# Bias induced drift and trapping on random combs and the Bethe lattice:
Fluctuation regime and first order phase transitions

Jesal D Kotak<EMAIL_ADDRESS>Mustansir Barma<EMAIL_ADDRESS>
###### Abstract

We study the competition between field-induced transport and trapping in a
disordered medium by studying biased random walks on random combs and the
bond-diluted Bethe lattice above the percolation threshold. While it is known
that the drift velocity vanishes above a critical threshold, here our focus is
on fluctuations, characterized by the variance of the transit times. On the
random comb, the variance is calculated exactly for a given realization of
disorder using a ‘forward transport’ limit which prohibits backward movement
along the backbone but allows an arbitrary number of excursions into random-
length branches. The disorder-averaged variance diverges at an earlier
threshold of the bias, implying a regime of anomalous fluctuations, although
the velocity is nonzero. Our results are verified numerically using a Monte
Carlo procedure that is adapted to account for ultra-slow returns from long
branches. On the Bethe lattice, we derive an upper bound for the critical
threshold bias for anomalous fluctuations of the mean transit time averaged
over disorder realizations. Finally, as for the passage to the vanishing
velocity regime, it is shown that the transition to the anomalous fluctuation
regime can change from continuous to first order depending on the distribution
of branch lengths.

###### keywords:

Disordered media , Random combs , Anomalous transport , First return time ,
Monte Carlo methods

###### PACS:

05.40.Fb , 05.40.-a , 05.10.Ln , 71.55.Jv

###### MSC:

[2020] 82D30 , 82C44 , 82C41 , 60K50 , 65C05

††journal: Physica A

[inst1]organization=TIFR Centre for Interdisciplinary Sciences, Tata Institute
of Fundamental Research,addressline=Gopanpally, city=Hyderabad,
postcode=500046, country=India

[inst2]organization=Indian Institute of Space Science and
Technology,addressline=Valiamala, city=Thiruvananthapuram, postcode=695547,
country=India

##  1 Introduction

Transport in disordered media is important in many settings [1, 2, 3] ranging
from conduction in disorder-doped semiconductors [4] and hydrodynamic
dispersion in porous media [3] to the diffusion of molecules in chromatography
[5, 6]. Especially interesting effects arise in the presence of a biasing
field, which drives the particles preferentially in one direction, inducing a
drift of the carriers. While a low value of bias induces a drift velocity
$v_{drift}$ proportional to the bias [7, 8, 9], larger values lead to
$v_{drift}$ becoming non-monotonic because of local traps which are easy to
reach but difficult to get out of. These traps correspond to local minima of
the potential energy; passage out of each trap typically involves activation,
a very slow process. Various aspects of the competition between drift and
trapping, both induced by the biasing field, and their effect on macroscopic
transport properties have been studied in [10, 11, 12, 13, 14, 15, 16, 17].

A good first model of a disordered medium is provided by the infinite spanning
cluster in the percolation problem, which forms above a critical value $p_{c}$
of the fraction $p$ of randomly occupied bonds. Considering the motion of a
biased random walker on this structure, it was argued that the macroscopic
drift velocity $v_{drift}$ is a non-monotonic function of bias, and in fact
vanishes above a critical value of the bias [10], later confirmed by Monte
Carlo simulations [18, 19, 20, 16]. Two types of traps were identified, namely
dead-end branches in the direction of the field, and backbends in infinite
paths along the backbone of the infinite cluster. Simple models were
introduced to study the effect of the two types of traps - the random comb
(RC) to model branches of random lengths, and a 1-D model with random
reversals (RR) of the easy direction, to model backbends along the backbone,
[10] (Fig. 1). In both RC and RR models it was found that $v_{drift}=0$ beyond
a critical value of the bias [10, 11, 13, 15, 14]. Moreover, in the RR model,
the diffusion constant was shown to diverge at a smaller critical value of
bias, indicating the presence of a regime with anomalously large spreads of
the profile in a range where $v_{drift}$ is non-zero. [11, 21, 22].

(a) Random comb (RC)

(b) Random reversals (RR)

Field Figure 1: Models for studying transport in disordered media

In this paper, we study fluctuations in time taken to cover a macroscopic
distance in the RC model with quenched disorder (branch lengths do not change
in time). We compute the variance of transit times and show that it diverges
at a critical bias even in the RC implying that there is indeed a second
regime where the variance of transit times is anomalously large although the
drift velocity is finite. This is established by considering a generalized RC
model with different strengths of bias along the backbone and the branches and
analyzing the dynamics exactly. In the limit where only forward movement is
allowed along the backbone, we compute two distinct types of variances in
transit times: (a) $\sigma_{temporal}^{2}$, coming from history-to-history
fluctuations and (b) $\sigma_{disorder}^{2}$, coming from sample-to-sample
fluctuations of the history-averaged mean transit time. The steady state
considerations that allowed $v_{drift}$ to be found in [13], can be used to
find $\sigma_{disorder}^{2}$ but not the variance $\sigma_{temporal}^{2}$,
which is relevant for the practical case of a single realization of quenched
disorder. To compute the latter, we need to follow the path of a single
walker, along the RC, allowing for multiple excursions into each branch, as
studied in [14] and [15], but circumventing the approximation made in these
works. We find that $\sigma_{temporal}^{2}$ and $\sigma_{disorder}^{2}$ are
different functions of the bias, yet they diverge at the same critical bias,
indicating the onset of a regime where fluctuations are anomalously large.

Thus there are three regimes of transport through the RC: (a) the Normal
Transport regime, for small values of bias where $v_{drift}$ and
$\sigma_{temporal}^{2}$ are finite, (b) the Anomalous Fluctuation (AF) regime,
which occurs for intermediate values of bias where $v_{drift}$ is non-zero but
$\sigma_{temporal}^{2}$ diverges and (c) the Vanishing Velocity (VV) regime
where $v_{drift}$ is zero and $\sigma_{temporal}^{2}$ remains divergent.
Figure 2 shows the differences in the profiles of particle position and
distribution of transit times in these three regimes.

(a) Particle position profile on backbone

(b) Transit time probability density

Figure 2: Profiles of particle position and transit time in the three regimes:
(i) Normal Transport (black), (ii) Anomalous Fluctuation (blue) and (iii)
Vanishing Velocity (red). (a) In the Normal Transport and the AF regimes the
particle covers a significant distance at $t=6400s$ while in the VV regime the
displacement is negligible. (b) The transit time probability density at
$N=460$ follows normal behaviour in the Normal Transport regime, whereas in
the AF and the VV regimes it has a fat tail. In (b) the distributions for the
AF and VV regimes are shifted by $500$ and $2000$ units along the x-axis,
respectively.

A second issue addressed in this paper concerns the order of the transition to
the strong fluctuation regime. In [17], it was pointed out that the transition
of $v_{drift}$ to zero may be continuous or first order, depending on whether
the distribution of branch lengths is exponential (as in [10, 13]), or whether
there are strong enough power-law corrections to the exponential form
characterised by the exponent $\lambda$ in Eq. (1). Interestingly, the
variance $\sigma_{temporal}^{2}$ shows a transition (first-order or
continuous) from a finite value to infinity, with the order of the transition
being the same as the transition to $v_{drift}=0$, even though the two
transitions occur at quite different values of the bias. Figure 3 depicts the
three regimes of transport along with the order of transitions to the AF and
VV regimes, as a function of bias and exponent $\lambda$.

Finally, we also analyze the fluctuations in transit time on the bond diluted
Bethe lattice. We analyze only $\sigma_{disorder}^{2}$ in this case and show
that it diverges at a critical value of the bias, implying the existence of a
second anomalous regime in this case as well.

$x\to\infty$$g=0$$x=2$$x=1$$x=0$$g=1$$x$$\lambda=0$$\lambda=1$$\lambda$Second
orderFirst orderSecond orderFirst orderNormalTransport$v_{drift}>0$
$\sigma_{temp}^{2}<\infty$AnomalousFluctuation$v_{drift}>0$
$\sigma_{temp}^{2}\to\infty$VanishingVelocity$v_{drift}=0$
$\sigma_{temp}^{2}\to\infty$ Figure 3: Depiction of the different regimes of
transport in the RC. Here $x=L_{g}/\xi$ with $L_{g}$ and $\xi$ being the
lengths which characterise the bias and disorder realizations respectively.
$\lambda$ is the power which appears in the disorder distribution Eq. (1).

##  2 Random Comb (RC) Model

###  2.1 Distribution of branch lengths

A comb is a structure consisting of a one-dimensional backbone with branches.
Combs with branches of infinite length are known to lead to anomalous
transport properties for random walkers [23]. The random comb (RC) under
consideration here is a disordered structure with branches of random lengths
much like the infinite cluster in percolation (Fig. 1). A realization of
disorder, R, is specified by the set of lengths
$\\{L_{1},L_{2},\dotsc,L_{N}\\}$ of branches which are attached to a backbone
of $N$ sites. These lengths are drawn independently from a distribution
$\Gamma(L)$, implying that the probability of occurrence of the configuration
is $P(\textbf{R})=\prod_{i=1}^{N}\Gamma(L_{i})$. The choice of $\Gamma(L)$ is
motivated by the form of the two-point correlation function $C(r)$ in the
percolation problem, thus taken to be

$\Gamma(L)=\frac{1}{Z}\,\frac{e^{-L/\xi}}{1+(L/L_{0})^{\lambda}}$ (1)

where $\xi$ is the correlation length and
$Z=\sum_{L=0}^{\infty}e^{-L/\xi}[1+(L/L_{0})^{\lambda}]^{-1}$ is a
normalization constant.

The above form for $\Gamma(L)$, proposed earlier in [17], is motivated by the
Ornstein-Zernike (OZ) form for $C(r)$ for $p>p_{c}$. The OZ value of $\lambda$
is $(d-1)/2$ for spatial dimensions $d\geq 3$ and $\lambda=2$ for $d=2$ [24,
25, 26]. On the RC it is instructive to consider the effect of different
values of $\lambda$ as it has a significant effect on the nature of the
transition as discussed in Section 4. The case $\lambda=0$, corresponding to a
purely exponential decay, was the form assumed in [10, 13], while the case
$\lambda=2$ was studied in [17].

A quantity $A(\textbf{R},t)$ may have different values in every temporal
evolution, or history, of particle motion in the same realization R with the
same initial conditions. The average over histories for a given R is denoted
by $\overline{A(\textbf{R},t)}$. Subsequently, one may take an average over
the disorder configurations, defined as
$\langle\overline{A(t)}\rangle=\sum_{\textbf{R}}\overline{A(\textbf{R},t)}P(\textbf{R})$.

