# Mean area of the convex hull of a run and tumble particle in two dimensions

Prashant Singh1, Anupam Kundu1, Satya N. Majumdar2 and Hendrik Schawe3
International Centre for Theoretical Sciences, Tata Institute of Fundamental
Research, Bengaluru 560089, India 1 LPTMS, CNRS, Univ. Paris-Sud, Université
Paris-Saclay, 91405, Orsay, France 2 Laboratoire de Physique Théorique et
Modélisation, UMR-8089 CNRS, CY Cergy Paris Université, France 3
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We investigate the statistics of the convex hull for a single run-and-tumble
particle in two dimensions. Run-and-tumble particle (RTP), also known as
persistent random walker, has gained significant interest in the recent years
due to its biological application in modelling the motion of bacteria. We
consider two different statistical ensembles depending on whether (i) the
total number of tumbles $n$ or (ii) the total observation time $t$ is kept
fixed. Benchmarking the results on perimeter, we study the statistical
properties of the area of the convex hull for RTP. Exploiting the connections
to extreme value statistics, we obtain exact analytical expressions for the
mean area for both ensembles. For fixed-$t$ ensemble, we show that the mean
possesses a scaling form in $\gamma t$ (with $\gamma$ being the tumbling rate)
and the corresponding scaling function is exactly computed. Interestingly, we
find that it exhibits crossover from $\sim t^{3}$ scaling at small times
$\left(t\ll\gamma^{-1}\right)$ to $\sim t$ scaling at large times
$\left(t\gg\gamma^{-1}\right)$. On the other hand, for fixed-$n$ ensemble, the
mean expectedly grows linearly with $n$ for $n\gg 1$. All our analytical
findings are supported with numerical simulations.

##  1 Introduction

Active matter refers to a class of driven non-equilibrium systems that
transduces systematic movement out of the supplied energy. Contrary to the
boundary-driven systems, the energy is exchanged at the local scale which
endows the constituent particles with self-propulsion [1, 2, 3]. As a result,
the dynamics of these systems break time-reversal symmetry and thus, violate
the detailed balance. Recently, the self-propulsion (or “active” nature) has
been harnessed to produce useful work for potential therapeutic applications
in various diseases like cancer and heart disease [4, 5]. Furthermore, they
reveal a plethora of complex features like clustering [6, 7], flocking [9, 8],
motility induced phase separation [10, 11, 12, 13], non-existence of equation
of states for pressure [14] and so on. Going beyond the theoretical studies,
the dynamics of active particles has been realised in many experiments based
on different phoretic effects [15, 16].

Run and tumble particle (RTP) has emerged as a quintessential model in
mimicking the dynamics of active particles. Previously known in the random
walk literature as persistent Brownian motion [17, 18], the RTP motion has
recently been quite extensively studied due to its biological application in
modelling the motion of bacteria like E Coli [19, 20, 10]. In this model, ,
the particle moves ballistically along a certain direction with constant speed
$v_{0}~{}(\geq 0)$ till a random time duration $\tau$ drawn from the
exponential distribution with constant rate $\gamma$, i.e. from the
distribution $\rho(\tau)=\gamma e^{-\gamma\tau}$. The event of ballistic
motion is referred to as a ‘run’. After the random time duration $\tau_{1}$,
the particle undergoes ‘tumbling’ in which it chooses a new direction
uniformly. In RTP models, the tumble events are assumed to occur
instantaneously. After the tumble event, it runs along this new direction with
constant speed $v_{0}$ for another random time duration $\tau_{2}$, again
drawn independently from $\rho(\tau)=\gamma e^{-\gamma\tau}$ . In this way,
the RTP moves in a series of runs interspersed by instantaneous tumbles that
occur after random time durations with rate $\gamma$. A schematic
representation of the trajectory is shown in Figure 1(a). Over the recent few
years, this model has been substantially studied and a variety of results are
known. Examples include - position distribution in free space as well as in
confining potential [21, 22, 23, 24, 25, 26, 27], condensation transition [28,
29, 30], persistent properties [31, 32, 33], extremal properties [34, 35],
path functionals [34, 36], current fluctuations [37], interacting multiple
RTPs [27, 38, 39, 40, 41], etc.

Figure 1: (a) Schematic representation of a typical trajectory of a RTP in two
dimensions with total number of runs $n=6$. The RTP moves in a series of runs
interspersed by instantaneous tumbles that occur after random times with rate
$\gamma$. (b) This figure shows the convex hull (red polygon) for the
trajectory on the left.

In this paper, we are interested in the statistics of convex hull for the RTP
in two dimensions. Consider a set of points
$(\vec{r}_{1},\vec{r}_{2},...,\vec{r}_{N})$ in two dimensions. For simplicity,
one can think of them as position of a particle at various instances of time.
Then, convex hull refers to the unique smallest convex polygon that encloses
all these points [42, 43] [see Figure 1(b)]. Now, for a stochastic process,
the set $(\vec{r}_{1},\vec{r}_{2},...,\vec{r}_{N})$ varies over realisations
which implies that the convex hull is also different for different
realisations. One is, then, interested in the statistical properties of this
random convex hull. In ecology, convex hull has been used in estimating the
extent over which the animals move during foraging or other activities [44].
Clearly, this knowledge is useful in designing and demarcating the
geographical territory for them. The properties of convex hull have been of
prime interest in the mathematics literature also [45, 46, 47, 48, 49]. In
physics, the mean area and mean perimeter of the convex hull are found to be
related to the subject of extreme value statistics [50]. Exploiting this
connection, the mean area and mean perimeter have been studied for a variety
of processes like Brownian motion [51], random acceleration [52], diffusion
with resetting [53], random walk and its generalisations [54, 55, 59, 60, 61,
56, 57, 58]. Extensions of these studies to higher dimensions and multi-
particle case have also been considered [62, 63, 64, 50]. Going beyond the
mean values, the entire distributions of the area and perimeter have also been
studied using sophisticated numerical techniques [55, 56, 57]. We refer to
[50] for a review on the convex hull problem.

Recently in [65], the mean perimeter of convex hull for RTP in a plane was
exactly computed for the two different ensembles - (i) fixed number of tumbles
$n$ and (ii) fixed observation time $t$ (discussed later). Here, we go beyond
this work to investigate the statistics of the area of convex hull for two
dimensional RTP. For both ensembles, we compute the mean area exactly. We
verify our analytical results numerically and, also study the variance and the
distribution of the area numerically.

The paper is organised as follows: In Sec. 2, we introduce the model and
summarize the main results of our paper. Sec. 3 contains a brief discussion on
convex hull problems for general $2$-$d$ stochastic processes. Analytic
calculations for mean area are presented in Sec. 4 for fixed-$n$ ensemble and
in Sec. 5 for fixed-$t$ ensemble. We devote Sec. 6 for the numerical study of
probability distribution of the area which is followed by the conclusion in
Sec. 7.

