Greens Functions for Random Resistor Networks Sayak Bhattacharjee1and Kabir Ramola2y 1Department of Physics Indian Institute of Technology Kanpur Kanpur 208016 India

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Green’s Functions for Random Resistor Networks

Sayak Bhattacharjee1, ∗and Kabir Ramola2, †

1Department of Physics, Indian Institute of Technology Kanpur, Kanpur 208016, India

2Tata Institute of Fundamental Research, Hyderabad 500107, India

(Dated: May 2, 2023)

We analyze random resistor networks through a study of lattice Green’s functions in arbitrary

dimensions. We develop a systematic disorder perturbation expansion to describe the weak disorder

regime of such a system. We use this formulation to compute ensemble averaged nodal voltages

and bond currents in a hierarchical fashion. We verify the validity of this expansion with direct

numerical simulations of a square lattice with resistances at each bond exponentially distributed.

Additionally, we construct a formalism to recursively obtain the exact Green’s functions for ﬁnitely

many disordered bonds. We provide explicit expressions for lattices with up to four disordered bonds,

which can be used to predict nodal voltage distributions for arbitrarily large disorder strengths.

Finally, we introduce a novel order parameter that measures the overlap between the bond current

and the optimal path (the path of least resistance), for a given resistance conﬁguration, which helps

to characterize the weak and strong disorder regimes of the system.

I. INTRODUCTION

Electrical networks have often been used to model a

wide variety of condensed matter phenomena, both in

steady-state as well as transient regimes. While they

have recently arisen as synthetic experimental test-beds

for topological quantum matter [1–3], a longstanding ap-

plication of electrical networks—in particular, resistor

networks—has been to model the conductivity of disor-

dered random media. This problem of fundamental inter-

est is relevant for transport measurements [4–7] and also

the study of critical phenomena [8–11]. In order to model

natural systems, microscopic disorder in such model sys-

tems is an important ingredient. Such situations often

require a subtle understanding of the properties of the

lattice Green’s function (which is related to the inverse

of the lattice Laplacian) [12,13], and therefore an in-

vestigation into techniques that can be used to compute

Green’s functions for disordered electrical networks rep-

resents a fundamental direction of theoretical as well as

experimental relevance.

Within this context, random resistor networks (RRN)

are a popular paradigm for modeling transport in disor-

dered media such as semiconductors [4,14–21], but also

in directed polymers [22–24], and porous rocks [25,26].

Formally deﬁned, an RRN is a network of resistors such

that the resistances are sampled from a probability distri-

bution, and the disorder strength may be controlled by a

tunable parameter [27–29]. Studies of such networks can

be approached through a multitude of ways, most primar-

ily by percolation theory [5,14,18,27,30–34], but also

through random walks [35,36], and optimization theory

[22,37–39].

While transport in directed polymers and porous rocks

is classical, transport in mesoscopic systems can often

∗sayakb@iitk.ac.in

†kramola@tifrh.res.in

be in the quantum (ballistic) regime, where the conduc-

tances are described by the Landauer-Büttinker formal-

ism [40]. However, we investigate RRNs in the Ohmic

limit, which is relevant for mesoscopic transport in the

diﬀusive regime, whenever the conduction length exceeds

the mean free path and phase coherence length of the

electronic wavefunction [40], as achieved in various con-

texts [41–43]. In fact, this formalism is valid whenever

the DC resistance can be deﬁned independent of the volt-

age between the terminals, which is also achieved in quan-

tum transport whenever eV and kBTare smaller than the

transmission coeﬃcient [44] (see for instance Refs. [45–

47]).

In this work, we consider an RRN with resistances sam-

pled from an exponentially wide distribution [19,28,48].

This model is also termed as the hopping percolation

model [5,21,31] and can be motivated from physical con-

siderations: the conductance between sites in disordered

media is often proportional to exp (−rij /r0−Eij /kBT)

where r0is a length scale for the decay of the wavefunc-

tion of the grains, rij is the distance and Eij is the energy

diﬀerence between two sites iand j. Thus, an exponen-

tial disorder of the form exp(axij )is natural: the strength

arepresents an energy and/or length scale of the disor-

dered system corresponding to the random variable xij

[21].