###  2.2 Particle Dynamics

We model the motion of particles on the disordered medium by a random walk. In
the absence of a driving field, the particles undergo diffusion on the lattice
[27]. The effect of an applied field is modelled by making the random walk
biased, with a larger transition rate along the field than against it. The
value of the bias in the backbone and branches may differ, reflecting the fact
that the applied field may have different components along the two (Fig. 4).

$\theta$BackboneBranchesE Figure 4: Generalized RC model with different
components of bias along the backbone and the branches.

Let $n=1,2,...,N$ be the coordinate of a site along the backbone, to which is
attached a branch of length $L_{n}$, and let $m=0,1,...,L_{n}$ label the sites
in the branch, with $m=0$ corresponding to the backbone site (Fig. 5). Let the
rate of transition between any two sites $a$ and $b$ be $W_{a,b}$. Therefore,
$W_{n,n+1}=w$ and $W_{n,n-1}=w^{\prime}$ are the forward and the backward
transition rates along the backbone, and $W_{m,m+1}=u$ and $W_{m,m-1}=v$ are
the forward and the backward rates, in the branches, respectively. Let
$p_{n,m}(t)$ be the probability that a particle is on site $(n,m)$ of the RC
at time $t$. If $L_{n}$ is non-zero, the time evolution of $p_{n,m}(t)$
follows the equations

$\begin{split}\frac{dp_{n,0}(t)}{dt}&=w\,p_{n-1,0}(t)+w^{\prime}\,p_{n+1,0}(t)+v\,p_{n,1}(t)-(w+u+w^{\prime})\,p_{n,0}(t),\\\
\frac{dp_{n,m}(t)}{dt}&=u\,p_{n,m-1}(t)+v\,p_{n,m+1}(t)-(u+v)p_{n,m}(t),\quad
0<m<L_{n},\\\
\frac{dp_{n,L_{n}}(t)}{dt}&=u\,p_{n,L_{n}-1}(t)-v\,p_{n,L_{n}}(t).\end{split}$
(2)

If $L_{n}=0$, the evolution follows

$\frac{dp_{n,0}(t)}{dt}=w\,p_{n-1,0}(t)+w^{\prime}\,p_{n+1,0}(t)-(w+w^{\prime})\,p_{n,0}(t).$
(3)

$(1,0)$$(2,0)$$\dotsc$$(n,0)$$\dotsc$$(N,0)$$(n,m)$$u$$v$$u$$v$$w$$w$$\gamma=u+v$$\gamma=v$$\gamma=u+w$$\gamma=w$
Figure 5: In the forward transport model ($w^{\prime}=0$) the transition
rates, $u$ and $v$ along branch bonds and $w$ along the backbone bonds, are
uniform. This corresponds to the inverse of mean waiting time $\gamma$ of a
random walker being different on different sites.

The rates of hopping ($u,v,w$ and $w^{\prime}$) are related to the strength of
the field $E$ and temperature $T$ through:

$\begin{split}\frac{v}{u}&=e^{-Ea\sin\theta/k_{B}T},\\\
\frac{w^{\prime}}{w}&=e^{-Ea\cos\theta/k_{B}T}\end{split}$ (4)

where $a$ is the lattice spacing (taken to be unity in the rest of the paper).
The bias in a branch may be parameterized by $g$, defined through
$u=W_{0}(1+g)$, $v=W_{0}(1-g)$, where $W_{0}$ is a constant. We assume a
uniform set of transition rates $(u,v,w$ and $w^{\prime})$ which ensures that
the particle density is uniform in the backbone in the steady state.

###  2.3 Forward Transport Model

The general problem of transport on the RC involves calculating the total
transit time along the backbone. Now any revisit to a site on the backbone
(after moving away from it, along the backbone) entails remembering the length
of the branch on the site in question, which makes the process non-Markovian.
In the spirit of a commonly used approximation, namely the ‘continuous-time
random walk’ (CTRW) as used in [28, 29], the non-Markovian property may be
avoided by considering a process in which a fresh branch length is drawn from
the given distribution, at each revisit to the site [15, 14]. One may ask how
good the CTRW approximation is. For a certain problem with disordered hopping
rates, it was shown that this approximation gives the exact answer for the
average probability of occupying a site at a certain time, but cannot properly
account for the variance of the probability [30]. A similar conclusion is
likely to hold for the CTRW-like approximation for the random comb.

Our approach is different; we consider a forward transport limit in which the
problem of memory does not arise at all: the particle is allowed to hop only
in the forward direction along the backbone so that revisits to a backbone
site $n$ are not allowed once the particle has jumped to site $n+1$. This
allows exact results to be obtained and obviates the necessity of making a
CTRW-like approximation in which a fresh length of the branch is drawn on
every revisit to site $n$. We work with the FTM as it offers simplicity and
tractability, avoiding any approximation, yet keeps the essence of the
problem, namely the effects of quenched disorder on dynamics, intact.

The FTM may be realized as follows. Consider applying a field (Fig. 4),
aligned almost along the backbone so that $\theta$ is very small. Now consider
the joint limit $\theta\to 0$, $E\to\infty$, such that $E\theta\sim$ constant.
This leads to $w^{\prime}/w\simeq 0$, hence disallowing backward motion along
the backbone. (Another way of achieving this limit is to keep the field
constant and reduce the temperature of the system $T\to 0$ with $\theta\to 0$
such that $\theta/T\sim$ constant.) Examples of systems where a “no-return”
condition may be met are hydrodynamic dispersion in porous media [3] and high-
performance liquid chromatography [6].

In the FTM, the hopping rate $w^{\prime}$ in Eqs. (2) and (3) vanishes. The
rates $u,v$ and $w$ may be related to the dynamics of a single particle on the
RC as illustrated in Fig. 5. Let $\gamma$ be the rate for particle movement
from a particular site ($\gamma$ is site-dependent). For any branch site
except the last, $\gamma=u+v$, with a probability $p=u/(u+v)$ for a downward
and probability $q=v/(u+v)$ for an upward hop. At the last site in any branch,
$\gamma=v$, and the probability of an upward hop is $1$. Likewise, on the
backbone, for a site with no branch, $\gamma=w$ with probability $1$ for
rightward motion, whereas on a site with a branch, $\gamma=u+w$ with a
probability $r=u/(u+w)$ of entering the branch, and $(1-r)=w/(u+w)$ of moving
rightward along the backbone. This choice of inhomogeneous values of $\gamma$
for the walker corresponds to uniform transition rates on backbone bonds - the
feature which ensures uniform density along the backbone, allowing the current
to be calculated in steady state. By contrast, if $\gamma$ is chosen to be
uniform (as in [15]), the transition rates on backbone bonds are non-uniform,
and determining the density profile is a nontrivial problem. The distribution
$\phi(t)$ of the hopping times is given by

$\phi(t)=\gamma e^{-\gamma t}.$ (5)

##  3 Fluctuations of the transit time

While characterizing transport, we may ask either for the distribution of the
location of a particle at a specified time $t$, or for the distribution of the
time taken for a particle to reach a specified site. In general, it is not
straightforward to go between these two descriptions [31], and we choose to
work with the latter.

In this section, we first discuss the steady state density profile on the RC,
and how it may be used to find $v_{drift}$. However, this approach does not
suffice to compute the variance $\sigma_{temporal}^{2}$ (defined in Sec. 3.2)
of transit times, which is the central quantity of interest in this paper. We
compute it in the FTM by following the trajectory of a walker along the
backbone by including multiple excursions into branches.

###  3.1 Steady state: Calculation of $v_{drift}$

Consider the forward transport model on a random comb with a transition rate
$w$ between sites $(n,n+1)$ of the backbone $n=1,2,\dotsc,N_{s}$, with
periodic boundary conditions. In the steady state reached at long times, the
current vanishes in every branch due to reflecting boundaries at branch tips,
whereas it has a uniform value $J$ in every bond of the backbone. This
suffices to find the full density profile, which in turn determines the
transit time, $T_{N_{s}}(\textbf{R})$, taken by a particle to traverse the
$N_{s}$ sites of the backbone in a given realization R. Following [13], if
$\rho_{0}$ is the density at site $(n,0)$, we have $\rho_{0}=J/w$ for all $n$.
The zero-current condition in the branch implies that,
$\rho_{m}=\rho_{0}\,e^{m/L_{g}}$, where

$L_{g}=\frac{1}{\log{(u/v)}}$ (6)

is a bias-induced length scale which describes the exponential growth of
density in the branches. The mean transit time in the steady state is related
to the overall number of particles $\mathcal{N}$ by
$\overline{T}_{N_{s}}(\textbf{R})=\mathcal{N}/J$. Thus as in [13], we find

$\overline{T}_{N_{s}}(\textbf{R})=\sum_{i=1}^{N_{s}}\overline{t}_{i}$ (7)

where,

$\overline{t}_{i}=\frac{1}{w}\frac{\exp[(L_{i}+1)/L_{g}]-1}{\exp(1/L_{g})-1}.$
(8)

So long as the disorder average of $\overline{t}_{i}$, over the distribution
(1), is finite (i.e. if $L_{g}>\xi$), the law of large numbers holds and
implies that $T_{N_{s}}(\textbf{R})/N_{s}$ converges to $\langle
T_{N_{s}}(\textbf{R})\rangle/N_{s}$ as $N_{s}\to\infty$. The answer is
independent of R and for the case of an exponentially decaying distribution of
branch lengths (i.e. $\lambda=0$ in (1)) is given by

$\displaystyle\frac{\langle\overline{T}_{N_{s}}\rangle}{N_{s}}$
$\displaystyle=\frac{1}{w\big{[}1-\exp{(1/L_{g}-1/\xi)}\big{]}},$
$\displaystyle L_{g}>\xi,$ (9) $\displaystyle=\infty,$ $\displaystyle
L_{g}\leq\xi.$

Correspondingly, the mean drift velocity is given by
$v_{drift}=N_{s}/\langle\overline{T}_{N_{s}}\rangle$. Therefore, in the steady
state, in the forward transport limit for the random comb, we have

$\displaystyle v_{drift}$
$\displaystyle=w\big{[}1-\exp(1/L_{g}-1/\xi)\big{]},$ $\displaystyle
L_{g}>\xi,$ (10) $\displaystyle=0,$ $\displaystyle L_{g}\leq\xi.$

The condition $v_{drift}=0$ defines the VV regime.