##  2 Model and summary of the results

We consider a RTP moving on a plane. Starting from the origin, the particle
chooses an angle $\phi_{1}$ (measured with respect to the $x$-axis) uniformly
from $[0,2\pi]$ and moves ballistically in that direction with a speed
$v_{0}$. The ballistic motion, referred to as ‘run’, persists along $\phi_{1}$
for a random time $\tau_{1}$ drawn from exponential distribution
$\rho(\tau)=\gamma e^{-\gamma\tau}$ with constant rate $\gamma$. After this,
the particle ‘tumbles’ instantaneously in which event it chooses a new
direction $\phi_{2}$ uniformly from $[0,2\pi]$. Then, it performs another run
for random time $\tau_{2}$ again drawn independently from $\rho(\tau)=\gamma
e^{-\gamma\tau}$. The motion continues in the form of the ballistic runs
interspersed by the instantaneous tumbles that occur after random times drawn
independently from exponential distribution. Let us focus on the $i$-th run
along the direction $\phi_{i}$. Denoting the displacement (position increment)
during $i$-th interval by $(x_{i},y_{i})$, we have

$\displaystyle x_{i}=v_{0}\tau_{i}\cos(\phi_{i}),$ (1) $\displaystyle
y_{i}=v_{0}\tau_{i}\sin(\phi_{i}),$ (2)

where $\tau_{i}$ is the time till which the $i$-th run lasts. The position of
the particle $(X_{i},Y_{i})$ after $i$-th run can be written in terms of
$(x_{i},y_{i})$ as

$\displaystyle X_{i}$ $\displaystyle=X_{i-1}+x_{i},$ (3) $\displaystyle Y_{i}$
$\displaystyle=Y_{i-1}+y_{i},$ (4)

where $i=1,2,...$ and we assume $(X_{0},Y_{0})=(0,0)$. As mentioned earlier,
we consider the motion of the particle in two different ensembles with - (i)
fixed $n$ and (ii) fixed $t$. In case (i), the particle undergoes a fixed
number of runs (say $n$) and we stop the process after $n$ runs have taken
place. The total observation time $t$ will fluctuate for different
realisations. Moreover, we consider the starting point as a tumble which makes
the number of runs equal to the number of tumblings and $n\geq 1$. On the
other hand, for ensemble (ii), we fix the total observation time $t$ and
therefore, the number of tumblings $n$ fluctuates for different realisations.

For these two ensembles, we look at the statistical properties of convex hull.
Recently, the mean perimeter of convex hull for this model was computed
exactly for the two ensembles and the distribution for the perimeter was
numerically studied [65]. Here, we investigate the statistics of area of the
convex hull both analytically and numerically. Using connection to the extreme
value statistics developed in [51, 50], we compute the mean area in the two
ensembles exactly. Next, we also investigate the variance and distribution of
the typical fluctuations of the area. Our main results are summarised below:

  1. 1.

For fixed $n$ ensemble, we find that the mean area $\langle A_{n}\rangle$ is
given by

$\displaystyle\langle
A_{n}\rangle=\frac{v_{0}^{2}}{2\gamma^{2}}\mathcal{S}_{n},~{}~{}n>1$ (5)

where the term $\mathcal{S}_{n}$ is given by

$\displaystyle\mathcal{S}_{n}$
$\displaystyle=\frac{2+\pi}{\sqrt{\pi}}\left[\frac{\Gamma\left(\frac{n-1}{2}-\left\lfloor\frac{n-3}{2}\right\rfloor\right)}{\Gamma\left(\frac{n}{2}-1+\lceil{\frac{3-n}{2}}\rceil\right)}+\frac{\Gamma\left(\frac{n}{2}+1-\left\lfloor\frac{n}{2}\right\rfloor\right)}{\Gamma\left(\frac{n+1}{2}-\left\lfloor\frac{n}{2}\right\rfloor\right)}-\frac{\Gamma\left(\frac{n+2}{2}\right)}{\Gamma\left(\frac{n+1}{2}\right)}-\frac{\Gamma\left(\frac{n+1}{2}\right)}{\Gamma\left(\frac{n}{2}\right)}\right]\Theta(n-1)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\sum_{m=1}^{n-1}\frac{\Gamma\left(\frac{n-m+1}{2}\right)}{\Gamma\left(\frac{n-m+2}{2}\right)}\left[\frac{\Gamma\left(2+\left\lfloor\frac{m-1}{2}\right\rfloor\right)}{\Gamma\left(\frac{3}{2}+\left\lfloor\frac{m-1}{2}\right\rfloor\right)}+\frac{\Gamma\left(\frac{3}{2}+\left\lfloor\frac{m}{2}\right\rfloor\right)}{\Gamma\left(1+\left\lfloor\frac{m}{2}\right\rfloor\right)}\right]$
(6)

Here $\left\lfloor z\right\rfloor~{}(\text{or }\lceil z\rceil)$ denotes the
greatest (or least) integer lesser (or greater) than or equal to $z$. For
large $n$, we find that $\mathcal{S}_{n}\simeq\pi n$ and inserting this in Eq.
(5) yields

$\displaystyle\langle A_{n}\rangle\simeq\frac{n\pi
v_{0}^{2}}{2\gamma^{2}},~{}~{}~{}~{}\text{as }n\to\infty.$ (7)

  2. 2.

On the other hand, for fixed $t$ ensemble, we find that the mean area $\langle
A(t)\rangle$ obeys the scaling relation

$\displaystyle\langle
A(t)\rangle=\frac{v_{0}^{2}}{2\gamma^{2}}\mathcal{J}(\gamma t),$ (8)

where the scaling function $\mathcal{J}(z)$ is exactly given by

$\displaystyle\mathcal{J}(w)=e^{-w}\sum_{n=2}^{\infty}\frac{\mathcal{S}_{n}}{\Gamma(n+2)}w^{n+1}.$
(9)

The scaling function $\mathcal{J}(w)$ displays the following asymptotic
behaviours:

$\displaystyle\mathcal{J}(w)$
$\displaystyle\simeq\frac{w^{3}}{3\pi}+\frac{w^{4}}{4\pi}\left(\frac{\pi}{6}-1\right),~{}~{}~{}~{}~{}~{}~{}\text{as
}w\to 0,$ (10) $\displaystyle\simeq\pi
w,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{as
}w\to\infty.$ (11)

Inserting these forms in Eq. (8) yields that $\langle A(t)\rangle$ exhibits
crossover from $\sim t^{3}$ scaling for $t\ll\gamma^{-1}$ to $\sim t$ scaling
for $t\gg\gamma^{-1}$:

$\displaystyle\langle A(t)\rangle$ $\displaystyle\simeq\frac{\gamma
v_{0}^{2}t^{3}}{6\pi},~{}~{}~{}~{}~{}\text{for }t\ll\gamma^{-1}$ (12)
$\displaystyle\simeq\frac{\pi v_{0}^{2}}{2\gamma}t,~{}~{}~{}~{}~{}~{}\text{for
}t\gg\gamma^{-1}.$ (13)

The comparisons of the analytic expressions in Eqs. (5) and (8) with numerical
simulation are illustrated, respectively, in Figures 2 and 3.