In exponentially disordered networks, past studies have

identiﬁed two disorder regimes—a weak disorder regime

when Laνand a strong disorder regime for Laν,

where νis the percolation connectedness exponent (ν=

4/3in two dimensions) [5,21,31,48,49]. The strong

disorder regime is characterized by optimal behavior: the

current distribution collapses to a self-similar fractal op-

timal path [39,48] (see Fig. 2for a visualization), whose

critical exponents (dopt = 1.22 in two dimensions) have

been numerically computed [22]. In the weak disorder

regime, the current distribution is delocalized through-

out the lattice and the optimal path (deﬁned as the path

of least resistance) is shown to be self-aﬃne with critical

exponents belonging to the universality class of directed

arXiv:2210.15562v3 [cond-mat.dis-nn] 30 Apr 2023

polymers [38,39]. This crossover from self-aﬃne to self-

similar is a characteristic of wide exponential disorder;

studies for Gaussian and uniform distributions give self-

aﬃne optimal paths across disorder strength [50]. Opti-

mal paths have, of course, been understood as equivalent

to domain walls in spin systems as well, where the im-

purities of the system help pin the wall to energetically

favourable sites in the system [51–53]. While critical ex-

ponents of the disorder regimes have been explored in

depth, a scalable analytical toolbox to analyse such dis-

ordered networks has not yet been developed.

In this article, we ﬁll this gap by demonstrating the use

of two analytic techniques to compute the lattice Green’s

functions of the disordered system. First, we construct

a perturbation theory that provides a hierarchical ex-

pansion for the nodal voltages in powers of the disorder

variables. Perturbative expansions have been attempted

in prior literature for computing the lattice conductance

[14,54], or in lattices with a small number of disordered

bonds [55], however, our formulated expansion helps to

explicitly compute disorder-averaged system observables

such as nodal voltages and bond currents by solving the

Dyson equation order by order. Similar perturbation ex-

pansions for Green’s functions have been helpful in study-

ing disordered crystalline media [56–62].

Next, we develop an exact formulation using a dyadic

perturbation of the lattice Green’s function, that can in

principle yield exact results for arbitrary disorder. The

dyadic bond formulation enables us to provide exact for-

mulae for the Green’s function for a ﬁnite number of dis-

ordered bonds in the lattice. Although similar ideas were

adopted previously for a single broken bond in the sys-

tem [55], we extend such a formulation to an arbitrary

number of bonds with disorder and are able to give an

analytically tractable expression for lattices with a small

number of disordered bonds. In particular, we provide

explicit formulae for lattices with up to four disordered

bonds.

We also perform numerical simulations that corrobo-

rate our theoretical results, as well as help us probe the

diﬀerent disorder regimes of the system. We compute

disorder-averaged nodal voltages with one, two and three

disordered bonds in the system, and study their ﬂuctu-

ations. The ﬂuctuations are shown to peak at a criti-

cal disorder strength and our numerics match perfectly

with the analytical predictions from exact formulae for

the Green’s functions. We also provide a novel order

parameter, which we term bond current ﬁdelity. This is

deﬁned as the overlap between the current distribution at

a particular disorder strength and the optimal path for

a particular resistance conﬁguration. We demonstrate

clear signatures of the weak and strong disorder regimes

in this order parameter, and its scalings are shown to be

in line with previously known critical exponents and our

analytical predictions using the Green’s function formal-

ism.

The rest of the paper is organized as follows. In Sec. II

we introduce the random resistor model and its lattice

i≡ |si⟩

j≡ |sj⟩

k≡ |sk⟩

|bα⟩

|bβ⟩

FIG. 1. The lattice convention used in this paper. Each

lattice site iis represented by a Ns=Lddimensional column

vector |sii, that forms an orthonormal basis. The basic lattice

translation vectors are denoted by {ˆem}with (1 ≤m≤d)

and to each site i, we assign the dbonds along ˆe, as shown

using the solid arrows along the bonds. Two representative

bond vectors are thus, |bαi ≡ |sii − |sjiand |bβi ≡ |sii − |ski,

where 1≤α, β ≤Nb= 2Ld.