###  3.2 Fluctuations of transit times: Calculation of $\sigma_{temporal}^{2}$

The variances $\sigma_{temporal}^{2}$ and $\sigma_{disorder}^{2}$ are defined
as follows

$\begin{split}\sigma_{temporal}^{2}&=\langle\overline{T_{N}^{2}(\textbf{R})}\rangle-\langle\overline{T_{N}(\textbf{R})}^{2}\rangle\\\
\sigma_{disorder}^{2}&=\langle\overline{T_{N}(\textbf{R})}^{2}\rangle-\langle\overline{T_{N}(\textbf{R})}\rangle^{2}.\end{split}$
(11)

The steady state approach suffices to calculate $\sigma_{disorder}^{2}$ (the
variance due to sample-to-sample fluctuations of the mean transit time across
realizations of disorder) but not $\sigma_{temporal}^{2}$ (the variance due to
history-to-history fluctuations of the total transit time in realization R,
finally averaged over all R). To find this, we need to track the full
trajectory of the particle through the system. We use the FTM, the key
simplifying feature of which is that the total transit time
$T_{N}(\textbf{R})$ is just the sum of residence times $\tau(L_{n})$ on the
individual backbone sites $n$, where $\tau(L_{n})$ is the time interval
between reaching site $n$ of the backbone and hopping from $n$ to $n+1$.
During the time it is resident on site $n$, the particle may make multiple
excursions into the attached branch if any. The residence time, therefore,
involves the sum of the first-return times corresponding to each excursion.
Therefore, we need the distribution of the first-return times on a single
branch.

We assume the backbone to be infinite and follow the particle till it reaches
the $N^{th}$ backbone site, with the initial condition that it is on the
backbone site $n=1$ at time $t=0$. The variance of transit time in disorder
realization R due to history-to-history fluctuations is

$\sigma_{N}^{2}(\textbf{R})=\overline{T_{N}^{2}(\textbf{R})}-\overline{T_{N}(\textbf{R})}^{2}.$
(12)

where $T_{N}(\textbf{R})=\sum_{i=1}^{N}\tau(L_{i})$. Let $\psi_{L}(t)$ denote
the distribution of the residence times. A typical excursion into the branch
involves the particle going from backbone site $m=0$ to branch site $m=1$
after waiting for some time at $m=0$. It may go further into the branch,
finally returning to $m=0$, marking the completion of one excursion, and may
perform several such excursions at $n$ before hopping to $n+1$. Let
$t^{(j)}(L)$ denote the total time taken for $j$ excursions in a branch of
length $L$ and let $f^{(j)}_{L}(t)$ be the distribution of these times. Thus,
$t^{(1)}(L)$ denotes the first return time to the backbone and
$f^{(1)}_{L}(t)$ denotes the distribution of first return times. Let us, for
clarity, denote them by $t_{fr}(L)$ and $f_{L}(t)$, respectively. The particle
may finally wait on the backbone site after the $j$ excursions, before hopping
to the next backbone site, therefore,

$\psi_{L}(t)=\sum_{j=0}^{\infty}\int_{0}^{t}dt_{1}\,r^{j}f^{(j)}_{L}(t_{1})(1-r)\phi(t-t_{1}).$
(13)

where $r$ is the probability of making an excursion into the branch.
Decomposing the total time taken for $j$ excursions into individual first
return times, we have

$f^{(j)}_{L}(t)=\int_{0}^{t}dt_{j}\dotsc\int_{0}^{t_{3}}dt_{2}\int_{0}^{t_{2}}dt_{1}\,f_{L}(t_{1})f_{L}(t_{2}-t_{1})\dotsc
f_{L}(t-t_{j-1}).$ (14)

###  3.3 First return times on a branch

Consider a single branch of length $L$ ($>0$). At $t=0$, let a particle be at
$m=0$. The distribution of first return times, $f_{L}(t)$, is found in terms
of $\phi(t)$, the waiting time density at $m=0$, and $Q_{L}(t)$, the
distribution of first passage times from branch site $(n,1)$ to the backbone
site $(n,0)$:

$f_{L}(t)=\int_{0}^{t}dt_{1}\,\phi(t_{1})Q_{L}(t-t_{1}).$ (15)

Let $\tilde{f}_{L}(s),\tilde{\phi}_{L}(s)$ and $\tilde{Q}_{L}(s)$ be the
Laplace transforms of $f_{L}(t)$, $\phi(t)$ and $Q_{L}(t)$, respectively, for
example

$\tilde{f}_{L}(s)=\int_{0}^{\infty}dt\,e^{-st}f_{L}(t).$ (16)

The convolution form of (15) implies that

$\tilde{f}_{L}(s)=\tilde{\phi}(s)\tilde{Q}_{L}(s)$ (17)

where

$\tilde{\phi}(s)=\frac{\gamma}{\gamma+s}$ (18)

follows from (5) and $\gamma=u+w$ at $m=0$. The expression for
$\tilde{Q}_{L}(s)$ was found exactly in [32],

$\tilde{Q}_{L}(s)=\Big{(}\frac{v}{u}\Big{)}^{1/2}\bigg{[}\frac{\sqrt{v/u}\sinh{L\zeta}-\sinh{(L-1)\zeta}}{\sqrt{v/u}\sinh{(L+1)\zeta}-\sinh{L\zeta}}\bigg{]}$
(19)

where

$\cosh{\zeta}=\frac{(u+v)}{\sqrt{4uv}}\Big{(}1+\frac{s}{u+v}\Big{)}.$ (20)

This result was later used in [15] within a CTRW-like approximation in which
branch lengths are drawn freshly at each revisit to a site.

Let $\overline{t}_{fr}(L)=-\big{[}\partial\log\tilde{f}_{L}(s)/\partial
s\big{]}_{s=0}$ be the history-averaged mean first return time. After
simplification we obtain

$\overline{t}_{fr}(L)=\frac{(u/v)^{L}-1}{u-v}+\frac{1}{u+w}.$ (21)

Similarly, we find the variance of the first return times by calculating the
second derivative of $\log\tilde{f}_{L}(s)$ at $s=0$,
$\sigma^{2}_{fr}(L)=\big{[}\partial^{2}\log\tilde{f}_{L}(s)/\partial
s^{2}\big{]}_{s=0}$, leading to

$\sigma_{fr}^{2}(L)=\frac{(u+v)(u/v)^{2L}-4L(u-v)(u/v)^{L}-(u+v)}{(u-v)^{3}}+\frac{1}{(u+w)^{2}}.$
(22)

In the limit $L\to\infty$, the mean $\overline{t}_{fr}(L)$ and the variance
$\sigma_{fr}^{2}(L)$ diverge, as expected for a biased random walk in an
infinite 1D system. Interesting effects coming from the finiteness of the
branch are seen in Monte Carlo simulations (Fig. 6). There are two time
scales. The first scale $\sim(u+v)/2(u-v)$ corresponds to the crossover from
diffusion to drift as the predominant mode of transport: the return times are
distributed as a power-law $\sim t^{-3/2}$ in the diffusive regime whereas the
distribution decays exponentially in the drift regime. There is a third
trapping regime in which the return times follow $\sim\exp(-t/\tau_{L})$, with
$\tau_{L}\sim\exp(L/L_{g})$. This is an outcome of the activation process
required to clear the barrier against the ‘easy-direction’ of bias. The very
large times required to cross this barrier pose a challenge for numerical
simulations, and a non-standard procedure had to be devised, as discussed in
A.

$10^{-6}$$10^{-5}$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$1$$10$$100$$1000$$10000$

$f_{L}(t)$

$t$$L=5$$10$$15$$60$

(a) Log-log scale

$10^{-6}$$10^{-5}$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$0$$50$$100$$150$$200$$t$$L=5$$10$$15$$60$

(b) Semi-log scale

Figure 6: Distribution of first return times for branches of different lengths
for the same value of bias ($g=0.244$). The leftmost curve (black) is the
distribution for an effectively infinite system and has a power-law
(diffusive) and exponentially decaying (drift) regimes. Since a branch is
finite, there is an additional ultra-slow exponentially decaying (trapping)
regime. For times beyond $t=2\times 10^{4}$ the procedure discussed in A is
used.

###  3.4 Residence time on a backbone site

Applying the convolution theorem to (13), we get

$\tilde{\psi}_{L}(s)=\sum_{j=0}^{\infty}r^{j}(1-r)\tilde{f}^{(j)}_{L}(s)\tilde{\phi}(s).$
(23)

With $\tilde{f}^{(j)}_{L}(s)=\big{[}\tilde{f}_{L}(s)\big{]}^{j}$ from (14) and
using (17), we obtain

$\tilde{\psi}_{L}(s)=\frac{(1-r)\tilde{\phi}(s)}{1-r\tilde{\phi}(s)\tilde{Q}_{L}(s)}.$
(24)

We find the mean and variance of the residence time by calculating the first
and second derivatives of $\log\tilde{\psi}_{L}(s)$ at $s=0$, with the results

$\overline{\tau}(L)=\frac{1}{w}\bigg{[}\frac{(u/v)^{L+1}-1}{(u/v)-1}\bigg{]},$
(25)
$\sigma^{2}_{Res}(L)=\big{[}u^{2}(u-v+2w)(u/v)^{2L}-2u(u-v)((2L+1)w+v)(u/v)^{L}\\\
-v(v^{2}+2uw-uv)\big{]}(u-v)^{-3}w^{-2}.$ (26)

Observing that the variance $\sigma_{Res}^{2}$ has contributions from
$\sigma_{fr}^{2}$ as well as from the fluctuations due to different means
corresponding to different $j$’s, we may write

$\sigma_{Res}^{2}(L)=\frac{1}{(u+w)^{2}}+\frac{r}{1-r}\sigma_{fr}^{2}(L)+\frac{r}{(1-r)^{2}}\overline{t}^{2}_{fr}(L).$
(27)

To highlight the $L$ dependence we may rewrite (26) as

$\sigma^{2}_{Res}(L)=Ae^{2L/L_{g}}-B((2L+1)w+v)e^{L/L_{g}}-C$ (28)

where $A=u^{2}(u-v+2w)[(u-v)^{3}w^{2}]^{-1}$, $B=2u[(u-v)^{2}w^{2}]^{-1}$ and
$C=v(v^{2}+2uw-uv)[(u-v)^{3}w^{2}]^{-1}$.

###  3.5 Total transit time across the backbone

The history-averaged total transit time for a realization R is
$\overline{T}_{N}(\textbf{R})=\sum_{i=1}^{N}\overline{\tau}(L_{i})$. Using
(25) we obtain the transit time, and note that it coincides with the steady
state transit times (7) and (8). The variance of total transit time for a
given realization is given by
$\sigma_{N}^{2}(\textbf{R})=\overline{T_{N}^{2}(\textbf{R})}-\overline{T_{N}(\textbf{R})}^{2}$,
which implies

$\sigma_{N}^{2}(\textbf{R})=\sum_{i=1}^{N}\sigma_{Res}^{2}(L_{i}).$ (29)

As long as the average of $\sigma_{Res}^{2}(L)$ over the disorder distribution
(1) is finite, the law of large number holds and implies that
$\sigma_{N}^{2}(\textbf{R})/N$ converges to
$\langle\sigma_{N}^{2}(\textbf{R})\rangle/N$ as $N\to\infty$.

