# Universal dynamical scaling laws in three-state quantum walks

P. R. N. Falcão<EMAIL_ADDRESS>Instituto de Física, Universidade
Federal de Alagoas, 57072-900 Maceió, AL, Brazil  A. R. C. Buarque  Instituto
de Física, Universidade Federal de Alagoas, 57072-900 Maceió, AL, Brazil  W.
S. Dias  Instituto de Física, Universidade Federal de Alagoas, 57072-900
Maceió, AL, Brazil  G. M. A. Almeida  Instituto de Física, Universidade
Federal de Alagoas, 57072-900 Maceió, AL, Brazil  M. L. Lyra  Instituto de
Física, Universidade Federal de Alagoas, 57072-900 Maceió, AL, Brazil

###### Abstract

We perform a finite-time scaling analysis over the detrapping point of a
three-state quantum walk on the line. The coin operator is parameterized by
$\rho$ that controls the wavepacket spreading velocity. The input state
prepared at the origin is set as symmetric linear combination of two
eigenstates of the coin operator with a characteristic mixing angle $\theta$,
one of them being the component that results in full spreading occurring at
$\theta_{c}(\rho)$ for which no fraction of the wavepacket remains trapped
near the initial position. We show that relevant quantities such as the
survival probability and the participation ratio assume single parameter
scaling forms at the vicinity of the detrapping angle $\theta_{c}$. In
particular, we show that the participation ratio grows linearly in time with a
logarithmic correction, thus shedding light on previous reports of sublinear
behavior.

##  I Introduction

For many decades, classical random walks have been essential for the
development of algorithms, as well for describing a wide range of physical
phenomena. The story is no different for their quantum counterpart, namely
quantum walks, since put forward by Aharonov et al. in aharonov1993 . They
have been crucial to the progress of quantum computing, including the design
of quantum algorithms shenvi2003 ; childs2004 ; childs2014 ; wong2015 ;
wong2018  for search problems and state transfer zhan2014 ; stefanak2016 ;
kurzynsky2011 , to name a few. One of the many advantages is the speedup
gained from the fact that the quantum walker spreads out ballistically due to
interference between its wavefunction amplitudes knight2003 . Moreover, it was
shown both continuous and discrete versions make for universal quantum
computation childs2009 ; lovett2010 .

In the discrete-time quantum walk, particularly, the walker evolves through a
predefined sequence of gates. The first gate is usually the coin operator that
superposes its internal degrees of freedom, followed by the translation
operator that moves it (say, left or right) depending on its internal state.
This goes on and on until the protocol reaches its goal kempe2003 ;
venegas2012 . The key point here is that there are endless forms to set those
rules and thus different routes to travel through the Hilbert space, what
drives progress in assessing hard computational problems vakulchyk2019 . From
that point of view, discrete-time quantum walks are also a powerful tool for
quantum simulation of complex phenomena such as quantum phase transitions
chandrashekar2008 ; wang2019 ; xwang2018 , nonlinear dynamics navarrete2007 ;
buarque2020 ; mendonca2020 ; buarque2021 , topological phases of matter
kitagawa2010 ; asboth2012 ; rakovszky2015 , localization obuse2011 ;
mendes2019 ; buarque2019 ; queiros2020 , and many others.