Laplacian formulation. We discuss the disorder perturba-

tion expansion for the Green’s function and compute the

nodal voltages perturbatively in Sec. III. In Sec. IV, we

introduce the exact formulation to obtain Green’s func-

tions for arbitrarily many bonds with disorder in the lat-

tice. This work is supplemented by numerical techniques

for nodal voltages and discussion of an order parameter

in Sec. V. Finally, we discuss and conclude the work in

Sec. VI, and present directions for future research.

II. RANDOM RESISTOR NETWORK

In this work, we consider a d-dimensional hypercu-

bic lattice of linear dimension Lwith periodic bound-

ary conditions, and resistors placed at each bond. There

are Ld(= Ns)sites and dLd(= Nb)bonds on such

ad-dimensional torus. While we consider a hypercubic

lattice with periodic boundary conditions, our formula-

tion can be easily adapted to other lattices with alternate

boundary conditions as well. Each lattice site is denoted

by a site index (in Latin alphabets) i≡(x1, x2, . . . , xd)

where {xk}(1 ≤k≤d)are the Cartesian coordinates of

the real space vector corresponding to site i(1≤i≤Ns).

For the formulation developed in this paper, we ﬁnd it

convenient to use bra-ket notation to denote the total

degrees of freedom on the lattice. The bras and kets

are vectors in Nsdimensional space. We deﬁne a basis

set of site vectors {|sii} that denote the sites so that

the vector |sii(denoted equivalently by site index i)

is the ith unit vector in Nsdimensional space. Thus,

hsj|sii=δij (1 ≤i, j ≤Ns)and the site vectors form a

complete orthonormal set, that is, Pi|sii hsi|= 1. A site

index subscript on a ket denotes the corresponding entry

of the column vector, for example, |siijis the jth entry of

this column vector. The notation hijiindicates that sites

iand jare connected by a bond on the lattice (nearest

neighbours). Analogous to the site vectors, it is also use-

ful to deﬁne bond vectors. We denote a (positive) unit

vector along a Cartesian axis by ˆe. Then, to each site i,

we unambiguously associate the dbonds along the posi-

tive Cartesian axes and denote the oriented bond along

ˆeby the notation hijiˆe. We can now deﬁne the bond

vectors for the bonds in the lattice (indexed by Greek

letters) by

|bαi:=|sii−|sjiwith hijiˆe.(1)

The bond index α(1≤α≤Nb) is equivalent to the tuple

(i, ˆe)which uniquely marks the bond starting at ialong

ˆe(see Fig. 1).

In the network, each of the bonds between two sites i

and jhas a resistance of magnitude Rij . For the voltages

at each site, we deﬁne a nodal voltage vector |Vi. We also

deﬁne a nodal current vector |Ii, which represents the al-

gebraic sum of currents exiting a site. At all sites iin the

lattice not connected to external leads, Kirchhoﬀ’s cur-

rent law trivially demands |Iii= 0. Thus, typically, the

nodal voltages are the most relevant (unknown) degrees

of freedom. To denote the bond currents, we deﬁne d

bond current vectors {|Jˆei} corresponding to the current

ﬂowing in the ˆedirection in the relevant bond assigned

to each site. The Nb-dimensional complete bond current

vector is denoted by |Ji≡|Jˆe1|Jˆe2|. . . |Jˆedi, where |sep-

arates the dblocks of the ket.

Our formulation in the following sections is indepen-

dent of the speciﬁcities of the external current or voltage

conﬁguration that sets up a steady-state in the lattice

(which we solve for). In conductivity measurements, a

popular choice is a bus-bar conﬁguration, where the volt-

ages are ﬁxed on two opposite sides [4]. In our numer-

ics, we consider a simpler conﬁguration—we ﬁx a current

source node iin and current sink node iout which input

and output unit current respectively. We thus set the

nodal current conﬁguration to be given by

|Iii:=δi,iin −δi,iout , , (2)

without loss of generality. We ﬁnd this choice particu-

larly useful to demonstrate current localization to opti-

mal paths in the circuit.