###  3.6 Disorder averaging: Exponential distribution, $\lambda=0$

Consider the disorder distribution $\Gamma(L)=(1-e^{-1/\xi})e^{-L/\xi}$,
corresponding to $\lambda=0$ in (1). The average of mean transit time over the
disorder is given by $\langle\overline{T}_{N}\rangle=\sum_{\textbf{R}}$
$P(\textbf{R})\overline{T}_{N}(\textbf{R})$ $=N\langle\overline{\tau}\rangle$,
if $L_{g}>\xi$, where
$\langle\overline{\tau}\rangle=\sum_{L}\overline{\tau}(L)\Gamma(L).$ Averaging
(25), we reproduce the steady state result (9). It then follows that the mean
drift velocity $N/\langle\overline{T}_{N}\rangle$ is identical to $v_{drift}$
in (10) from the steady state argument.

If $\langle\sigma_{Res}^{2}\rangle$ is finite, then using (12) and the fact
that branches are independent and identically distributed, we have

$\frac{\sigma_{temporal}^{2}}{N}=\langle\sigma_{Res}^{2}\rangle.$ (30)

Using (28), we find

$\displaystyle\frac{\sigma_{temporal}^{2}}{N}$
$\displaystyle=\frac{1+(2w/v)e^{1/L_{g}-1/\xi}+(2w/v-1)e^{2/L_{g}-2/\xi}}{w^{2}\big{[}1-e^{1/L_{g}-1/\xi}\big{]}^{2}\big{[}1-e^{2/L_{g}-1/\xi}\big{]}},$
$\displaystyle L_{g}>2\xi,$ (31) $\displaystyle=\infty,$ $\displaystyle
L_{g}\leq 2\xi.$

The exact expression (31) shows that $\sigma_{temporal}^{2}$ increases as a
function of bias $g$ and in fact diverges for $L_{g}\leq 2\xi$ (Fig. 7). The
regime $\xi<L_{g}\leq 2\xi$, where the history-to-history fluctuations of
transit time are anomalously large, even though $v_{drift}$ is finite, is the
Anomalous Fluctuation regime. The regime $L_{g}<\xi$ corresponds to
$v_{drift}=0$ and defines the Vanishing Velocity regime. Fig.8 shows the
variation of the mean transit time and the variance $\sigma^{2}_{temporal}$ as
a function of $N$ in the normal, AF and VV regimes.

For each realization of disorder, the history-averaged mean transit time
itself is different. The quantity
$\sigma_{disorder}^{2}=\langle\overline{T}_{N}^{2}\rangle-\langle\overline{T}_{N}\rangle^{2}$
is the contribution of different mean values to the variance. Using (25), we
find

$\displaystyle\sigma_{disorder}^{2}$
$\displaystyle=\frac{Ne^{(2/L_{g}-1/\xi)}\big{[}1-e^{-1/\xi}\big{]}}{w^{2}\big{[}1-e^{(1/L_{g}-1/\xi)}\big{]}^{2}\big{[}1-e^{(2/L_{g}-1/\xi)}\big{]}},$
$\displaystyle L_{g}>2\xi$ (32) $\displaystyle=\infty,$ $\displaystyle
L_{g}\leq 2\xi.$

Notice that the $\sigma_{disorder}^{2}$ diverges for $L_{g}\leq 2\xi$ as does
$\sigma_{temporal}^{2}$. Though they are different functions of bias, their
qualitative behaviour is the same (Fig. 7). A third type of variance,
$\sigma_{3}^{2}=\langle\overline{T_{N}^{2}}\rangle-\langle\overline{T_{N}}\rangle^{2}$
$=\langle\,\overline{(T_{N}-\langle\overline{T_{N}}\rangle)^{2}}\,\rangle$,
accounts for fluctuations of transit time about the disorder averaged mean
$\langle\overline{T_{N}}\rangle$. We note that
$\sigma_{3}^{2}=\sigma_{temporal}^{2}+\sigma_{disorder}^{2}$, and do not
discuss it further.

Within the FTM we see that the mean drift velocity, $v_{drift}$, is a
monotonically decreasing function of $g$ (Fig. 9(a)) as there is no regime of
diffusive motion as $g\to 0$. By making the transition rate $w$, along the
backbone, a function of $g$, one may engineer a non-monotonic behaviour of
$v_{drift}$ (Fig. 9(b)).

Figure 7: Mean drift velocity and variances $\sigma_{temporal}^{2}$ and
$\sigma_{disorder}^{2}$ vs bias $g$. The critical values $g_{1}$ and $g_{2}$
mark the onset of the VV and AF regimes respectively. The dots represent the
values from Monte Carlo simulations and the dashed black lines represent the
theoretically expected curves (Eqs. (10), (31) and (32)) with $N=2000$. The
two curves on the left are scaled by factors $k_{1}=3N$ and $k_{2}=3N/20$ for
clarity.

(a) Mean transit time $\langle\overline{T}_{N}\rangle$ vs $N$

(b) Variance $\sigma_{temporal}^{2}$ vs $N$

Figure 8: Breakdown of the central limit theorem in AF regime and breakdown of
the law of large numbers in VV regime. The solid dots represent the values
from Monte Carlo simulations averaged over $10^{3}$ realizations and
$6.4\times 10^{5}$ histories. (a) The dashed lines represent $y\sim N$ in the
Normal Transport and the AF regimes, and $y\sim N^{x}$ in the VV regime. (b)
The dashed lines represents $N$ in the Normal Transport and $N^{2x}$ in the AF
and the VV regimes. In (b) each of the solid lines represent disorder average
of $\sigma_{temporal}^{2}$ over $200$ realizations highlighting the strong
fluctuations brought in by disorder.

The variance of transit times, $\sigma_{temporal}^{2}/N$, is finite for
$L_{g}>2\xi$ which implies that the central limit theorem holds resulting in a
Gaussian distribution of transit times in the normal transport regime. For
$L_{g}\leq 2\xi$, we enter the Anomalous Fluctuation regime where the variance
diverges and the central limit theorem breaks down. The transit time
distribution then follows a stable distribution with index $x=L_{g}/\xi$ [13].
Figure 8(b) shows that the variance grows linearly with system size $N$ in the
normal transport regime, while it grows faster than linearly in the AF regime.
In the Vanishing Velocity regime, i.e. $L_{g}\leq\xi$, the mean transit time
also diverges and the law of large numbers breaks down (Fig. 8(a)). The
typical transit time follows a Levy stable distribution of degree $x$ with a
norming constant $N^{1/x}$ in the VV regime [12, 13].

##  4 Continuous and first order phase transitions in the RC

With a purely exponential distribution of branch lengths ($\lambda=0$ in Eq.
(1)) we have seen that $\langle\overline{T}_{N}\rangle$ diverges continuously
at $g_{1}$ (where $L_{g_{1}}=\xi$ from Eq. (9)), and that
$\sigma_{temporal}^{2}$ and $\sigma_{disorder}^{2}$ diverge continuously at
$g_{2}$ (where $L_{g_{2}}=2\xi$ from Eqs. (31) and (32)) implying that the
transitions to the VV and AF regimes are of the second order.

As pointed out in [17], if the power law exponent $\lambda=2$, the transition
at $g_{1}$ becomes discontinuous, with $\langle\overline{T}_{N}\rangle$ being
finite at $g_{1}$. This is verified from the expression

$\langle\overline{T}_{N}\rangle=\frac{N}{w(e^{1/L_{g}}-1)}\Bigg{[}\frac{e^{1/L_{g}}}{Z}S\bigg{(}\frac{1}{\xi}-\frac{1}{L_{g}},\lambda\bigg{)}-1\Bigg{]}$
(33)

where

$S(\alpha,\lambda)=\sum_{L=0}^{\infty}\frac{e^{-\alpha
L}}{1+(L/L_{0})^{\lambda}}.$ (34)

At the transition point $L_{g_{1}}=\xi$, we see that
$\langle\overline{T}_{N}\rangle$ diverges if $\lambda\leq 1$ and converges if
$\lambda>1$. Thus, the transition is continuous for $\lambda\leq 1$ and first
order for $\lambda>1$ (Fig. 9). The analogous question arises about the order
of the transition at $g_{2}$, where $L_{g_{2}}=2\xi$. We have,

$\sigma_{temporal}^{2}=N\Bigg{[}\frac{A}{Z}\,S\bigg{(}\frac{1}{\xi}-\frac{2}{L_{g}},\lambda\bigg{)}-\frac{B(w+v)}{Z}S\bigg{(}\frac{1}{\xi}-\frac{1}{L_{g}},\lambda\bigg{)}\\\
+\frac{2Bw}{Z}S^{\prime}\bigg{(}\frac{1}{\xi}-\frac{1}{L_{g}},\lambda\bigg{)}-C\Bigg{]}$
(35)

where $S^{\prime}(\alpha,\lambda)=\partial S(\alpha,\lambda)/\partial\alpha$
and $A,B$ and $C$ were defined in Section 3.4. Setting $L_{g_{2}}=2\xi$, we
see that $\sigma_{temporal}^{2}$ diverges if $\lambda\leq 1$, implying that
this transition is continuous, while $\sigma_{temporal}^{2}$ is finite for
$\lambda>1$, implying a first order jump (Fig. 9). The situation is summarized
succinctly in Fig. 3.

(a) $w=$ constant

(b) $w=g$

Figure 9: When $\lambda>1$ ($=2$ here), a first order phase transition is
observed in $v_{drift}$ and $\sigma_{temporal}^{2}$. The variance
$\sigma_{temporal}^{2}$ has been scaled down by a factor $k=5\times 10^{-3}$
for clarity.

##  5 Fluctuations in transit time in the diluted Bethe lattice

Consider biased diffusion on a randomly diluted Bethe lattice in which a
certain fraction of the bonds have been removed randomly. Earlier it was shown
[13] that beyond a certain value of the bias $g_{1}^{\prime}$, the mean
transit time along the backbone diverges, implying a vanishing drift velocity.
Thus $g_{1}^{\prime}$ is an upper bound on the critical bias $g_{1}$ for the
transition to the VV regime. In this section, we will show that beyond a value
$g_{2}^{\prime}$, the fluctuations become anomalous. We study the steady state
properties and analyze only $\sigma_{disorder}^{2}$. We find an upper bound
$g_{2}^{\prime}$ on the value of bias where it diverges.