A remarkable feature is that in most of the cases pointed out above, the
quantum walk setups giving rise to that kind of phenomena are simple two-state
models on a line. We may get even more of it by adding, for instance, extra
internal degrees of freedom. In this context, the three-state Grover walk
inui2005  has attracted some attention due to its many ubiquitous properties,
one of them being an intrinsic localization taking place at the input site,
something that just does not occur to the classical random walker. In this
Grover walk, the walker can move to the left, right, and stay at the same
position. The additional internal degree of freedom is responsible for the
generation of a constant eigenvalue in Fourier space which leads to a
nonvanishing probability at the initial input position. Thus, the probability
to find the walker at this position will saturate to a constant value in the
long time regime. inui2005 ; falkner2014 ; konno2014 . Two generalizations of
the Grover walk was obtained in stefanak2012  by working out families of
three-state coin operators that maintain localization. The same authors
investigated the quantum walk dynamics in detail in stefanak2014 , discussing
the role played by each eigenvector on the dynamics via an analytical
approach. Moreover, it was shown in wong2017  that a class of lackadaisical
quantum walks (where each site features an integer number of self-loops) is
equivalent to one of those continuous deformations of the three-state Grover
coin proposed in stefanak2012 . Due to its faster ballistic dispersion, this
kind of quantum walk can boost search algorithms in general graphs, as
demonstrated in wong2015 ; wong2018 . More recently, an implementation of
three-state quantum walks in a quantum circuit model was proposed in saha2021
.

In this work, we seek to unveil some similarities between a general form of
the three-state Grover walk with other classes of discrete-time quantum walks
by looking for dynamical scaling laws at the vicinity of the threshold where
localization is suppressed. To do so, we define a set of effective two-
component initial states whose relative phase is defined by a mixing angle
$\theta$. At the threshold $\theta=\theta_{c}$ the dynamics resembles that of
the standard two-state Hadamard quantum walk for it evolves ballistically with
no localized component. We show that the survival probability and the
participation ratio satisfies universal dynamical scaling laws in the vicinity
of $\theta_{c}$ for any value of the coin deformation parameter. We also
obtain an analytical expression for the participation ratio featuring a
logarithm correction. This has never been reported in the quantum walk
literature and actually explains the emergence of the effective sublinear time
evolution seen in omanakuttan2018 .

##  II Model

We consider a quantum walker propagating through an infinite one-dimensional
lattice. The Hilbert space of the system is made up of two parts, namely
$H=H_{p}\otimes H_{c}$, where $H_{p}$ accounts for the position space
$\left\\{{|x\rangle}\right\\}$, with $x\in\mathbb{Z}$, and $H_{c}$ is a three-
dimensional coin space, spanned by
$\left\\{{|L\rangle=(1,0,0)^{T},|S\rangle=(0,1,0)^{T},|R\rangle=(0,0,1)^{T}}\right\\}$.
The walker evolves in discrete time steps via
$|\psi(t+1)\rangle=\hat{U}|\psi(t)\rangle$, where
$\hat{U}=\hat{S}\cdot[\hat{C}\otimes\mathbb{I}]$ is a unitary operator, with
$\mathbb{I}$ being the identity operator acting on $H_{p}$. At each step,
following the coin operator $\hat{C}$ responsible for mixing the internal
degrees of freedom, the operator

$\displaystyle\hat{S}=\sum_{x=-\infty}^{\infty}$
$\displaystyle\bigg{[}|x-1\rangle\langle x|\otimes|L\rangle\langle
L|+|x\rangle\langle x|\otimes|S\rangle\langle S|$ (1)
$\displaystyle+|x+1\rangle\langle x|\otimes|R\rangle\langle R\mid\bigg{]}$

shifts each position state to the left, right, or none, according to its
related internal state. The probability to find the walker at position $x$ at
a given step $t$ is given by $P_{x}(t)=\sum_{c}|(\langle x|\otimes\langle
c|)|\psi(t)\rangle|^{2}$, with $c\in\\{L,S,R\\}$.

The three-state coin operator we consider here is a one-parameter class of the
Grover diffusion coin that preserves localization (see stefanak2012  for
details), namely

$\hat{C}(\rho)=\begin{pmatrix}-\rho^{2}&\rho\sqrt{2-2\rho^{2}}&1-\rho^{2}\\\
\rho\sqrt{2-2\rho^{2}}&2\rho^{2}-1&\rho\sqrt{2-2\rho^{2}}\\\
1-\rho^{2}&\rho\sqrt{2-2\rho^{2}}&-\rho^{2}\end{pmatrix},$ (2)

where the coin parameter $\rho\in(0,1)$ determines the speed of ballistic
dispersion. That means one will find the peaks at $\pm\rho t$ after $t$ steps.
Note that $\rho=0$ yields trivial dynamics, as $\hat{C}$ is reduced to the
permutation matrix, meaning that the walker will get stuck between the origin
and its nearest neighbors. On the other hand, if $\rho=1$ the coin operator
gets the diagonal form and no shuffling will occur during evolution. The
standard, unweighted Grover coin inui2005  is recovered when
$\rho=1/\sqrt{3}$.