  3. 3.

We have also studied the variance and the distribution of the area in both
ensembles numerically. We found that for large time ($n$ in fixed tumble
ensemble and $t$ in fixed time ensemble) the variance of the area grows
quadratically with time. In addition we found that the central part of the
distribution (describing the typical fluctuations around the mean) for large
time possesses a scaling form when the area is scaled by its mean and the
scaling form matches with that of the Brownian motion, as expected.

In what follows, we derive the results for the mean area explicitly and study
it’s typical fluctuations numerically.

##  3 Mean area of the convex hull

Let us begin by briefly summarising the central idea to compute the mean area
of the convex hull for two dimensional stochastic processes. A more detailed
account of this idea is given in [51, 50]. Based on the knowledge of the
Cauchy’s formulae for closed curve [66], it was shown that the mean area and
mean perimeter for random convex hull are related to the subject of extreme
value statistics. To see this connection, consider a closed curve
$\mathcal{C}$ parametrised by the points
$\\{(\mathcal{X}(s),\mathcal{Y}(s))\\}$ on its boundary where $s$ is the arc
length. For the curve $\mathcal{C}$, we now define the support function
$\mathcal{M}(\theta)$ along the direction $\theta$ (with respect to $x$-axis)
as

$\displaystyle\mathcal{M}(\theta)=\underset{s\in\mathcal{C}}{\text{max}}\left[\mathcal{X}(s)\cos\theta+\mathcal{Y}(s)\sin\theta\right].$
(14)

Geometrically, the support function $\mathcal{M}(\theta)$ represents the
maximum extension of the curve $\mathcal{C}$ along the direction $\theta$.
Interestingly, the perimeter and the area of the domain enclosed by
$\mathcal{C}$ are given in terms of $\mathcal{M}(\theta)$ by Cauchy’s formula
as

$\displaystyle L$ $\displaystyle=\int_{0}^{2\pi}d\theta~{}M(\theta),$ (15)
$\displaystyle A$
$\displaystyle=\frac{1}{2}\int_{0}^{2\pi}d\theta\left[\mathcal{M}^{2}(\theta)-\left(\mathcal{M}^{\prime}(\theta)\right)^{2}\right].$
(16)

To elaborate further, let us, for simplicity, consider a discrete time
stochastic process of $n$ steps. Let the positions of the particle at
successive (discrete) times of a realisation are denoted by
$\\{(X_{i},Y_{i})\\}$ where $i=1,2,...,n$. We further consider that
$\mathcal{C}$ now represents the convex hull corresponding to the points
$\\{(X_{i},Y_{i})\\}$. To construct the support function $\mathcal{M}(\theta)$
for $\mathcal{C}$, one clearly needs $\\{(\mathcal{X}(s),\mathcal{Y}(s))\\}$
which is a difficult task. However, it was shown in [51, 50] that this problem
can be circumvented by noting the fact that $\mathcal{M}(\theta)$ is also the
maximum of the projections of all points $\\{(X_{i},Y_{i})\\}$ along the
direction $\theta$. One can now write the support function
$\mathcal{M}(\theta)$ as

$\displaystyle\mathcal{M}(\theta)$ $\displaystyle=\underset{1\leq i\leq
n}{\text{max}}\left[X_{i}\cos\theta+Y_{i}\sin\theta\right].$ (17)

Using this form for $M(\theta)$ in Eqs. (15) and (16)and then taking average
over different realisations one gets the mean perimeter and the mean area of
the convex hull $\mathcal{C}$. Since, we are interested in area only, we
provide below the expression of mean area which follows directly from Eq.
(16):

$\displaystyle\langle
A_{n}\rangle=\frac{1}{2}\int_{0}^{2\pi}d\theta\left[\langle\mathcal{M}^{2}(\theta)\rangle-\langle\left(\mathcal{M}^{\prime}(\theta)\right)^{2}\rangle\right].$
(18)

To proceed further, we assume that the maximum in Eq. (17) is attained in the
$k^{*}$-th step which enables us to write $\mathcal{M}(\theta)$ and
$\mathcal{M}^{\prime}(\theta)$ as

$\displaystyle\mathcal{M}(\theta)=X_{k^{*}}\cos\theta+Y_{k^{*}}\sin\theta,$
(19)
$\displaystyle\mathcal{M}^{\prime}(\theta)=-X_{k^{*}}\sin\theta+Y_{k^{*}}\cos\theta.$
(20)

For isotropic processes, the suppport function
$\langle\mathcal{M}^{2}(\theta)\rangle$ and $\langle\mathcal{M}^{\prime
2}(\theta)\rangle$ are independent of $\theta$ and we can consider just the
direction $\theta=0$. For this case, the mean area in Eq. (18) becomes

$\displaystyle\langle A_{n}\rangle=\pi\left[\langle M_{n}^{2}\rangle-{\langle
Y_{k^{*}}^{2}\rangle(n)}\right],$ (21)

where $M_{n}=\text{max}\left[{X_{1},X_{2},...,X_{n}}\right]$ is the maximum
displacement along the $x$-axis and $Y_{k^{*}}$ is the abscissa at $k^{*}$-th
time-step at which the maximum $M_{n}$ along $x$-direction is reached. Later,
Eq. (21) will be useful in calculating the mean area of the convex hull for
RTP in fixed $n$ ensemble.

Although Eq. (21) is derived for discrete time isotropic stochastic process,
one can derive an analogous formula for the mean area of $\mathcal{C}$ in the
continuous time case [50]. For this case, the mean area of the convex hull
reads

$\displaystyle\langle A(t)\rangle=\pi\left[\langle M^{2}(t)\rangle-{\langle
Y(t_{m})^{2}\rangle(t)}\right],$ (22)

where $M(t)$ is the maximum of the $x$-coordinate till observation time $t$
i.e. $M(t)=\text{max}[\\{X(\tau)\\},~{}\forall 0\leq\tau\leq t]$ and $t_{m}$
is the time at which maximum $M(t)$ is reached. Also, $Y(t_{m})$ represents
the $y$-coordinate of the RTP at time $t_{m}$ in trajectory of duration $t$.
Once again, Eq. (22) will be useful in computing the mean area for RTP in
fixed $t$ ensemble.

Before closing this section, we remark that the formulae of mean area in Eqs.
(21) and (22) apply to general isotropic $2$-$d$ stochastic process. In the
following, we use these formulae to compute the mean area of $\mathcal{C}$ for
the RTP model in two dimension. We first compute the mean area for the
fixed-$n$ ensemble and then focus on the fixed-$t$ ensemble.