In the Bethe lattice under consideration, each site has an even coordination
number, $2z$. The bonds are classified into two types based on their alignment
with respect to the ‘easy direction’ set by the field, as illustrated in Fig.
10 for $z=2$.

Field Figure 10: Easy directions in the Bethe lattice for z = 2.

Consider diluting the structure by randomly removing a fraction of $q=(1-p)$
of all bonds. Above the percolation threshold, $p_{c}=1/(2z-1)$, infinite
clusters form in which each site is connected to infinity through at least one
path. Sites that have at least two ways to connect to infinity constitute the
backbone. Finite clusters connected at a single site to the backbone are
called branches.

In order to find the total transit time along the backbone in the steady
state, we use the fact that $\overline{T}=\mathcal{N}/J$, where $\mathcal{N}$
is total number of particles and $J$ is the current injected into the backbone
in the steady state. Let $\rho_{n}$ be the density of particles at a backbone
site $n$. The density at a branch site $m$ is related to $\rho_{n}$ through

$\rho_{m}=e^{-r_{nm}/L_{g}}\rho_{n}$ (36)

where $r_{nm}$ is the difference between total number of up bonds and down
bonds connecting $n$ to $m$. The total number of particles in a branch
$\beta$, associated with site $n$ is then just $\rho_{n}$ times a factor
$F_{n\beta}$, given by

$F_{n\beta}=\sum_{m\in\beta}e^{-r_{nm}/L_{g}}.$ (37)

The total number of possible configurations of a branch depend only on whether
it is an up branch ($r_{n1}=+1$) or a down branch ($r_{n1}=-1$). The average
of $F_{\pm}$ over all the up or down branches is given by

$\langle F_{\pm}\rangle=\sum_{m}e^{-r_{nm}/L_{g}}\langle\eta_{nm}\rangle.$
(38)

The quantity $\eta_{nm}$ is an indicator function such that $\eta_{nm}=1$ if
site $m$ belongs to branch $\beta$ and $0$ otherwise. Taking an average over
all the possible configurations of $\beta$, we obtain

$\displaystyle\langle F_{\pm}\rangle$
$\displaystyle=p\big{[}\mathcal{Q}(p)\big{]}^{2z-1}\sum_{k=1}^{2}\frac{C_{k}^{\pm}(g)}{1-\lambda_{k}(g)p\big{[}\mathcal{Q}(p)\big{]}^{2z-2}},$
$\displaystyle g<g_{1}^{\prime},$ (39) $\displaystyle=\infty,$ $\displaystyle
g\geq g_{1}^{\prime}.$

Details of derivation are given in B, here $g_{0}$ is the solution of the
equation $\lambda_{1}(g)p\big{[}\mathcal{Q}(p)\big{]}^{2z-2}=1$, and the
functions $\lambda_{k}(g),\mathcal{Q}(p)$ and $C_{k}^{\pm}(g)$ are defined in
B.

The evaluation of $\sigma_{disorder}^{2}$ involves calculating

$\Delta F_{\pm}^{2}=\langle F_{\pm}^{2}\rangle-\langle F_{\pm}\rangle^{2}.$
(40)

The first term on the right is given by

$\langle
F_{\pm}^{2}\rangle=\sum_{m}e^{-2r_{nm}/L_{g}}\langle\eta_{nm}\rangle+\sum_{m\neq
m^{\prime}}e^{-(r_{nm}+r_{nm^{\prime}})/L_{g}}\langle\eta_{nm}\eta_{nm^{\prime}}\rangle.$
(41)

From Eqs. (38) and (41) and using the independence of $\eta_{nm}$ and
$\eta_{nm^{\prime}}$, we obtain

$\Delta
F_{\pm}^{2}=\sum_{m}e^{-2r_{nm}/L_{g}}\langle\eta_{nm}\rangle(1-\langle\eta_{nm}\rangle).$
(42)

The final result for $\Delta F_{\pm}^{2}$ is

$\displaystyle\Delta F_{\pm}^{2}=$ $\displaystyle
p\big{[}\mathcal{Q}(p)\big{]}^{2z-1}\sum_{k=1}^{2}\frac{D_{k}^{\pm}}{1-\alpha_{k}(g)p\big{[}\mathcal{Q}(p)\big{]}^{2z-2}}$
(43)
$\displaystyle-p^{2}\big{[}\mathcal{Q}(p)\big{]}^{4z-2}\sum_{k=1}^{2}\frac{D_{k}^{\pm}}{1-\alpha_{k}(g)p^{2}\big{[}\mathcal{Q}(p)\big{]}^{4z-4}},$
$\displaystyle\textrm{if}\,g<g_{2}^{\prime},$ $\displaystyle=$
$\displaystyle\infty,$ $\displaystyle\textrm{if}\,g\geq g_{2}^{\prime}$

where $g_{2}^{\prime}$ is the solution of the equation
$\alpha_{1}(g)p\big{[}\mathcal{Q}(p)\big{]}^{2z-2}=1$. As shown in the B,
$\alpha_{k}(L_{g})=\lambda_{k}(L_{g}/2)$ and
$D_{k}^{\pm}(L_{g})=C_{k}^{\pm}(L_{g}/2)$, which implies that
$L_{g_{2}^{\prime}}=2L_{g_{1}^{\prime}}$ and hence
$g_{2}^{\prime}<g_{1}^{\prime}$. Therefore, the variance of mean transit times
diverges at a lower value of bias than the mean transit time itself, pointing
to the existence of a distinct regime of anomalous fluctuations in the dilute
Bethe lattice.

In the analysis above, we have accounted only for trapping by branches. There
is the possibility of trapping by valleys along the backbone, which means that
$g_{2}^{\prime}$ is, in principle, an upper bound to the true threshold for
the AF regime.

##  6 Conclusions

We may draw two broad conclusions from our study of field-induced drift and
trapping for biased random walkers on the random comb: first, that there is a
separate anomalous fluctuation (AF) regime between the normal transport and
vanishing velocity (VV) regimes, as indicated by a divergence of the exactly
computed variance of transit times; and secondly, that the transition to the
AF regime changes from continuous to first order if the power law corrections
to the exponential form (Eq. (1)) are strong enough. These conclusions are
summarized in the phase diagram in Fig. 3.

In the regime where the variance is finite, we calculated
$\sigma_{temporal}^{2}$ and $\sigma_{disorder}^{2}$ exactly, in the forward
transport limit on the random comb. In this limit, the transit time is simply
the sum of the residence times on each backbone site. In order to ensure a
uniform transition rate in each bond of the backbone, we showed that rates for
random walk motion need to be site-dependent. Uniformity of backbone-bond
rates implies that the steady state particle density is uniform along the
backbone and rises exponentially in the branches.

The distribution of first return times in a branch, which is a prerequisite to
finding the distribution of residence times, itself shows three distinct
behaviours, corresponding to three smoothly connected regimes: diffusive, with
a distribution $\sim 1/t^{3/2}$; drift, associated with a rapidly decaying
exponential distribution; and a trapping regime with a distribution of ultra-
slow return times which grow exponentially with branch length, reflecting the
difficulty of crossing the bias-induced barrier. These points are brought out
clearly by numerical simulations, as shown in Fig. 6. A non-standard
procedure, described in A, had to be followed for the numerical simulations in
view of the extremely large times involved.

The occurrence of the vanishing velocity regime, known from earlier work [13,
14, 15] is associated with the breakdown of the law of large numbers, as the
mean residence time on a site becomes infinite. Likewise, the AF regime is
associated with the breakdown of the central limit theorem, owing to a
diverging variance of the residence time distribution. In [12], it was argued
that the time-distance relation in the VV regime asymptotically follows $T\sim
R^{1/x}$ with $x=L_{g}/\xi$. In the AF regime, we expect that asymptotically,
$T\sim R/v_{drift}$, but $\Delta T^{2}\sim R^{2/x}$.

The order of the transitions is sensitive to the distribution of branch
lengths for the random comb. It becomes first order at the bias thresholds of
both the AF and VV regimes once $\lambda$ exceeds $1$.

Finally, we obtained an upper bound on the critical bias for the variance
$\sigma_{disorder}^{2}$ to diverge on the Bethe lattice with bond dilution.
This indicates the presence of an anomalous fluctuation regime for this
structure as well.

We conclude by pointing to interesting open problems involving other types of
stochastically moving particles on the random comb. Two examples: (a)
interacting particles with mutual exclusion and (b) non-interacting active
particles, undergoing run-and-tumble motion. Problem (a) has been studied
earlier [33, 17] and it has been argued that there is no VV regime, as
branches essentially fill up and their lengths are screened, thereby limiting
mean times spent in a branch. It would be interesting to carry out a study of
fluctuations in this system, to see if there is an AF regime. For case (b) one
needs to ascertain whether there is a VV regime at all, and even if not,
whether there is an AF regime.

## Acknowledgements

This work forms part of a thesis submitted by J.D.K. in partial fulfilment of
the requirement for the degree of Master of Science awarded by Indian
Institute of Space Science and Technology, Thiruvananthapuram. We acknowledge
useful discussions with Arghya Das, Kabir Ramola, Stephy Jose and Navdeep
Rana. J.D.K. would like to thank TIFR Hyderabad for hospitality and IIST for
academic support. M.B. acknowledges support under the DAE Homi Bhabha Chair
Professorship of the Department of Atomic Energy, India. This project was
funded by intramural funds at TIFR Hyderabad from the Department of Atomic
Energy (DAE), Government of India.

##  Appendix A Numerical procedure: Simulating random walks on random combs

We perform numerical simulations to get an insight into random walk dynamics
on random combs. The complete procedure is divided into three steps as
follows.

###  1 Generating realizations of disorder

We generate and store multiple different realizations of disorder randomly
from the distribution (1). Each realization contains $9000$ branches and we
generate $1000$ such realizations. The maximum branch length, $L_{max}$,
across the realizations is recorded. With the value $\xi=1$, we found
$L_{max}=16$.

###  2 First return times in branches: Setting up a Library

A library contains a collection of first return times for branch lengths
$L=1,2,\dotsc,L_{max}$ for $10^{6}$ different histories. The first return
times on a given branch length, corresponding to different histories, are
stored sequentially in a single column. Each column corresponds to a different
branch length $L$. Multiple libraries are generated, each with a different
value of $g$.