At this point, we shall work out the eigenvectors of the coin operator defined
above, as expressing the walker input state in such basis will simplify the
analysis considerably. Straightforward algebra leads to

$\displaystyle|\sigma^{+}\rangle$
$\displaystyle=\sqrt{\frac{1-\rho^{2}}{2}}|L\rangle+\rho|S\rangle+\sqrt{\frac{1-\rho^{2}}{2}}|R\rangle,$
(3) $\displaystyle|\sigma_{1}^{-}\rangle$
$\displaystyle=\frac{\rho}{\sqrt{2}}|L\rangle-\sqrt{1-\rho^{2}}|S\rangle+\frac{\rho}{\sqrt{2}}|R\rangle,$
(4) $\displaystyle|\sigma_{2}^{-}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}(|L\rangle-|R\rangle),$ (5)

corresponding to eigenvalues 1,-1,-1, respectively. An initial state prepared
at $x=0$ can thus be conveniently written as
$|\psi(t=0)\rangle=|0\rangle\otimes|\psi_{c}\rangle$, with

$|\psi_{c}\rangle=\alpha|\sigma^{+}\rangle+\beta|\sigma_{1}^{-}\rangle+\gamma|\sigma_{2}^{-}\rangle.$
(6)

Each of these parts plays very specific roles in the dynamics, as shown in
stefanak2014  in great detail. In particular, the initial state
$|\sigma_{1}^{-}\rangle$ alone does not lead to localization at the origin.
Rather, its propagation resembles that of the two-state Hadamard quantum walk.
Outcomes for each coin state can be assessed analytically via Fourier analysis
and weak-limit theorems grimmet2004 . One can then approximate the spatial
probability distribution in the long-time limit in terms of the group velocity
density $\omega(\nu)$. This was worked out first for the special case
featuring $\rho=1/\sqrt{3}$ falkner2014  and later for arbitrary $\rho$ using
the Riemann-Lebesgue lemma machida2015 . For
$|\psi_{c}\rangle=|\sigma_{1}^{-}\rangle$, the group velocity density reads
stefanak2014

$\omega(\nu)=\frac{\sqrt{1-\rho^{2}}}{\pi(1-\nu^{2})\sqrt{\rho^{2}-\nu^{2}}},$
(7)

where $\nu=x/t$. The probability distribution for finite $t$ follows directly
as $P_{x}(t)=\omega(\nu)/t$. This readily indicates that the probability to
find the particle at the origin goes to zero with time.

##  III Results

Our primary goal here is to look after dynamical scaling laws at the vicinity
of the point where localization starts to take place. It means that the
initial coin state $|\psi_{c}\rangle$ must overlap with at least another
eigenstate besides $|\sigma_{1}^{-}\rangle$. If we set at the origin ($x=0$) a
symmetric initial state of the form

$|\psi(t=0)\rangle=|0\rangle\otimes\left(\cos\theta|S\rangle+\sin\theta|\phi\rangle\right),$
(8)

with $|\phi\rangle=(|L\rangle+|R\rangle)/\sqrt{2}$ and $\theta\in[0,\pi]$, the
coefficients of Eq. (6) become
$\alpha=(\rho\cos\theta+\sqrt{1-\rho^{2}}\sin\theta)$,
$\beta=(\rho\sin\theta-\sqrt{1-\rho^{2}}\cos\theta)$, and $\gamma=0$. We thus
define a characteristic angle $\theta_{c}=\cos^{-1}(-\sqrt{1-\rho^{2}})$,
which is the angle for which $\alpha=0$ and
$|\psi_{0}\rangle=|\sigma_{1}^{-}\rangle$. By definition,
$\theta_{c}\in[\pi/2,\pi]$.