##  4 Mean area for fixed-$n$ ensemble

Let us first look at the RTP in fixed-$n$ ensemble where the total number of
runs $n$ is fixed but the total time $t$ varies for different realisations. As
indicated by Eq. (21), we need the maximum $M_{n}$ of the $x$-coordinate
trajectory and the corresponding abscissa $Y_{k^{*}}(n)$ to compute the mean
area $\langle A_{n}\rangle$. Recall from Eqs. (3) and (4), the position
coordinates of the RTP performs random walks with correlated increments
(jumps) $(x_{i},y_{i})$ are given in Eqs. (1) and (2). Also recall that we
have chosen the initial position of the RTP to be $(X_{0},Y_{0})=(0,0)$. Let
us first compute the joint probability distribution $p(x_{i},y_{i},\tau_{i})$
of the increments $x_{i}$ and $y_{i}$ along, respectively, the $x$ and $y$
directions, and time duration $\tau_{i}$ for the $i$-th run. Since the RTP
moves ballistically during the time $\tau_{i}$, we have
$v_{0}\tau_{i}=\sqrt{x_{i}^{2}+y_{i}^{2}}$. Also, the time $\tau_{i}$ is
exponentially distributed $\rho(\tau_{i})=\gamma e^{-\gamma\tau_{i}}$. This
enables us to write the joint distribution $p(x,y,\tau)$ as

$\displaystyle p(x,y,\tau)=\frac{\gamma
e^{-\gamma\tau}}{\pi}~{}\delta(v_{0}^{2}\tau^{2}-x^{2}-y^{2}),$ (23)

where the factor $1/\pi$ comes from the normalisation condition. Finally,
integrating $p(x,y,\tau)$ over $\tau$, we get the joint distribution of the
increments $x$ and $y$ as

$\displaystyle p(x,y)=\frac{\gamma}{2\pi
v_{0}~{}\sqrt{x^{2}+y^{2}}}~{}~{}\text{exp}\left(-\frac{\gamma}{v}\sqrt{x^{2}+y^{2}}\right).$
(24)

Notice that, the problem of run and tumble motion now got mapped to a model of
random walks in two dimensions with the jump distribution $p(x,y)$ given in
Eq. (24). Such mapping have been considered in [32] to study the persistent
properties of RTP. The advantage now is that one can use Eq. (21), true for
discrete time processes, also for RTP. In what follows, we use this equation
to compute the mean area of the convex hull for fixed-$n$ ensemble. From Eq.
(21), we see that this reduces to the problem of computing $\langle
M_{n}^{2}\rangle$ and $\langle Y_{k^{*}}^{2}\rangle(n)$ which we calculate
below.

###  4.1 Computation of $\langle M_{n}^{2}\rangle$

In order to compute the second moment of the maximum
$M_{n}=\text{max}\\{X_{0},X_{1},X_{2},...,X_{n}\\}$ of the $x$-component of a
given trajectory of $n$ steps, we first recall that
$\\{X_{0},X_{1},X_{2},...,X_{n}\\}$ denote just a one dimensional random walk
trajectory such that $X_{i}=X_{i-1}+x_{i}$. The increment $x_{i}$ is
distributed according to the probability distribution $p_{1}(x_{i})$ which is
obtained by integrating the joint distribution $p(x_{i},y_{i})$ in Eq. (24)
over all $y_{i}$. The resulting expression reads

$\displaystyle p_{1}(x)=\int_{-\infty}^{\infty}dy~{}p(x,y)=\frac{\gamma}{\pi
v_{0}}K_{0}\left(\frac{\gamma|x|}{v_{0}}\right),$ (25)

where $K_{\nu}(z)$ is the modified Bessel function of second kind. Note that
$p_{1}(x)$ is both symmetric and continuous. Hence the random walker is
characterised by the identical and independent increments $\\{x_{i}\\}$ drawn
from symmetric and continuous distribution $p_{1}(x_{i})$.

To calculate $\langle M_{n}^{2}\rangle$, we use the Pollaczek-Spitzer formula
[67, 46] which characterises the maximum $M_{n}$ for a random walk with
identical and independent increments drawn from symmetric and continuous
distribution. If $Q_{n}(M)=\text{Prob}[M_{n}\leq M]$ denotes the cumulative
probability of $M_{n}$, then according to the Pollaczek-Spitzer formula,
$Q_{n}(M)$ satisfies [54, 68]:

$\displaystyle\sum_{n=0}^{\infty}z^{n}\langle e^{-\lambda
M_{n}}\rangle=\sum_{n=0}^{\infty}z^{n}\int_{0}^{\infty}dMe^{-\lambda
M}Q_{n}^{\prime}(M)=\frac{\phi(z,\lambda)}{\sqrt{1-z}},$ (26)

where $0\leq z\leq 1$ and $\lambda\geq 0$ and the function $\phi(z,\lambda)$
is defined as

$\displaystyle\phi(z,\lambda)=\text{exp}\left(-\frac{\lambda}{\pi}\int_{0}^{\infty}d\xi~{}\frac{\ln(1-z\hat{p}_{1}(\xi))}{\lambda^{2}+k^{2}}\right),$
(27)

with $\hat{p}(\xi)$ being the Fourier transform of $p(x)$ given by

$\displaystyle\hat{p}_{1}(\xi)=\int_{-\infty}^{\infty}dx~{}e^{i\xi
x}p_{1}(x)~{}=\frac{1}{\sqrt{1+\xi^{2}\sigma^{2}}}.$ (28)

We have inserted $p_{1}(x)$ from Eq. (25) in writing $\hat{p}_{1}(\xi)$ and
defined $\sigma=\frac{v_{0}}{\gamma}$. One can suitably use Eq. (26) to
compute all moments of $M_{n}$. In fact, the Pollaczek-Spitzer formula was
used in [69] to determine the generating functions for all moments of $M_{n}$.
For the first two moments, one can show that

$\displaystyle h^{(1)}(z)=\sum_{n=0}^{\infty}z^{n}\langle
M_{n}\rangle=\frac{1}{\pi(1-z)}\int_{0}^{\infty}\frac{d\xi}{\xi^{2}}~{}\ln\left(\frac{1-z\hat{p}_{1}(\xi)}{1-z}\right),$
(29) $\displaystyle h^{(2)}(z)=\sum_{n=0}^{\infty}z^{n}\langle
M_{n}^{2}\rangle=(1-z)\left[h^{(1)}(z)\right]^{2}+\frac{\sigma^{2}z}{2(1-z)^{2}}.$
(30)

By appropriately differentiating $h^{(2)}(z)$ with respect to $z$, it is
straightforward to show that the second moment $\langle M_{n}^{2}\rangle$ can
be expressed completely in terms of the first moment $\langle M_{n}\rangle$ as

$\displaystyle\langle M_{n}^{2}\rangle=\sum_{m=1}^{n-1}\langle
M_{m}\rangle\left[\langle M_{n-m}\rangle-\langle
M_{n-m-1}\rangle\right]+\frac{n\sigma^{2}}{2}.$ (31)

Expanding the right hand side of Eq. (29) one can in principle compute
$\langle M_{n}\rangle$. Using Kac’s formula [45] for mean maximum
displacement, it was recently computed explicitly in [65] where it was shown
that