The first return times for a branch of length $L$ with bias $g$ are generated
as follows. Let the particle be at $m=0$, at $t=0$. We draw a time randomly
from the distribution $\phi(t)$, with $\gamma=u+w$, and update the particle
position to $m=1$. Again a time $t$ is randomly drawn from $\phi(t)$, but with
$\gamma=u+v$ this time, and the position updated to $m\pm 1$ with a
probability $p$ or $q$ (Section 2). The process is continued until $m=0$ again
or if $t$ crosses a preset value $t_{max}$. In the later case, the time is
stored as $t_{max}$. The total time taken for the excursion is stored and the
process is repeated for $10^{6}$ histories.

In case a time $t_{max}$ is drawn from the library (for computing the
residence time), we draw another time randomly from the normalized
distribution $1/\tau_{L}\exp(-t/\tau_{L})$ and add it to $t_{max}$. The value
of $t_{max}$ is chosen to be large enough to always be in the trapping regime,
where the times are governed by $\sim A\exp(-t/\tau_{L})$, $A$ being some
constant.

The time $\tau_{L}$ is determined by fitting an exponential curve to the
numerically obtained profiles of first-return times (Fig. 6). We find that
$\tau_{L}=C(g)e^{L/L_{g}}$, where $C(g)$ is a constant for a given $g$.
Fitting the curve of $\tau_{L}$ vs $L$ to an exponential we determine the
value of $C(g)$. This curve is extrapolated to determine $\tau_{L}$ for large
$L$.

###  3 Total transit time: Summing the residence times

As it progresses along the backbone, the typical trajectory of a particle on
the RC involves performing multiple excursions into a branch, waiting at the
corresponding backbone site, hopping to the next backbone site and repeating
it for $N$ backbone sites. In a particular history, the number of excursions
$n$ at a given site is chosen from the distribution $r^{n}(1-r)$ (Fig. 5) and
$n$ first return times are drawn from the library. These times are summed up
and a waiting time, drawn from $\phi(t)=(u+w)\exp(-(u+w)t)$, is added to the
sum to obtain the residence time.

In a given realization of disorder, the residence time is computed for all the
branches and added to obtain the total transit time. We perform $10^{4}$
histories over a given realization of disorder and average over $10^{3}$
realizations of disorder. The procedure is repeated for each of the $13$
different values of $g$.

##  Appendix B Calculations on the Bethe lattice

Here we evaluate the sum (42) using the method proposed in [13]. The quantity
$\eta_{nm}$ is the indicator function, the average of which is given by

$\langle\eta_{nm}\rangle=p^{N}\big{[}\mathcal{Q}(p)\big{]}^{(2z-2)N+1}$ (44)

where $\mathcal{Q}(p)$ is the probability that all the paths in a given
direction from an occupied site are finite and $N$ is the total number of
bonds between $n$ and $m$, which follows

$\mathcal{Q}(p)=q+p\big{[}\mathcal{Q}(p)\big{]}^{2z-1}.$ (45)

For simplicity, let us assign each bond, a pseudospin $s_{i}$ such that
$s_{i}=+1$ for an ‘up’ bond and $s_{i}=-1$ for a ‘down’ bond. Therefore,
$N=\sum_{k\in\beta}|s_{k}|$ and $r_{nm}=\sum_{k}s_{k}$. At every branch site
$m+1$ there are $z$ ways of having $s_{m+1}=s_{m}$ and $z-1$ ways of having
$s_{m+1}=-s_{m}$. Let $n(s_{m+1},s_{m})$ be such that

$\begin{split}n(s_{m+1},s_{m})&=z,\qquad\quad\quad s_{m+1}=s_{m},\\\
&=z-1,\qquad\,s_{m+1}=-s_{m}.\end{split}$ (46)

It is found that

$\langle
F_{s_{1}}\rangle=\mathcal{Q}(p)\sum_{N=1}^{\infty}\nu^{N}(C_{1}^{\pm}\lambda_{1}^{N-1}+C_{2}^{\pm}\lambda_{2}^{N-1})$
(47)

where $\lambda_{1}(g)$ and $\lambda_{2}(g)$ are the eigenvalues of the
transfer matrix $T_{1}$

$T_{1}=\begin{pmatrix}z\,e^{-h}&z-1\\\ z-1&z\,e^{h}\end{pmatrix}$ (48)

with $\lambda_{1}>\lambda_{2}$,

$\lambda_{k}(g)=z\cosh h\pm\big{[}z^{2}\sinh^{2}h+(z-1)^{2}\big{]}^{1/2}$ (49)

and the coefficients $C_{1}^{\pm}$ and $C_{2}^{\pm}$ are given by

$C_{k}^{\pm}=\frac{(z-1)(\lambda_{k}-e^{\mp h})}{2(\lambda_{k}-z\cosh
h)(\lambda_{k}-z\,e^{\mp h})}$ (50)

where we have substituted $h=1/L_{g}$ and
$\nu=p\big{[}\mathcal{Q}(p)\big{]}^{(2z-2)}$. The sum in (47) converges for
$\lambda_{1}(g)\,p\big{[}\mathcal{Q}(p)\big{]}^{2z-2}<1$, and is given by

$\langle
F_{\pm}\rangle=p\big{[}\mathcal{Q}(p)\big{]}^{2z-1}\sum_{k=1}^{2}\frac{C_{k}^{\pm}(g)}{1-\lambda_{k}(g)p\big{[}\mathcal{Q}(p)\big{]}^{2z-2}}.$
(51)

Let $g_{1}^{\prime}$ be the solution of
$\lambda_{1}(g)\,p\big{[}\mathcal{Q}(p)\big{]}^{2z-2}=1$. The variance of
transit time is given by,

$\displaystyle\Delta F_{\pm}^{2}$
$\displaystyle=\sum_{m}e^{-2r_{nm}/L_{g}}\langle\eta_{nm}\rangle\big{(}1-\langle\eta_{nm}\rangle\big{)}$
(52)
$\displaystyle=\sum_{N=1}^{\infty}\mathcal{Q}\nu^{N}\big{(}1-\mathcal{Q}\nu^{N}\big{)}\sum_{\\{s_{2},..,s_{N}\\}}\prod_{k=1}^{N-1}n(s_{k+1},s_{k})e^{2(s_{1}+\dotsc+s_{N})/L_{g}}.$

Using transfer matrix $T_{2}$, obtained by replacing $h\to 2h$ in $T_{1}$, the
above reduces to

$\Delta
F_{\pm}^{2}=\sum_{N=1}^{\infty}\mathcal{Q}(p)\nu^{N}\big{(}1-\mathcal{Q}(p)\nu^{N}\big{)}\sum_{s_{N}=\pm
1}e^{-(s_{1}+s_{N})h}\langle s_{1}|T_{2}^{N-1}|s_{N}\rangle.$ (53)

Therefore,

$\Delta
F_{\pm}^{2}=\sum_{N=1}^{\infty}\mathcal{Q}(p)\nu^{N}\big{(}1-\mathcal{Q}(p)\nu^{N}\big{)}(D_{1}^{\pm}\alpha_{1}^{N-1}+D_{2}^{\pm}\alpha_{2}^{N-1})$
(54)

where $\alpha_{1}(g)>\alpha_{2}(g)$ are the eigenvalues of $T_{2}$,

$\alpha_{k}(g)=z\cosh 2h\pm\big{[}z^{2}\sinh^{2}2h+(z-1)^{2}\big{]}^{1/2}$
(55)

and the coefficients $D_{1}^{\pm}$ and $D_{2}^{\pm}$ are given by

$D_{k}^{\pm}=\frac{(z-1)(\alpha_{k}-e^{\mp 2h})}{2(\alpha_{k}-z\cosh
2h)(\alpha_{k}-z\,e^{\mp 2h})}.$ (56)

The series is convergent for $\alpha_{1}(g)p\big{[}Q(p)\big{]}^{2z-2}<1$. Let
$g_{2}^{\prime}$ be the solution for the latter with an equality sign, thus
for $g<g_{2}^{\prime}$,

$\displaystyle\Delta F_{\pm}^{2}=$ $\displaystyle
p\big{[}\mathcal{Q}(p)\big{]}^{2z-1}\sum_{k=1}^{2}\frac{D_{k}^{\pm}}{1-\alpha_{k}(g)p\big{[}\mathcal{Q}(p)\big{]}^{2z-2}}$
(57)
$\displaystyle-p^{2}\big{[}\mathcal{Q}(p)\big{]}^{4z-2}\sum_{k=1}^{2}\frac{D_{k}^{\pm}}{1-\alpha_{k}(g)p^{2}\big{[}\mathcal{Q}(p)\big{]}^{4z-4}}.$

Clearly, $\alpha_{k}(h)=\lambda_{k}(2h)$, hence,
$\alpha_{k}(L_{g})=\lambda_{k}(L_{g}/2)$. Therefore, the value of $L_{g}$
obtained in this case will be twice that obtained in the former case and hence
$g_{2}^{\prime}<g_{1}^{\prime}$.

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# Indoor Propagation Measurements with Sekisui Transparent Reflectors at
28/39/120/144 GHz

Chethan K. Anjinappa, Ashwini P. Ganesh, Ozgur Ozdemir, Kris Ridenour, Wahab
Khawaja, İsmail Güvenç,
Hiroyuki Nomoto†, and Yasuaki Ide†
Department of Electrical and Computer Engineering, NC State University,
Raleigh, NC 27606
†Research & Development Institute, High Performance Plastics Company, Sekisui
Chemical CO.,LTD.
e-mail: {canjina, apondey, oozdemi, kwrideno, wkhawaja<EMAIL_ADDRESS>
†{hiroyuki.nomoto<EMAIL_ADDRESS>This work has been supported in
part by the National Science Foundation (NSF) through the award CNS-1916766 as
well as the industry membership fees from the Broadband Wireless Access Center
(BWAC) I/UCRC Center.

###### Abstract

One of the critical challenges of operating with the terahertz or millimeter-
wave wireless networks is the necessity of at least a strong non-line-of-sight
(NLoS) reflected path to form a stable link. Recent studies have shown that an
economical way of enhancing/improving these NLoS links is by using passive
metallic reflectors that provide strong reflections. However, despite its
inherent radio advantage, metals can dramatically influence the landscape’s
appearance - especially the indoor environment. A conceptual view of escaping
this is by using transparent reflectors. In this work, for the very first
time, we evaluate the wireless propagation characteristics of passive
transparent reflectors in an indoor environment at 28 GHz, 39 GHz, 120 GHz,
and 144 GHz bands. In particular, we investigate the penetration loss and the
reflection characteristics at different frequencies and compare them against
the other common indoor materials such as ceiling tile, clear glass, drywall,
plywood, and metal. The measurement results suggest that the transparent
reflector, apart from an obvious advantage of transparency, has a higher
penetration loss than the common indoor materials (excluding metal) and
performs similarly to metal in terms of reflection. Our experimental results
directly translate to better reflection performance and preserving the radio
waves within the environment than common indoor materials, with potential
applications in controlled wireless communication.