We are now ready to track the dynamics of such input walker state featuring a
mixing angle $\theta$ between eigenstates that lead to localized and
delocalized behavior stefanak2014 . We set open boundary conditions and sizes
$N$ large enough so as to avoid the wavefunction from reaching the edges.
Figure 1 shows the probability $P_{x}(t)$ evolution in space and time for
three representative values of $\theta$, including $\theta_{c}$, and
$\rho=1/\sqrt{3}$, which is the case of the standard Grover walk inui2005 . It
clearly depicts that when $\theta=\theta_{c}$ the wavepacket tends to a
delocalized regime, much like the two-state Hadamard quantum walk. As soon as
$|\sigma^{+}\rangle$ is taken into account ($\theta\neq\theta_{c}$) a finite
fraction of the wavepacket remains trapped in the origin.

Figure 1: Time evolution of occupation probability $P_{x}$ for the three-state
Grover Walk ($\rho=1/\sqrt{3}$) over $100$ time-steps for distinct values of
$\theta$. In general, part of the wavepacket remains trapped at the origin,
except at $\theta_{c}$ which is when the initial coin state corresponds to
$|\sigma_{1}^{-}\rangle$.

Figure 2: (a) Time evolution of the survival probability (SP) for three mixing
angles and $\rho=1/\sqrt{3}$. In the absence of localization ($\theta_{c}$),
SP decays as $1/t$. (b) The participation ratio (PR) at $\theta_{c}$. It grows
according to the analytical prediction
$\text{PR}(\theta_{c})=at/[b+\text{ln}(t)]$, with $a=3.798$ and $b=1.54$. The
inset shows the time-dependence of the effective power-law exponent,
evidencing the slow convergence to the linear behavior

To better characterize the dynamical behavior in the vicinity of $\theta_{c}$,
we shall take a look at the survival probability $\mathrm{SP}(t)\equiv
P_{x=0}(t)$ in Fig. 2(a). In the long-time regime, the survival probability
saturates at a finite value due to onset of localization while $\text{SP}\sim
t^{-1}$ when the mixing angle is right at $\theta_{c}$, as expected. This
power-law decay in time of the survival probability is consistent with Eq.
(7).

Another powerful tool to portray the walker dynamics is the participation
ratio

$\text{PR}(t)=\frac{1}{\sum_{x}P_{x}(t)^{2}},$ (9)

which accounts for the fraction of the wavepacket spreading along the lattice.
Whenever the system has a localized component, the participation ratio
converges to a finite value, since this term will take over the sum in Eq.
(9). In the absence of that, it will become of the order of the spacial
extension of the spreading wavepacket and, as such, continuously grow in time.
Since the three-state quantum walk has an intrinsic localization in the
initial site for all $\theta\neq\theta_{c}$, one expects PR to saturate in the
long-time regime. When the initial coin state is exactly
$|\sigma_{1}^{-}\rangle$, the long-time behavior of the participation ratio
can be extracted from Eq. (7). Assuming that the spacial extension of the
wavepacket becomes too large for long times, the discrete sum required to
compute PR can be replaced by a continuous integral, with the asymptotic
spacial distribution $\omega({\nu})/t$. Integration by partial fractions then
gives (see details in the Appendix)

$\text{PR}(t)=\frac{at}{b+\text{ln}(t)},$ (10)

where $a=a(\rho)$ and $b=b(\rho)$ are constants. This linear growth with
logarithmic correction is illustrated in Fig. 2(b) for the Grover walk.

Figure 3: (a) Survival probability as a function of $\theta$ evaluated at
distinct time steps, with $\tau_{0}=2000$. It converges quickly to a finite
value, except at the characteristic angle $\theta_{c}$. (b) Collapse of SP in
the vicinity of $\theta_{c}$, showing a universal scaling law.