$\displaystyle\langle
M_{n}\rangle=\frac{\sigma}{2\sqrt{\pi}}\sum_{j=1}^{n}\frac{\Gamma\left(\frac{j+1}{2}\right)}{\Gamma\left(\frac{j+2}{2}\right)}.$
(32)

Using this expression, we get

$\displaystyle\langle M_{n-m}\rangle-\langle
M_{n-m-1}\rangle=\frac{\sigma}{2\sqrt{\pi}}~{}\frac{\Gamma\left(\frac{n-m+1}{2}\right)}{\Gamma\left(\frac{n-m+2}{2}\right)},~{}~{}n>1.$
(33)

Finally inserting Eqs. (32) and (33) in the expression of $\langle
M_{n}^{2}\rangle$ in Eq. (31) we get

$\displaystyle\langle
M_{n}^{2}\rangle=\frac{v_{0}^{2}}{2\gamma^{2}}\left(\frac{\mathcal{S}_{n}}{\pi}+n\right),~{}~{}n>1$
(34)

with $\sigma=\frac{v_{0}}{\gamma}$ and

$\displaystyle\mathcal{S}_{n}$
$\displaystyle=\frac{\sqrt{\pi}}{\sigma}\sum_{m=1}^{n-1}\frac{\Gamma\left(\frac{n-m+1}{2}\right)}{\Gamma\left(\frac{n-m+2}{2}\right)}\langle
M_{m}\rangle.$ (35)

Inserting $\langle M_{m}\rangle$ from Eq. (32) in the above equation and
simplifying, one gets the explicit expressions of $\mathcal{S}_{n}$ given in
Eq. (6).

###  4.2 Computation of $\langle Y_{k^{*}}^{2}\rangle(n)$

We now compute the other term $\langle Y_{k^{*}}^{2}\rangle(n)$ in the
expression of the mean area in Eq. (21). To calculate this we first compute
the joint distribution
$\mathcal{P}(Y,k^{*}|n)=\text{Prob.}[Y_{k^{*}}=Y,k^{*}|n]$ and then compute
the second moment of the displacement $Y_{k^{*}}$ of the particle along $y$
direction at step $k^{*}$ in which the particle reaches it’s maximum along the
$x$ direction in a walk of $n$-steps. It is possible to show that one can
compute this joint distribution for a general $2$-d discrete time random walk
where the position coordinates $(X_{i},Y_{i})$ at $i^{\rm th}$ step evolves,
starting from $X_{0}=0,~{}Y_{0}=0$, as $X_{i}=X_{i-1}+x_{i}$ and
$Y_{i}=Y_{i-1}+y_{i}$ with the jump increments $(x_{i},y_{i})$ at different
steps are drawn independently from the common distribution $p(x,y)$. Note that
at a given step, the increments $x$ and $y$ can be correlated.

To proceed let us define marginal distribution of the $y$-increment

$\displaystyle p_{2}(y)=\int_{-\infty}^{\infty}dx~{}p(x,y),$ (36)

and the marginal distribution of the $y$-coordinate $Y_{k}$ of the walker at
step $k$

$\displaystyle
P(Y,k)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}...\int_{-\infty}^{\infty}$
$\displaystyle
dy_{1}dy_{2}...dy_{k}~{}\delta\left(Y-\sum_{i=1}^{k}y_{i}\right)~{}\prod_{i=1}^{k}p_{2}(y_{i}).$
(37)

Now consider any trajectory in $2$-d up to step $n$. Let $k^{*}$ denote the
time at which the the $X_{i}$’s achieve their maximum $M_{n}$ and $Y_{k^{*}}$
denote the $y$-coordinate exactly at step $k^{*}$. Recall, we want to compute
the joint distribution of $Y_{k^{*}}$ and $k^{*}$, given the total number of
steps $n$, i.e., $\mathcal{P}(Y,k^{*}|n)$. We show that this joint
distribution is given by

$\mathcal{P}(Y,k^{*}|n)=q_{k^{*}}\,q_{n-k^{*}}P(Y,k^{*})\,,$ (38)

where $P(Y,k^{*})$ is defined in Eq. (37) and $q_{n}={2n\choose n}\,2^{-2n}$
is the Sparre-Andersen survival probability of a $1$-d random walk with
arbitrary symmetric and continuous jump distribution [70, 69].

To prove the claim in Eq. (38), we start with the joint probability
distribution for $M_{n}=M,~{}Y_{k^{*}}=Y$ and $k^{*}$ denoted by
$\mathscr{P}(M,Y,k^{*}|n)$. This joint probability of the $n$-step walk can be
expressed as a multi-dimensional integral

$\displaystyle\mathscr{P}(M,Y,k^{*}|n)=\int\vec{dx}~{}\vec{dy}~{}\mathcal{Z}_{k^{*}}\big{(}M,Y,\\{x_{i}\\},\\{y_{i}\\}\big{)}\prod_{j=1}^{n}p(x_{j},y_{j}),$
(39)

with $\mathcal{Z}_{k^{*}}\left(M,Y,\\{x_{i}\\},\\{y_{i}\\}\right)$ defined as

$\displaystyle\begin{split}\mathcal{Z}_{k^{*}}\big{(}M,Y,\\{x_{i}\\},\\{y_{i}\\}\big{)}=&~{}\Theta(M)\Theta(M-X_{1})\Theta(M-X_{2})...\Theta\left(M-X_{k^{*}-1}\right)\times\\\
&\delta\left(M-X_{k^{*}}\right)~{}\delta\left(Y-Y_{k^{*}}\right)\Theta\left(M-X_{k^{*}+1}\right)...\Theta\left(M-X_{n}\right),\end{split}$
(40)

where $X_{i}=\sum_{j=1}^{i}x_{j}$ and $Y_{i}=\sum_{j=1}^{i}y_{j}$ (recall we
have chosen $X_{0}=Y_{0}=0$). Here $\Theta(n)$ is the Heaviside theta
function. The function
$\mathcal{Z}_{k^{*}}\big{(}M,Y,\\{x_{i}\\},\\{y_{i}\\}\big{)}$ ensures that
$X_{k^{*}}=M$ and $Y_{k^{*}}=Y$ while all other $\\{X_{i}\\}$ are smaller than
$M$. Finally, we integrate over all $\\{x_{i},y_{i}\\}$ with appropriate joint
distribution $\prod_{i=1}^{n}p(x_{i},y_{i})$. For simplicity, we have used the
short-hand notation $\vec{dx}=dx_{1}dx_{2}...dx_{n}$ and
$\vec{dy}=dy_{1}dy_{2}...dy_{n}$.

Since we are interested in the joint distribution $\mathcal{P}(Y,k^{*}|n)$ of
$Y_{k^{*}}=Y$ and $k^{*}$, we integrate $\mathscr{P}(M,Y,k^{*}|n)$ in Eq. (39)
over $M$ i.e.