###### Index Terms:

6G, B5G, Indoor coverage, mmWave, non-line-of-sight (NLOS), Sub-THz,
Transparent reflectors.

##  I Introduction

Millimeter wave (mmWave) and Terahertz bands are crucial enablers of beyond 5G
(B5G)/6G wireless communications. In particular, the THz communication is
recognized as one of the nine key communication technology enablers [1], due
to its large bandwidth and its support of new possibilities ranging from
augmented/virtual reality to high-resolution positioning and autonomous
driving [2]. One of the numerous challenges in a THz/mmWave network, is the
requirement of the line of sight (LoS) or at least a strong first-order
reflected non-LoS (NLoS) path to form a link, as diffraction is not
significant at these frequencies. Particularly the availability of the NLoS
link depends on the reflection profile of a scatterer, material, and the
frequency of operation.

There are numerous solutions proposed in the literature for mmWave indoor and
outdoor NLoS coverage enhancement. These include utilizing high sensitivity
receivers, high transmit power, use of multiple access points, and beam-
forming using antenna arrays. However, all these solutions have limitations
when there are blockage and NLoS propagation [3]. Re-configurable reflecting
surfaces comprising of meta-materials and active repeaters for mmWave coverage
enhancement have recently attracted extensive interest in the literature [4,
5]. However, maintaining a large number of active repeaters and re-
configurable reflecting surfaces is expensive, and the acquisition of channel
state information is a complex problem [6].

A simple and economical solution for mmWave coverage enhancement at indoor and
outdoor scenarios is by using passive metallic reflectors [7, 3]. The metallic
reflectors of different shapes and sizes can be selected based on the coverage
requirement. For example, in [8], a flat metallic sheet helps to achieve
directional coverage at 28 GHz deep in the corridor, whereas cylindrical and
spherical reflectors help in a uniform but short-area corridor coverage. The
different shapes and size reflectors can be made part of the interior/exterior
design. Earlier simulation and measurement studies, such as [3, 7, 9], have
shown that objects such as metal act as perfect reflectors, enabling strong
reflections for directional NLoS communication. Thus, the deployment of the
passive metallic reflectors can create a favorable propagation environment by
introducing new multipath components (MPCs) to the channel and increasing the
overall spatial diversity of the MPCs. However, metals will have a dramatic
influence on appearance, especially the indoor environment, spoiling the
landscape. For instance, attaching metals to windows blocks sunlight and view;
one can think of many other possibilities.

A viable option to counter the above disadvantage is the use of transparent
reflectors. These passive reflectors are not only transparent by design but
are also thin, flexible, low-cost, and lightweight [10]. Thus, they offer a
more sustainable and easy-to-deploy alternative without altering the
appearance of the surface. Even though this is fascinating from a conceptual
view, how effective the transparent reflectors are in terms of radio
propagation needs to be evaluated from a practical point of view. With these
in perspective, the main goal of this paper is to investigate the critical
wireless propagation characteristics of transparent reflectors. In particular,
we characterize the penetration loss (attenuation through the material), a
measure signifying the leakiness of a material, and reflection properties of
the transparent reflector and evaluate its efficacy against the commonly used
indoor material [11]. In terms of the operational frequency, we are primarily
interested in the commonly considered mmWave bands of 28 GHz and 39 GHz, and
the popular sub-THz bands of 120 GHz and 144 GHz. To the best of our
knowledge, this is the first work to evaluate transparent reflectors for
wireless communication at mmWave and sub-THz bands. Experimental results
demonstrate the promising performance of the transparent reflectors compared
to commonly found indoor materials with potential applications to diverse
scenarios in the 5G and 6G wireless communications, such as indoor and outdoor
coverage improvement and enhancement.

##  II System Model

This section describes the details of the transparent reflectors and other
indoor materials used in our measurements. We also briefly overview the
channel sounder and its capabilities, and pivotal details for better
understanding the penetration loss and reflection experiments.

###  II-A Transparent Reflectors

Transparent reflectors are flexible radio wave reflection films and are an
economical and more sustainable alternative to conventional methods of
increasing the signal strength. In terms of the construction, a transparent
film consists of a meta-material layer that is structured to diffusely
reflects high-frequency radio waves, a highly transparent adhesive, a special
coating to protect the film’s surface, and an adhesive layer at the bottom of
the surface. This is illustrated graphically in Fig. 1 [10]. These reflectors
when placed on any surface such as a wall or ceiling will reflect all the
incident rays enabling them to enhance and improve coverage. In this paper, we
use transparent reflectors manufactured by Sekisui Chemical, Inc.

Figure 1:  Transparent reflector film construction and appearance.

###  II-B Common Indoor Materials

In this work, along with characterizing the transparent reflector radio
properties, we also compare it against other commonly used indoor materials
such as ceiling tile, clear glass, drywall, plywood, and metal (stainless
steel), which are shown in Fig. 2. Each of these materials has varying
dimensions quantified in Table I. In particular, these thicknesses play a
pivotal role in penetration loss characterization. A normalized penetration
loss is also presented in Section IV-A which quantifies the attenuation per
unit thickness. Further, the reflection characteristics of the transparent
reflector will only be compared against the metal, as it is an ideal
reflector; other materials are ignored due to space constraints and we will
include in our future work. In this context, we also like to point the reader
to our prior work [11] where we have characterized penetration loss alone (and
not reflection) for the above-discussed materials except the metal and
transparent reflector; a key contribution of this work.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2: Common indoor materials and transparent reflector. Material |  Dimension (in cm) |  Thickness (in mm)
---|---|---
Ceiling Tile |  121.92 x 60.9 |  14.34
Clear Glass |  76.2 × 91.4 |  2.23
Drywall |  121.92 x 243.8 |  16.30
Plywood |  121.92 x 57.91 |  13.20
Metal |  60.96 x 91.44 |  0.50
Transparent reflector |  41.91 x 41.91 |  0.42
Table I: Materials used in the measurements.

(a)

(b)

(c)

Figure 3: Experimental setup for characterizing (a) Penetration loss (b)
Reflection: Open door measurement setup with transmitter and reflector (c)
Reflection: Open door measurement setup with receiver.

###  II-C Channel Sounder and Equipment Details

All the measurements were conducted using a channel sounder based on NI’s PXI
platform [12] that consists of PXIe- 1085 transmitter/receiver (Tx/Rx)
chassis, Tx/Rx radio heads, and a FS725 Rubidium (Rb) clock. All the
processing is done by the PXIe-7902/8880 (FPGA/host PC inside the chassis).
Further, the PXIe-3610/3630 perform the D/A and A/D conversions, whereas, the
PXIe-3620 is responsible for the intermediate frequency (IF) up-conversion.. A
series of samplings and conversions are performed through the PXIe- 1085
chassis. The channel sounder transmits a 2048 bit length Zadoff-Chu sequence
that is filtered by a root raised cosine (RRC) filter. The generated samples
are then fed into FPGA PXIe-7902 that undergo digital to analog conversion at
PXIe-3610 converter at a sampling rate of $fs$ = 3.072 GS/s. The base-band
signal is then up-converted to the IF signal in the PXIe-3620 module. For 28
GHz and 39 GHz, an IF of 10.56 GHz is used. The 120 GHz and 144 GHz uses an IF
of 12.6 GHz. The Tx/Rx radio heads further up-convert the IF signals to high-
frequency radio signals.

At the 28 GHz setup, the Tx radio head used was NI mmRH-3642 and NI mmRH-3652
was the Rx radio head. An LO signal at 3.52 GHz is multiplied by three to
generate IF signal at 10.56 GHz. Further, on the Rx end the high-frequency RF
signal is down-converted to IF, by the radio-heads. The horn antennas
SAR1725-34KF-E2 connected to the Tx and the Rx radio head have an antenna gain
of 17 dBi Gain, 3 dB beam-width of 26 degrees, and 24 degrees in the E-plane
and H-plane, respectively. The antenna system has a maximum linear dimension
of 4.072 cm with a far-field of 30.97 cm.

Similarly, at 39 GHz, the Tx radio head used was NI mmRH-3643 and NI mmRH-3653
was the Rx radio head. IF signal generation is similar to our 28 GHz setup.
The horn antennas SAR2013-222F-E2 connected to the Tx and the Rx radio head
have an antenna gain of 20 dBi Gain, 3dB beam-width of 15 degrees, and 16
degrees in the E-plane and H-plane, respectively. The antenna system has a
maximum linear dimension of 4.384 cm with a far-field of 50.005 cm.

At 120 GHz and 144 GHz, the Tx radio head used was VDI WR6.5CCU and VDI
WR6.5CCD was the Rx radio head. In this case the LO generated at 4.2 GHz is
multiplied by 3 to generate IF signal at 12.6 GHz. Band-pass filters
WR6.5BPFE116-123 and WR6.5BPFE140-148 are used for 120 GHz and 144 GHz,
respectively. The horn antennas VDI WR-6.5 connected to the Tx and the Rx
radio head has an antenna gain of 21 dBi Gain, 3 dB beam-width of 13 degrees
in both E-plane and H-plane, respectively. The antenna system has a maximum
linear dimension of 1.080 cm with a far-field of 11.205 cm. For more details
consult [11].

##  III Characterizing Penetration and Reflection

This section describes the measurement setup used for characterizing the
penetration loss of each of the materials discussed in Section II and the
reflection characteristics of the transparent and metal reflector. All the
measurements were carried out in an indoor environment in Engineering Building
2 at NC State University, Raleigh, NC.

###  III-A Penetration Loss: Indoor Measurement

Fig. 3(a) displays the indoor measurement setup for the penetration loss
experiment. The Tx and Rx radio heads were mounted on the cardboard at 10
inches from the ground and were separated by a distance of $d\approx 1$ m.
Both the Tx/Rx were beam aligned manually such that the radio heads center
pointing to each other with a 0∘ incident angle, corresponding to the LoS
scenario with no blockage/obstruction. At first, we collected the LoS
measurement data (without any blockage) and identified the LoS path component
(earliest amongst all the possible MPCs) and extracted the averaged-power
$P_{\text{Rx}}^{\text{LoS}}$ associated with the LoS path for each measurement
point, given by

$P_{\text{Rx}}^{\text{LoS}}\text{[dB]}=P_{\text{Tx}}+G_{\text{Tx}}+G_{\text{Rx}}-PL_{\text{FS}},$
(1)

where $P_{\text{Tx}},G_{\text{Tx}}$, and $G_{\text{Rx}}$ are the transmit
power, the gain at the Tx, and the gain at the Rx, respectively. The term
$PL_{\text{FS}}=20\log_{10}\left(\frac{4\pi df}{c}\right)$ denote the free
space path loss, with $f$ and $c$ denoting the frequency of operation and
speed of light, respectively.