Figure 4: (a) Participation ratio as a function of $\theta$ for distinct time
steps, with $\tau_{0}=2000$. (b) Data collapse unveiling a universal scaling
law in the vicinity of $\theta_{c}$.

It is important to stress here that the presence of such logarithmic
correction has to be taken into account carefully as the convergence to the
ultimate linear power-law growth is very slow. Numerically, the effective
power-law behavior is estimated by the slope of the
$\ln{\mathrm{PR}}\times\ln{t}$ curve in a given time interval. As the presence
of a logarithmic correction implies in a slowly varying effective exponent it
is useful to write $\mathrm{PR}\propto t^{\lambda(t)}$, with

$\lambda(t)=\frac{d\ln{\mathrm{PR}}}{d\ln{t}},$ (11)

in which a term proportional to the rate of variation of $\lambda(t)$ was
disregarded assuming $d(\ln{\lambda})/d(\ln{t})<<\ln{\lambda}/\ln{t}$. The
asymptotic behavior given by Eq. (10) implies that
$\lambda(t)=1-1/(b+\ln{t})$, evidencing the very slow logarithmic convergence
$\lambda(t\rightarrow\infty)\rightarrow 1$.

In the inset of Fig. 2(b) we show the time evolution of the $\lambda$ computed
via numerical calculation of PR alongside the one derived from Eq. (10). Note
that the asymptotic convergence to $\lambda=1$ is actually very slow. As a
matter of fact, such logarithmic correction on the time-evolution of the
participation ratio is quite common in quantum walks with no localized
component (e.g. two-state quantum walks) featuring group-velocity distribution
singularities such as those present in Eq. (7). Furthermore, we would also
like highlight that the sublinear time evolution of the participation ratio
reported in previous studies omanakuttan2018  is nothing more than an
effective description limited to a specific time interval. It hinders the
logarithmic correction unveiled here, which is the true asymptotic behavior.

Let us now turn our attention to the dynamics occurring at the vicinity of
$\theta_{c}$. In Fig. 3(a), we show the survival probability of the Grover
walk at distinct times as a function of $\theta$ so as to cover a variety of
initial states. Once again, SP saturates to a fixed value except at the
vicinity of the critical parameter $\theta_{c}$, that is where it scales as
$\text{SP}\sim t^{-1}$. In this region data can be collapsed into a single
universal curve by relying on the fact that in the long time limit, SP decays
as $\text{SP}\propto(\theta-\theta_{c})^{2}$. This quadratic behavior takes
place as the overlap between $|\sigma^{+}\rangle$ and the initial state
$\alpha$ vanishes linearly. In association with the $1/t$ decay at
$\theta_{c}$ discussed before, we can thus rewrite the survival probability in
a single-parameter scaling form as

$\text{SP}(\theta,t)=t^{-1}f[(\theta-\theta_{c})t^{1/2}],$ (12)

with $f(0)$ being a constant and $f(\eta\gg 1)\propto\eta^{2}$. This is
depicted in 3(b), where we plot $t\times\text{SP}$ as a function of the
scaling variable $(\theta-\theta_{c})t^{1/2}$. The collapse resulting from the
curves obtained from distinct time steps corroborates the proposed single
parameter scaling in the vicinity of $\theta_{c}$. Most importantly, this is
independent of the coin parameter $\rho$ and thus can be considered as a
universal dynamical scaling law for the present three-state quantum walk
model.

