path rebalancing uses smaller amounts, is more pronounced in the next experiment.
As $1$ Satoshi payments are not too useful in a real world setting we checked if the overall amounts that can be routed increase in a better balanced network.

\begin{figure}
 \centering
 \includegraphics[width=6cm]{code/vs/fig/imba_vs_median_payment_size.png}
 \caption{The imbalance of the network is plotted against the median value of payments that can be fulfilled between all pairs of nodes on the first attempt along the cheapest path. Note that failed payments are part of the statistics and counted as payments which can send $0$ Satoshi.}
 \label{fig:imba_vs_payment_size}
\end{figure}

In~\cref{fig:imba_vs_payment_size} we compare the imbalance of the network with the median possible payment amount which is computed as follows:
For all pairs of shortest paths on the base fee graph we look at the amount that could be forwarded along that path.
Finally we take the median of those values.
This means that at least $50\%$ of payment pairs are able to forward this amount.
Again we see that the lower the imbalance score becomes the higher the median possible payment size gets.
This suggests that statistically a more balanced network is able to successfully route higher payment amounts on the first try.
All four strategies are achieving a median possible payment amount of roughly $50000$ Satoshi.
This means that in a balanced network $50\%$ of payment pairs are able to successfully conduct a payment of $0.5$ mBTC along the cheapest path.
The result also indicates that the strategy for probing rebalancing cycles seems to have little to no impact to the final abilities of the network to perform payments.

\begin{figure}
 \centering
 \includegraphics[width=6cm]{code/vs/fig/maximum_payable_amount_all_pair_chepest_paths_balanced_network.png}
 \caption{The maximal payable amounts on all pairs of cheapest paths on the initial imbalanced network and on the network after applying the friend of a friend rebalancing strategy.}
 \label{fig:cdf_paymentsize}
\end{figure}

We look a little bit closer at the data point of the most balanced network with the friend of a friend strategy and compare it to the imbalanced network.
Instead of only looking at the median we study the cumulative distribution function of the histogram in~\cref{fig:cdf_paymentsize}.
We can see that in the imbalanced network almost $88.8\%$ of the paths are not able to forward a single Satoshi (meaning a $11.2\%$ success rate).
This number drops below $1.7\%$ (meaning a $98.3\%$ success rate) for the balanced network.

For the friend of a friend network we examine the distribution of Gini coefficients with the initial distribution of Gini coefficients.
We provide the cumulative distribution function of both distributions with in~\cref{fig:cdf_gini}.

\begin{figure}
 \centering
 \includegraphics[width=6cm]{code/vs/fig/comparison distribution of Ginicoefficients.png}
 \caption{Comparing the distribution of the node's Gini coefficients before and after rebalancing with the friend of a friend strategy.}
 \label{fig:cdf_gini}
 \end{figure}

The Kolmogorov-Smirnoff distance reaches a value of $0.74$.
This verifies the statistical significance of the rebalancing heuristic in comparison to staying with the imbalanced network.
The median value of the Gini coefficients in the best balanced network we could acquire is about $0.15$.
In comparison, on the imbalanced network the median Gini coefficient is $0.5$.

Our results show that nodes do not need to strive for $\nu_u = 0.5$ and perfect local balance of their channels in order to produce high success rate of random payments.

% =========================================================================================
\section{Future Work}
\label{sec:future}

As the simulation had consumed a lot of computational resources, we have not check if the greedy heuristic is stable under concurrent payments and rebalancing operations taking place.
In a similar gist simulations on larger networks would be interesting but require more significant computational resources and time to be conducted.

The current experiments have assumed a stable network topology.
However, every channel opening or closure, as well as every single payment, will change the imbalance of the network.
It is therefore necessary to study how it adapts if the topology changes in real time due to opening or closing channels as well as a reallocation of funds due to payments that are taking place.
In particular, it would be interesting to see how this heuristic works in combination with JIT-Routing as this would unbalance nodes to provide liquidity which our heuristic would then redistribute over the friend of a friend network.
This would occur along all nodes on a path in intersecting friend of a friend networks.
It would be interesting to see how this algorithm behaves together with Atomic Multipath Payments which are in the style of~\cite{bagaria2019boomerang} redundant but along our results carry at most $50000$ Satoshi.

% =======================================================================
\section{Discussion}
\label{sec:conclusion}

Computing the rebalancing cycles is the most expensive operation in our simulation.
If the greedy heuristic is implemented in a real payment channel network, it would only be
done locally by each node, which in itself is not expensive.
This holds true if those cycles will only be searched within the friend of a friend network which is typically small.
As the results of the different strategies for finding cycles did not vary and for the previous reason we propose to stick to rebalancing in the friend of a friend network and introducing a communications protocol between nodes to collaboratively find such rebalancing cycles by sharing some information.

% ==========================================================================================
\section{Acknowledgements}
\label{sec:ack}

We are grateful to Vivek Bagaria and Joachim Neu for an interesting discussion about their and our work that initiated this research. We thank Stefan Richter for helpful comments on early drafts of this work. A special thank goes to Thomas Gottron, Christian Decker and Dmytro Piatkivskyi.
Finally we thank the donors to \url{https://tallyco.in/s/lnbook} and Patreons supporting open knowledge.

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