As stated before, the theory holds for any arbitrary

voltage or current source-sink conﬁguration. In this par-

ticular work, we present simulations for iin ≡(−(L−

1)/2,0) and iout ≡(0,0) on the square lattice in two

dimensions, so that the current enters at the midpoint

of the left boundary and exits at the center node of the

lattice (see Fig. 2for reference). This source-sink conﬁg-

uration forces the system size Lto be an odd integer.

Lattice Laplacian Formulation

We assume that the bond resistances in the RRN are

independent of the voltage diﬀerence between the sites

−12 −8−4 0 4 8 12

−12

−8

−4

0.05

0.10

0.15

0.20

0.25

0.30

−12 −8−4 0 4 8 12

−12

−8

−4

0.2

0.4

0.6

0.8

−12 −8−4 0 4 8 12

−12

−8

−4

0.0

0.2

0.4

0.6

0.8

−12 −8−4 0 4 8 12

−12

−8

−4

0.0

0.2

0.4

0.6

0.8

(a) (b)

a=1 a =15

a=30 a =45

FIG. 2. Current distributions for a 25 ×25 lattice with ex-

ponential disorder (see Sec. III) at disorder strengths of (a)

a= 1, (b) a= 15, (c) a= 30 and (d) a= 45. The resistances

at each bond are distributed as Rij :=eaxij , with the random

variables {xij }remaining ﬁxed as the disorder strength ais

increased. The current source and sink are at (−12,0) and

(0,0) respectively. Convergence to an optimal path (dark red

path in (d)) can be clearly observed with increasing disorder

strength.

(as in Ohm’s law), and obtain the following,

|Jˆeii=|Vii− |Vij

Rij

with hijiˆe.(3)

Now, due to local charge conservation in the steady state,

we apply Kirchhoﬀ’s current law, given by

|Iii=X

jwith hiji

|Vii− |Vij

Rij

.(4)

where jwith hijiindicates a sum over index jwhen-

ever iand jare connected by a bond. We study RRNs

where the bond resistances are perturbed from a mean

resistance R0, which without loss of generality we can set

equal to 1 unit. We can recast Eq. (4) in the following

linear algebraic form

L|Vi+|Ii= 0,(5)

where Lis the lattice Laplacian (or conductance matrix)

[63]. Explicitly, the Laplacian is given by

[L]ij :=









−Pjwith hiji(Rij )−1if i=j

(Rij )−1if hiji

0otherwise.

(6)

Observe that when all resistances are equal to R0,Lre-

duces to the usual circulant form of the lattice Lapla-

cian of a d-dimensional torus, as expected. Clearly,

Eq. (5) can be solved by inverting the Laplacian, thus

|Vi=−L−1|Ii. Thus our basic object of study is the

lattice Green’s function given by

G≡ −L−1,(7)

which provides all the system properties. We must be

careful to note that due to the sum rule implemented

by Kirchoﬀ’s current law [64], the Laplacian is a non-

invertible matrix, and hence, must be inverted by pro-

jecting out the zero mode |0i= (1 1 . . . 1)Tof the Lapla-

cian. Speciﬁcally, the Green’s function and the Laplacian

are related by

LG =GL ≡ −(1− |0i h0|).(8)

Since the voltages in the system are equivalent upto an

arbitrary constant, this Green’s function can be used as

an inverse without concern. While, in principle, such

an inversion may be performed numerically, in Sec. III

and IV, we construct analytic techniques to compute the

Green’s function for a disordered lattice in terms of the

Green’s function for the perfect lattice.

We now demonstrate how to compute the bond cur-

rents generally. The relationship in Eq. (3) may be recast

into the following linear algebraic form

Dˆe|Vi+|Jˆei= 0,(9)

where the diﬀerence matrices are given explicitly by

[Dˆe]ij :=Rhijiˆe−1









−1if i=j

1if hijiˆe

0otherwise

(10)

where Rhijiˆeis the resistance on hijiˆe. Again, notice that

when all the resistances are equal to R0, this generalized

diﬀerence matrix becomes the usual diﬀerence operator

for a d-dimensional torus. Thus, once the voltages are

known, the bond currents can be simply computed using

|Jˆei=−Dˆe|Vi.