We also analyze the $\text{PR}=\text{PR}(\theta)$ for distinct time steps as
illustrated in Fig. 4(a). As discussed previously, PR saturates to a finite
value due to the localization occurring at the origin. It is only at
$\theta_{c}$ that PR grows linearly with the reported logarithmic correction.
However, finite-time corrections take place close to $\theta_{c}$. In the
long-time regime $t\rightarrow\infty$, PR shall diverge as
$\text{PR}\propto(\theta-\theta_{c})^{-4}$. Such law results from the very
definition of PR that takes the squared survival probability as input. The
participation ratio can also be written in a single parameter scaling form
that incorporates both asymptotic behaviors, such as

$\text{PR}(\theta,t)=\tilde{t}g[(\theta-\theta_{c}){\tilde{t}}^{1/4}],$ (13)

where $\tilde{t}=t/(b+\ln{t})$, $g(0)$ is a constant, and $g(\eta\gg
1)\propto\eta^{-4}$. Data collapse is shown in Fig. 4(b), where we plot
$\text{PR}/\tilde{t}$ as a function of the scaling variable
$(\theta-\theta_{c})\tilde{t}^{1/4}$. Once again, the collapse is accurate
thus supporting both the single-parameter dynamic scaling hypothesis and the
logarithmic correction to the relevant time scale.

Last but not least, we report the dependence of the wavepacket spreading
dynamics on the coin parameter $\rho$. A density plot of the participation
ratio as a function of both $\rho$ and $\theta$ evaluated at the long-time
regime is shown in Fig. 5. It becomes large at the locus
$\theta_{c}=\cos^{-1}(-{\sqrt{1-\rho^{2}}})$. However, the degree of
wavepacket delocalization in nonuniform and reaches its maximum when the coin
parameter approaches that of the Grover walk, $\rho=1/\sqrt{3}$.

Figure 5: Density plot of the participation ratio PR in the $\rho$ vs $\theta$
parameter space evaluated at $t=10^{4}$ (long-time regime). Delocalization is
pronounced at $\theta_{c}=\cos^{-1}(-{\sqrt{1-\rho^{2}}})$, with maximum
wavepacket spreading being reached as the Grover walk condition
$\rho=1/\sqrt{3}$ is approached.

##  IV Concluding remarks

We have studied the dynamics of a general three-state quantum walk, where the
coin operator allows for a continuous tunning from a non-propagating regime
$\rho=0$ to a non-mixing regime $\rho=1$ between the left and right side
components. We set as input an effective two-level state whose relative phase
parametrized by an angle $\theta$ weights the contribution of the coin
eigenstates that leads to delocalized and localized dynamical regimes.

Full delocalization was shown to occur only at $\theta=\theta_{c}$, where the
survival probability $\text{SP}\propto 1/t$. A detailed analysis of the
wavepacket dynamics was then carried out around that point and we unveiled a
single-parameter dynamical scaling law of the form $\text{SP}=t^{-1}f(\eta)$,
with $\eta=(\theta-\theta_{c})t^{1/2}$. The scaling function $f(\eta\gg
1)\propto\eta^{2}$ accounts for the long-time quadratic decay of the survival
probability as $\theta_{c}$ is approached.

We also showed that the participation ratio PR grows linearly in time with a
logarithmic correction of the form $\text{PR}\propto t/(b+\ln{t})$. This
feature is present in delocalized quantum walks, such as the two-state
Hadamard quantum walk, having an asymptotic distribution with group-velocity
distribution singularities as found in Eq. (7). This logarithmic correction
uncovers the actual mechanism underneath the effective sublinear dynamics
reported in omanakuttan2018 , for instance.

The participation ratio was found to satisfy a single-parameter scaling law
$\text{PR}\propto\tilde{t}g(\eta)$ with $\tilde{t}=t/(b+\ln{t})$ and
$\eta=(\theta-\theta_{c})\tilde{t}^{1/4}$. The scaling function $g(\eta\gg
1)\propto\eta^{4}$, thus accounting for the long-time divergence of
$\text{PR}\propto(\theta-\theta_{c})^{-4}$ in the vicinity of $\theta_{c}$.

The above scaling laws worked out here are universal and hold for all values
of the coin parameter $\rho$ for the three-state quantum walk. However, the
degree of delocalization at $\theta_{c}$ depends nonmonotonicaly on $\rho$,
reaching the maximum as the Grover walk condition $\rho=1/\sqrt{3}$ is
fulfilled. The finite-time scaling analysis employed here can be extended to
bring about hidden features underlying several families of quantum walks,
including those whose dynamics is already well established. This is of great
importance for the design of quantum algorithms. Seeking for universal
dynamical laws in discrete-time quantum walks also make for a better
understanding over their connections with Hamiltonian models, allowing for
progress in physical implementations.