The penetration loss (signal attenuation) through a material depends mainly on
the material characteristics, frequency of operation, and the incident angle
[13]. We characterize the penetration loss of different materials at
frequencies discussed in Section II-B. To do so, we place the material of
interest midway (at 0.5 m) between the Tx and Rx at an incident angle of 0∘,
as shown in Fig. 3(a). We leave the experiments for different incident angles
as future work. It is also worth mentioning that another interesting future
work direction is the optimal deployment of transparent reflectors to enhance
coverage [14]. Similar to the LoS measurement data, we collected measurement
points with obstruction via the material and measured the power
$P_{\text{Rx}}^{\text{OLoS}}$ associated with the obstructed LoS, given by

$P_{\text{Rx}}^{\text{OLoS}}\text{[dB]}=P_{\text{Tx}}+G_{\text{Tx}}+G_{\text{Rx}}-PL_{\text{FS}}-L_{\text{Pen.}}.$
(2)

The term $L_{\text{Pen.}}$ denotes the penetration loss induced by the
material signifying the attenuation introduced due to obstruction. Note that
the signal attenuation through a material depends on the thickness of the
material; see Section IV-A for normalized attenuation results. Finally, the
term $L_{\text{Pen.}}$ can be estimated by subtracting the averaged obstructed
LoS power $P_{\text{Rx}}^{\text{OLoS}}$ from the averaged (unobstructed) LoS
power $P_{\text{Rx}}^{\text{LoS}}$. That is,

$L_{\text{Pen.}}\text{[dB]}=\leavevmode\nobreak\
P_{\text{Rx}}^{\text{LoS}}-P_{\text{Rx}}^{\text{OLoS}}.$ (3)

Examples of the above phenomenon appear in Fig. 4. The left and right figures
show the measurements obtained for the LoS and the transparent material at 39
GHz, respectively. As demonstrated, we only consider the LoS path and
obstructed LoS that arrive at the identical time delay for calculating
$L_{\text{Pen.}}$, ignoring the reflections (ground and other reflections from
the background and the material itself). We also emphasize that there was no
beam spillage problem as the size of the materials was larger than the beam
area [15].

Figure 4: $L_{Pen.}$ calculation at 39 GHz: (left) LOS and (right) transparent
reflector measurement. Each delay index corresponds to a delay of 0.65 ns.

###  III-B Reflection: Open Door Measurement

In this sub-section, we describe the experimental setup for characterizing the
reflection property of the material. The measurement setup is a practical
communication scenario (open-door indoor environment), where the Tx is outside
the room, and the Rx is inside the room, with the LoS path obstructed by the
wall. To extend the coverage, we deploy passive reflectors to provide the NLoS
link between the Tx/Rx, as illustrated in Fig. 3(b). For the current
experiment, we compare the transparent reflector against the metal reflector
alone (as metal is the perfect conductor). We emphasize that, unlike the
previous experiment with small distances, beam spillage would be an issue here
due to larger distances between the Tx/reflector/Rx, respectively; consult
[15] for details. Thus, reflectors of different sizes would have different
beam spillage ratios. For a fair comparison, the metal reflector size was
trimmed down to a similar size as the transparent reflector (thickness is the
same as in Table I). Even though the beam spillage phenomenon exists, it would
be the same for both cases. Further, the reflectors were mounted on a tripod
such that it forms 45∘ incident and reflected angle between the Tx/Rx,
respectively. The distance between Tx/reflector center and reflector center/Rx
was 1.40 m and 3.43 m, respectively. Further, the broadband pyramidal
absorbers, each of dimension $24^{{}^{\prime}}\times 24^{{}^{\prime}}\times
4^{{}^{\prime}}$, were deployed near the tripod to avoid unnecessary
reflections from the tripod.

In this experiment, both the Tx and the Rx are equipped with a rotating gimbal
to measure the received power from different directions. This helps in
characterizing (to some extent) the diffuse scattering phenomenon that happens
in radio bands [16]. We chose the gimbal rotation range of 90∘ to -90∘ at the
Tx and 140∘ to -140∘ at Rx, respectively. A 5∘ resolution was maintained at
both Tx/Rx gimbal, this separation assured a trade-off between the complete
coverage and experiment run-time. We take the same approach as in the previous
section to characterize the reflectivity of metal and transparent reflectors.
Unlike the simple analytical characterization of the penetration loss, the
reflection power depends on multiple factors such as the incident/departure
angles formed between the reflector and Tx/Rx, the orientation of the
reflector, the beam area formed on the reflector, and the area of the
reflector [15, 14]. Thus, we avoid the theoretical framework in this study and
calculate the total received power. This simplifies the way to characterize
the material reflectivity, as presented in Section IV-B.

(a)

(b)

Figure 5: Penetration loss results at mmWave and sub-THz frequencies.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

(p)

(q)

(r)

(s)

(t)

Figure 6: 28 GHz: first row (a)-(e), 39 GHz: second row (f)-(j), 120 GHz:
third row (k)-(o), 144 GHz: fourth row (p)-(t). The colormap indicates the
received signal power with units in dBm. For 120 GHz and 144 GHz, the colormap
indicates relative power as the power levels are not calibrated.

##  IV Experimental Results

In this section, we present the experimental results and draw critical
insights into the propagation characteristics of the transparent reflector. We
do so by comparing it against the performance of commonly found indoor
materials. In general, we treat the metal reflector as a benchmark due to its
ideal propagation characteristics. For the measurements, the transmit power
$P_{\text{Tx}}$ was set to -10 dBm for all the experiments unless mentioned
otherwise, and the bandwidth of the transmitted signal was set to be 1.5 GHz.
All the results obtained in this section are averaged over the multiple
snapshots of each measurement.

###  IV-A Penetration Loss Results

Fig. 5 illustrates the averaged $L_{\text{Pen.}}$ results of different
materials and is obtained as discussed in Section III-A. It should be
emphasized that Fig. 5(a) shows the absolute $L_{\text{Pen.}}$, whereas, Fig.
5(b) denote the normalized $L_{\text{Pen.}}$ per centimeter thickness of the
material. As seen in Fig. 5(a), the ceiling tile offers the lowest attenuation
at all four frequencies, followed by other materials (such as clear glass,
drywall, and plywood) with different attenuation factors not exceeding 7.5 dB
at all four frequencies. Amongst all the materials, metal has the highest
$L_{\text{Pen.}}$ followed by the transparent reflector. The high
$L_{\text{Pen.}}$ of metal is obvious because of its small penetration depth
and skin depth. At 28 GHz and 39 GHz, metal reflector resulted in a weak
obstructed LoS. However, at 120 GHz and 144 GHz, there was no obstructed LoS
(no MPC) through the metal, even at the maximum power of 10 dBm supported by
the sub-THz equipment. Thus, we have capped the $L_{\text{Pen.}}$ as 70 dB
(maximum dynamic range supported by the system) in Fig. 5(a), which can be
treated as a lower bound.

In general, the trend is intuitive: higher the frequency higher the
attenuation of the material, excluding metal and transparent reflector. Fig.
5(b) reveals a similar trend as demonstrated in [11]. An interesting and
important result is the $L_{\text{Pen.}}$ of the transparent reflector.
Despite being transparent, it offers higher penetration loss than common
indoor materials except for metal, with an obvious cosmetic advantage. For
example, the ceiling tile alone in an indoor setup would leak the power if the
Tx-beam is pointed upward towards the ceiling tile. However, covering the
ceiling tile with transparent reflectors would not leak the power due to high
penetration loss, but will rather preserve the radio waves within the room
resulting in a better radio signal strength.

###  IV-B Reflection Results

Fig. 6 illustrates the heatmap of the total power received at the Rx for each
of the Tx/Rx rotating gimbal angle bin. The figure contains four rows, where
each row contains the results for each frequency, in the order of 28 GHz, 39
GHz, 120 GHz, and 144 GHz, respectively. Further, each row contains five
columns, where the first column corresponds to no reflector (no tripod)
scenario, followed by the heatmaps of received power with the metal and
transparent reflector, respectively. The last two columns denote the
contribution from the metal and the transparent reflector contribution
(MRC/TRC) alone, respectively. These two plots help to compare the performance
of the reflectors and they are acquired by subtracting the results for the
metal/transparent reflector from the results with no reflector, respectively.

A pragmatic assessment of the first column across all the four frequencies
reveals that there is a strong obstructed LoS path penetrating through the
wall peaking at a Tx/Rx angle of (-30∘, 90∘); see Fig. 3(b)-(c) for the
obstructed wall and the measurement setup. Further, as the operating frequency
increases, the number of Tx/Rx angle combinations for possible NLoS links
decreases due to the increased path loss. We emphasize that the absolute power
levels across different frequencies depend on the varying capabilities at
those frequencies, ranging from different antenna gains and beamwidths,
different beam spillage ratio, among others (see Section II-C).

Heatmaps in Columns 2 and 3 reveal that having a reflector in the evaluated
scenario provides an NLoS link with a similar performance as the obstructed
link at all four frequencies (power peaking at a Tx/Rx angle of (0∘, 0∘) ).
This signifies the promising benefits of passive reflectors at mmWave and sub-
THz bands. Amongst the metal and transparent reflector, at 28 GHz and 120 GHz,
TRC contributes in a slightly higher power than MRC, and vice-versa at 39 GHz
and 144 GHz. The inconsistency in result may be due to the manual replacement
of the reflectors during measurements that may have caused a slight change in
the orientation. Nevertheless, the maximum power deviation does not exceed 4
dB. Overall, these preliminary but encouraging results suggest that the
transparent reflectors perform similar to the metal reflector. Our other
indoor experimental results (not presented in this paper) are in good
agreement with our presented results and have shown a similar trend.

##  V Conclusion

In this work, we demonstrated for the first time in the literature that
transparent reflectors can be used as passive reflectors with similar
performance as metallic reflectors for enhancing the wireless coverage at
mmWave and sub-THz bands. In particular, we performed measurements at 28 GHz,
39 GHz, 120 GHz, and 144 GHz, characterized penetration loss of common indoor
materials, and compared reflection characteristics of the transparent
reflector against the metal reflector. The measurements results suggest that
the transparent reflector, apart from an obvious advantage of transparency,
has higher penetration loss than the indoor materials such as ceiling tile,
clear glass, drywall, and plywood, and aids in preserving the radio waves
within the indoor environment. Also, it performs similarly to metal reflectors
in terms of reflection. Our future work includes experiments to characterize
the various radio properties of the transparent reflectors at different
geometrical positions.

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