Our formulations in the following sections will be in-

dependent of the explicit choice of disorder in the resis-

tances. As motivated in the introduction, however, we

intend to study the crossover from the regimes of weak

to strong disorder in the hopping percolation model [48],

which is obtained by setting

Rij :=eaxij ,(11)

where xij ∈(0,1) (and iand jshare a bond) is a uni-

formly distributed random variable and acontrols the

strength of the disorder. The limit a→0yields a lat-

tice with zero disorder (perfect lattice), while a→ ∞

provides the strong disorder limit. For ease of analytical

calculations, we ﬁnd it convenient to introduce the scalar

variables {ζij }which represent the disorder in the bond

resistances. The explicit relationship considered is given

Rij := (1 −ζij )−1.(12)

Then, from Eqs. (11) and (12), one can ﬁnd that the

distribution of the variables ζis given by (for a≥0)

f(ζ) = a−1(1 −ζ)−1for 0< ζ < 1−e−a.(13)

This also implies that the resistances obey an inverse

probability distribution, that is, f(R)=1/(aR)for 1≤

R≤ea. The moments of the disorder ζcan also be

computed exactly; in particular, the ﬁrst three moments

are given by hζi= (−1 + a+e−a)/a,hζ2i= (−3 +

2a+ 4e−a−e−2a)/(2a), and hζ3i= (−11 + 6a+ 18e−a−

9e−2a+2e−3a)/(6a)where h·i denotes a disorder ensemble

average.

III. DISORDER PERTURBATION EXPANSION

For the weak disorder regime, that is, a small deviation

from the perfect lattice, we can compute the degrees of

freedom accurately using a perturbation expansion in the

disorder. We control such a perturbation by a tuning

parameter λ, such that 0<λ<1and λ= 0 corresponds

to the zero disorder state. Therefore, we associate the

tuning parameter to the disorder ζso that the resistances

are redeﬁned as Rij ≡(1 −λζij )−1. We consider linear

perturbations on the Laplacian and diﬀerence operators

as follows

L:=L(0) +λL(1),(14a)

Dˆe:=D(0)

ˆe+λD(1)

ˆe.(14b)

Note, the explicit forms of the perfect lattice’s Laplacian

L(0) and diﬀerence operator D(0)

ˆecan be obtained from

Eqs. (6) and (10) by setting all resistance magnitudes

Rij to 1.

A linear order perturbation in these operators of the

network is complete, and should induce perturbation ex-

pansions (in λ) upto all higher orders for the system vari-

ables. Thus, we assume

|Vi:=|Vi(0) +λ|Vi(1) +λ2|Vi(2) +O(λ3),(15a)

|Jˆei:=|Jˆei(0) +λ|Jˆei(1) +λ2|Jˆei(2) +O(λ3)(15b)

where the superscript denotes the order of the expan-

sion; naturally, the (0) index denotes the values of the

quantities in the zero disorder state.

Applying the constraint that the above equations must

obey Ohm’s and Kirchhoﬀ’s laws at each order of λ, we

obtain a hierarchical scheme to explicitly determine the

higher order corrections to each of the above quantities.

We ﬁrst use Eqs. (4) and (5) to determine the corrections

to the Laplacian matrices and the nodal voltages. Using

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Green'sFunctionsforRandomResistorNetworksSayakBhattacharjee1,andKabirRamola2,y1DepartmentofPhysics,IndianInstituteofTechnologyKanpur,Kanpur208016,India2TataInstituteofFundamentalResearch,Hyderabad500107,India(Dated:May2,2023)WeanalyzerandomresistornetworksthroughastudyoflatticeGreen'sfunctionsinarb...

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Greens Functions for Random Resistor Networks Sayak Bhattacharjee1and Kabir Ramola2y 1Department of Physics Indian Institute of Technology Kanpur Kanpur 208016 India

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