##  V Acknowledgments

This work was partially supported by CAPES (Coordenação de Aperfeiçoamento de
Pessoal do Nível Superior), CNPq (Conselho Nacional de Desenvolvimento
Científico e Tecnológico), and FAPEAL (Fundação de Apoio à Pesquisa do Estado
de Alagoas).

## Appendix

Figure 6: (a) Sublinear time-evolution of $\delta(t)=\rho t-x_{m}$, evidencing
$\delta\propto t^{1/3}$; (b) Time-dependence of the occupation probability at
the wave-front $P_{x_{m}}$ indicating its power-law decay $P_{x_{m}}\propto
t^{-2/3}$.

In this appendix, we carry out the derivation of Eq. (10). Considering the
initial state $|\sigma_{1}^{-}\rangle$, the long-time behavior of the
participation ratio can be analytically extracted using the group-velocity
density given by Eq. (7). The inverse participation ratio is given by

$[\mathrm{PR}(t)]^{-1}=\sum_{x}P_{x}(t)^{2}=\sum_{x}[\omega(\nu)/t]^{2},$ (14)

with $\nu=x/t$ and the sum extending over the wave-packet support whose fronts
evolve ballistically in time with velocity $\rho$. The wave-front positions
are at $x_{m}=\pm(\rho t-\delta)$, with $\delta$ being a positive sublinear
correction. In Fig. 6(a) we depict the numerically obtained time-dependence
$\delta(t)\propto t^{1/3}$ for which the wave-front was considered to be the
position of the wave distribution maximum. Such sublinear correction actually
avoids a true divergence of the probability density at the wave-front.
According to Eq. (7), $\omega(\nu_{m}=x_{m}/t)\propto\sqrt{t/\delta(t)}$ in
the long-time regime. This implies that the probability density at the wave-
front is $P_{x{{}_{m}}}(t)\propto t^{-2/3}$ as made clear in Fig. 6(b).
Replacing the sum over $x$ with a sum over $\nu$ in Eq. 14, we get

$[\mathrm{PR}(t)]^{-1}=\sum_{\nu=-(\rho-\delta/t)}^{\rho-\delta/t}[\omega(\nu)/t]^{2},$
(15)

where the interval between successive values of $\nu$ is $\Delta\nu=1/t$.
Therefore, in the long-time regime, the last sum can be turned into an
integral of the form

$\displaystyle[\mathrm{PR}(t)]^{-1}$ $\displaystyle=$
$\displaystyle\int_{-(\rho-\delta/t)}^{\rho-\delta/t}\frac{\omega^{2}}{t}d\nu$
$\displaystyle=$
$\displaystyle\frac{1-\rho^{2}}{\pi^{2}t}\int_{-(\rho-\delta/t)}^{\rho-\delta/t}\frac{d\nu}{(1-\nu^{2})^{2}(\rho^{2}-\nu^{2})}.$
(16)

One can then highlight its time-dependence via the partial fraction
decomposition

$\frac{1}{(1-\nu^{2})^{2}(\rho^{2}-\nu^{2})}=\frac{A}{(1-\nu^{2})^{2}}+\frac{B}{1-\nu^{2}}+\frac{C}{\rho-\nu}+\frac{D}{\rho+\nu}.$
(17)

The integrals of the first two terms converge to finite values as
$t\rightarrow\infty$ while the last two diverge as $\ln{t}$ due to the group-
velocity distribution singularities at $\nu=\pm\rho$. Consequently, the long-
time inverse participation ratio decays as
$[\mathrm{PR}(t)]^{-1}=(1/at)(b+\ln{t})$, unveling a logarithmic correction to
the linear long-time evolution of the participation ratio.

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