$\displaystyle\mathcal{P}(Y,k^{*}|n)=\int_{0}^{\infty}dM~{}\mathscr{P}(M,Y,k^{*}|n).$
(41)

To proceed further we take the Fourier transformation with respect to $Y$

$\displaystyle\bar{\mathcal{P}}\left(\xi,k^{*}|n\right)$
$\displaystyle=\int_{-\infty}^{\infty}dYe^{i\xi
Y}\int_{0}^{\infty}dM~{}\mathscr{P}(M,Y,k^{*}|n),$ (42)

and perform some algebraic simplications in Eq. (39). We relegate the details
of the calculations to A and present only the final result here. The final
expression reads

$\displaystyle\bar{\mathcal{P}}\left(\xi,k^{*}|n\right)$
$\displaystyle=q_{k^{*}}q_{n-k^{*}}~{}\left[\hat{p}_{2}(\xi)\right]^{k^{*}},$
(43)

where $\hat{p}_{2}(\xi)$ in Eq. (43) represents the Fourier transformation of
the marginal distribution $p_{2}(y)$ of $y$-increment [see Eq. (36)] and is
defined by

$\displaystyle\hat{p}_{2}(\xi)=\int_{-\infty}^{\infty}dy~{}e^{i\xi
y}p_{2}(y).$ (44)

The term $q_{n}={2n\choose n}\,2^{-2n}$, as mentioned earlier, is the survival
probability of a random walker in one dimension starting from the origin and
with jumps drawn independently from a symmetric and continuous distribution.
Note that the term $\left[\hat{p}_{2}(\xi)\right]^{k^{*}}$ in Eq. (43) is
actually the Fourier transform of the marginal distribution $P(Y,k^{*})$ of
the $y$-coordinate at step $k^{*}$ as can be easily seen from Eq. (37). Hence,
performing inverse Fourier transform on both sides of Eq. (43) one arrives at
the result in Eq. (38). Notice that the expression of $\mathcal{P}(Y,k^{*}|n)$
in Eq. (38) appears naturally in the following form:

$\displaystyle\begin{split}\mathcal{P}(Y,k^{*}|n)=\text{Prob.}[Y_{k^{*}}=Y]~{}\times~{}\text{Prob.}[M_{n}~{}\text{occurs~{}at~{}step}~{}k^{*}~{}\text{in}~{}n\text{-step~{}walk}].\end{split}$
(45)

This result is quite universal and holds true for any joint distribution
$p(x,y)$ as long as it is symmetric and continuous in $x$. This universality
is a consequence of Sparre Anderson theorem [70, 69]. In addition if the joint
distribution $p(x,y)$ is such that $\langle y^{2}\rangle$ is finite, one finds

$\displaystyle\langle Y_{k^{*}}^{2}\rangle(n)$
$\displaystyle=\sum_{k^{*}=1}^{n}\langle
Y_{k^{*}}^{2}\rangle~{}\times~{}q_{k^{*}}q_{n-k^{*}}$ (46)
$\displaystyle=.\sum_{k^{*}=1}^{n}\langle
y^{2}\rangle~{}k^{*}~{}q_{k^{*}}q_{n-k^{*}}$ (47) $\displaystyle=\langle
y^{2}\rangle~{}\frac{n}{2}.$ (48)

where we have used $q_{k}=2^{-2k}\binom{2k}{k}$. Note that this result is also
universal.

Recall, in this paper, we are interested to compute $\langle
Y_{k^{*}}^{2}\rangle(n)$ for RTP in which case the joint distribution $p(x,y)$
is given in Eq. (24). For this distribution one has

$\displaystyle p_{2}(y)=\frac{\gamma}{\pi
v_{0}}K_{0}\left(\frac{\gamma|y|}{v_{0}}\right),$ (49)

with $\langle y^{2}\rangle=\frac{v_{0}^{2}}{\gamma^{2}}$ which gives

$\displaystyle\langle Y_{k^{*}}^{2}\rangle(n)=\frac{v_{0}^{2}}{2\gamma^{2}}n.$
(50)

Figure 2: Comparison of the mean area of the convex hull for a RTP in
fixed-$n$ ensemble (left) and fixed-$t$ ensemble (right) with numerical
simulations. The corresponding analytic expressions are given in Eqs. (5) and
(8) respectively. Parameters chosen are $v_{0}=1$, $\gamma=1.5$ for left panel
and $v_{0}=1$, $\gamma=1$ for right panel.

Figure 3: Comparison of the mean area for fixed-$n$ (left) and fixed-$t$
(right) ensembles with the numerical simulations. For fixed-$n$ ensemble
(left), we have denoted $\alpha_{n}=\langle A_{n}\rangle/n\sigma^{2}$ with
$\sigma=v_{0}/\gamma$ and $\langle A_{n}\rangle$ given in Eq. (5). On the
other hand, for fixed-$t$ ensemble (right), we have introduced the notation
$\alpha(t)=\langle A(t)\rangle/\gamma t\sigma^{2}$ with $\langle A(t)\rangle$
given in Eq. (8). For both panels, we have used $v_{0}=1$.

###  4.3 Mean area $\langle A_{n}\rangle$

Substituting $\langle Y_{k^{*}}^{2}\rangle(n)$ from Eq. (50) along with
$\langle M_{n}^{2}\rangle$ from Eq. (34) in the expression of $\langle
A_{n}\rangle$ in Eq. (21), we obtain mean area of the convex hull for $2$-d
isotropic run and tumble motion in fixed-$n$ ensemble as quoted in Eq. (5).
For large $n$, we find $\mathcal{S}_{n}\simeq\pi n$ (see B for proof) which
yields the asymptotic form of $\langle A_{n}\rangle$ in Eq. (5) as

$\displaystyle\langle
A_{n}\rangle\simeq\frac{n\pi}{2}\sigma^{2},~{}~{}~{}~{}~{}~{}~{}~{}\text{as
}n\to\infty.$ (51)

This matches with the mean area of the convex hull of a discrete two
dimensional random walk of $n$ steps for any jump distribution with a finite
variance $\sigma^{2}$ [54]. In Figure 2 (left panel), we have compared our
analytic result of $\langle A_{n}\rangle$ in Eq. (5) with the simulation
results. We observe an excellent agreement between them. To construct convex
hull numerically, we deploy the Andrew’s monotone chain algorithm [71] which
is further expedited with Akl’s heuristic [72]. Then, to calculate the area,
we denote the $m$ vertices of the convex hull as
$\\{\bar{X}_{i},\bar{Y}_{i}\\},~{}1\leq i\leq m$ in order of their Cartesian
coordinates and use

$\displaystyle
A=\frac{1}{2}\sum_{i=0}^{m-1}\left(\bar{Y}_{i}+\bar{Y}_{i+1}\right)\left(\bar{X}_{i}-\bar{X}_{i+1}\right),$
(52)

with $(\bar{X}_{0},\bar{Y}_{0})=(\bar{X}_{m},\bar{Y}_{m})$. Finally, we
estimate the mean area using the simple sampling where we generate $10^{4}$
realisations of RTP, construct area of each of them using Eq. (52) and then
take the average.

To compare the mean area for different parameters, we rescale $\langle
A_{n}\rangle$ in Eq. (5) with $\sigma^{2}$ where $\sigma=v_{0}/\gamma$.
Moreover, from Eq. (51), we see that $\langle A_{n}\rangle$ scales linearly
with $n$ for $n\gg 1$. Therefore, we also rescale $\langle A_{n}\rangle$ with
$n$ to remove the asymptotic growth with respect to $n$ for proper
visualisation. Defining

$\displaystyle\alpha_{n}=\frac{\langle A_{n}\rangle}{n\sigma^{2}},$ (53)

one expects $\alpha_{n}$ to be independent of $v_{0}$ and $\gamma$ via Eq.
(5). Also, $\alpha_{n}$ should saturate to the value
$\frac{\pi}{2}\left(=1.571..\right)$ for $n\to\infty$. Indeed, in figure 3
(left panel), we observe that $\alpha_{n}$ is identical for two different
values of $\gamma$, namely $\gamma=1$ and $\gamma=0.5$. Moreoever, it
approaches the value $\frac{\pi}{2}$ as we go to higher values of $n$. This
comparison of $\alpha_{n}$ for two different values of $\gamma$ provides
another verification of $\langle A_{n}\rangle$ in Eq. (5).

##  5 Fixed-$t$ ensemble

The previous section dealt with the mean area of the convex hull for RTP in
fixed-$n$ ensemble. We now consider the mean area in fixed-$t$ ensemble where
the observation time $t$ is fixed but the number of runs $n$ varies from
sample to sample. For this case also, we show that the run and tumble model
can be suitably mapped to a random walker in two dimensions which is then used
to calculate exactly the mean area $\langle A(t)\rangle$ via Eq. (22). To
begin with, let us consider a realisation of RTP with $n$ runs where the
$i$-th run lasts for time $\tau_{i}$ with position increments $x_{i}$ and
$y_{i}$. Since at the end of each run except the $n$-th one, the RTP
encounters a tumbling, the times $\\{\tau_{i}\\}~{}\text{for}~{}1\leq
i\leq(n-1)$ are all drawn independently from the exponential distribution
$\rho(\tau_{i})=\gamma e^{-\gamma\tau_{i}}$. Therefore the joint distribution
$p(x_{i},y_{i},\tau_{i})$ with $1\leq i\leq(n-1)$ is given by Eq. (23). On the
other hand, during the last interval $\tau_{n}$, the RTP does not encounter
any tumble: the probability of which is $e^{-\gamma\tau_{n}}$. Hence, the
corresponidng joint distribution is

$\displaystyle
p_{\text{last}}(x_{n},y_{n},\tau_{n})=\frac{e^{-\gamma\tau_{n}}}{\pi}~{}\delta(v_{0}^{2}\tau_{n}^{2}-x_{n}^{2}-y_{n}^{2})=\frac{1}{\gamma}~{}p(x_{n},y_{n},\tau_{n}),$
(54)

where $p(x_{n},y_{n},\tau_{n})$ is given by Eq. (23). We emphasise that unlike
in the fixed-$n$ case, here the runs are correlated due to the constraint of
fixed $t$. To see this more clearly, we write the grand joint distribution of
$\\{x_{i}\\},~{}\\{y_{i}\\}$ and $n$ below:

$\displaystyle
P\big{(}\\{x_{i}\\},\\{y_{i}\\},n|t\big{)}=\int_{0}^{t}d\tau_{1}\int_{0}^{t}d\tau_{2},...,\int_{0}^{t}d\tau_{n}~{}\frac{1}{\gamma}\left[\prod_{i=1}^{n}p(x_{i},y_{i},\tau_{i})\right]\delta\left(\sum_{i=1}^{n}\tau_{i}-t\right).$
(55)

To get rid of the $\delta$-function, we take Laplace transformation with
respect to $t~{}(\to s)$

$\displaystyle\int_{0}^{\infty}dt~{}e^{-st}~{}P\big{(}\\{x_{i}\\},\\{y_{i}\\},n|t\big{)}$
$\displaystyle=\frac{1}{\gamma}\left[\prod_{i=1}^{n}\frac{\gamma~{}\text{exp}\left(-\frac{(\gamma+s)}{v}\sqrt{x_{i}^{2}+y_{i}^{2}}\right)}{2\pi
v_{0}~{}\sqrt{x_{i}^{2}+y_{i}^{2}}}\right],$ (56)

which we rewrite as

$\displaystyle\int_{0}^{\infty}dt~{}e^{-st}~{}P\big{(}\\{x_{i}\\},\\{y_{i}\\},n|t\big{)}$
$\displaystyle=\frac{1}{\gamma}\left(\frac{\gamma}{\gamma+s}\right)^{n}~{}\left[\prod_{i=1}^{n}g_{s}(x_{i},y_{i})\right],$
(57) $\displaystyle\text{with }~{}~{}~{}~{}~{}~{}~{}g_{s}(x,y)$
$\displaystyle=\frac{(\gamma+s)~{}\text{exp}\left(-\frac{(\gamma+s)}{v}\sqrt{x_{i}^{2}+y_{i}^{2}}\right)}{2\pi
v_{0}\sqrt{x^{2}+y^{2}}}.$ (58)

Finally, inverting the Laplace transform in Eq. (57), the grand joint
distribution $P\big{(}\\{x_{i}\\},\\{y_{i}\\},n|t\big{)}$ can be formally
written as

$\displaystyle
P\big{(}\\{x_{i}\\},\\{y_{i}\\},n|t\big{)}=\int_{\Gamma}\frac{ds}{2\pi
i}~{}e^{st}~{}\frac{1}{\gamma}\left(\frac{\gamma}{\gamma+s}\right)^{n}~{}\left[\prod_{i=1}^{n}g_{s}(x_{i},y_{i})\right],$
(59)

where $\Gamma$ is the Bromwich contour in the complex $s$ plane. Note that the
function $g_{s}(x,y)$ given in Eq. (58) can be interpreted as a probability
distribution as it is positive over full $(x,y)$ plane and normalised to
unity. As a result the term inside the square bracket in the integrand of the
Eq. (59) can be interpreted as the joint distribution of the increments
$x_{i}$ and $y_{i}$ of a random walker in two dimension in steps
$i=1,2,...,n$. In the context of RTP such mapping to random walk problem was
observed earlier [65, 35] and exploited to study the survival probability in
higher dimension [35]. In this paper we follow a similar calculation using
this mapping and compute the mean area $\langle A(t)\rangle$ of the convex
hull by employing the formula in Eq. (22). As seen in this formula, we need to
calculate $\langle M^{2}(t)\rangle$ and $\langle Y(t_{m})^{2}\rangle(t)$ to
compute $\langle A(t)\rangle$. In the following, we use the joint distribution
$P\big{(}\\{x_{i}\\},\\{y_{i}\\},n|t\big{)}$ in Eq. (59) to calculate these
two quantities explicitly.

###  5.1 Computation of $\langle M^{2}(t)\rangle$

Let us begin with the computation of $\langle M^{2}(t)\rangle$ where $M(t)$ is
the maximum of the $x$-co ordinate of RTP up to observation time $t$. For
this, we need to compute the statistics of the maximum of a $1d$ random walker
for fixed $t$ with $n$ jumps in the $x$-coordinates: $\\{x_{i}\\}$ for
$i=1,2,...,n$. The joint distribution of increments $\\{x_{i}\\}$ can be
obtained by integrating $P\big{(}\\{x_{i}\\},\\{y_{i}\\},n|t\big{)}$ in Eq.
(59) over all $\\{y_{i}\\}$ as

$\displaystyle P_{x}\big{(}\\{x_{i}\\},n|t\big{)}$
$\displaystyle=\int_{-\infty}^{\infty}dy_{1}dy_{2}..dy_{n}~{}P\big{(}\\{x_{i}\\},\\{y_{i}\\},n|t\big{)},$
(60) $\displaystyle=\int_{\Gamma}\frac{ds}{2\pi
i}~{}e^{st}~{}\frac{1}{\gamma}\left(\frac{\gamma}{\gamma+s}\right)^{n}~{}\left[\prod_{i=1}^{n}g_{s}(x_{i})\right],$
(61) $\displaystyle\text{with }~{}~{}~{}~{}~{}~{}~{}g_{s}(x)$
$\displaystyle=\int_{-\infty}^{\infty}dy~{}g_{s}(x,y)=\frac{(\gamma+s)}{\pi
v_{0}}K_{0}\left(\frac{(\gamma+s)|x|}{v_{0}}\right).$ (62)

Using this expression of $P_{x}\big{(}\\{x_{i}\\},n|t\big{)}$, we now proceed
to calculate the statistics of the maximum $M(t)$. To this end, we define
$Q(M,n|t)$ as the probability that $X_{i}<M$ for $1\leq i\leq n$, where
$X_{i}=\sum_{j=1}^{i}x_{j}$. It is easy to realise that $Q(M,n|t)$ is actually
the survival probability that the walker with $n$ steps up to time $t$ has not
crossed $X=M$. Formally, this is given as

$\displaystyle Q(M,n|t)$
$\displaystyle=\int_{-\infty}^{\infty}dx_{1}...\int_{-\infty}^{\infty}dx_{n}~{}\text{Prob.}\left[X_{1}<M,X_{2}<M,...,X_{n}<M,n|t\right],$
(63)
$\displaystyle=\int_{-\infty}^{\infty}dx_{1}...\int_{-\infty}^{\infty}dx_{n}~{}\Theta(M-X_{1})..\Theta(M-X_{n})~{}P_{x}\big{(}\\{x_{i}\\},n|t\big{)}$
(64)

Note that $Q(M,n|t)$ is also the probability that the maximum displacement of
the $1d$ random walk with $n$ steps up to time $t$ is $\leq M$.
Differentiating $Q(M,n|t)$ with $M$ gives the joint probability distribution
for $M$ and $n$ which can then be used to calculate $\langle M^{2}(t)\rangle$.
The formal expression of $\langle M^{2}(t)\rangle$ reads

$\displaystyle\langle M^{2}(t)\rangle$
$\displaystyle=\sum_{n=1}^{\infty}\int_{0}^{\infty}dM~{}M^{2}~{}\partial_{M}Q(M,n|t),$
(65) $\displaystyle=\int_{\Gamma}\frac{ds}{2\pi
i}~{}e^{st}~{}\frac{1}{\gamma}\left(\frac{\gamma}{\gamma+s}\right)^{n}~{}\langle
M_{s}^{2}(n)\rangle$ (66)

where $\langle M_{s}^{2}(n)\rangle$ is

$\displaystyle\langle M_{s}^{2}(n)\rangle$
$\displaystyle=\int_{0}^{\infty}dM~{}M^{2}~{}\partial_{M}Q_{s}(M,n),~{}~{}~{}~{}~{}\text{with}$
(67) $\displaystyle Q_{s}(M,n)$
$\displaystyle=\int_{-\infty}^{\infty}dx_{1}...\int_{-\infty}^{\infty}dx_{n}~{}\Theta(M-X_{1})..\Theta(M-X_{n})~{}\left[\prod_{i=1}^{n}g_{s}(x_{i})\right].$
(68)

Here $\int_{-\infty}^{\infty}dx~{}g_{s}(x)=1$ which can be verified easily
from Eq. (62). Hence $Q_{s}(M,n)$ can be deciphered as the cumulative
distribution that the maximum is less than $M$ up to $n$ steps for an
auxilliary $1d$ random walk with identical and independent jumps which follow
symmetric and continuous distribution $g_{s}(x)$ given in Eq. (62).
Consequently, $\langle M_{s}^{2}(n)\rangle$ is the second moment of the
maximum $M_{s}(n)$ of the auxillary random walk which can be calculated using
the Pollaczek Spitzer formulae in Eqs. (29) and (30) as done for fixed-$n$
ensemble. To avoid repetition, we present the details of this calculation in C
and write only the final expression of $\langle M_{s}^{2}(n)\rangle$ here
which reads

$\displaystyle\langle
M_{s}^{2}(n)\rangle=\frac{v_{0}^{2}}{2(\gamma+s)^{2}}\left(\frac{\mathcal{S}_{n}}{\pi}+n\right),$
(69)

where $\mathcal{S}_{n}$ is given in Eq. (6). Substituting $\langle
M_{s}^{2}(n)\rangle$ in the expression of $\langle M^{2}(t)\rangle$ in Eq.
(66) and performing the inverse Laplace transformation gives

$\displaystyle\langle
M^{2}(t)\rangle=\frac{v_{0}^{2}}{2\gamma^{2}}\left[e^{-\gamma t}-1+\gamma
t+\frac{e^{-\gamma
t}}{\pi}\sum_{n=1}^{\infty}\frac{\mathcal{S}_{n}}{\Gamma(n+2)}(\gamma
t)^{n+1}\right].$ (70)

###  5.2 Computation of $\langle Y(t_{m})^{2}\rangle(t)$

We next calculate $\langle Y(t_{m})^{2}\rangle(t)$ for the mean area $\langle
A(t)\rangle$ in Eq. (22). Recall that $Y(t_{m}|t)$ is the $y$-coordinate of
the RTP at time $t_{m}$ when the maximum $M(t)$ of the $x$-coordinate is
attained in a trajectory of duration $t$. To calculate $\langle
Y(t_{m})^{2}\rangle(t)$, we first notice that for a trajectory of duration $t$
in the fixed-$t$ ensemble also the maximum in the $x$-direction occurs at the
end of some complete jump step, say $k^{*}$ which is a function of the total
number of jumps $n$ occurring in time $t$. Of course the number of jumps $n$
is a random quantity and consequently so is $k^{*}$ as they change from
realisation to realisation and also they are functions of $t$. Hence denoting
the time at the end of step $k^{*}$ by $t_{m}$, we can write
$Y(t_{m})=\sum_{i=1}^{k^{*}}y_{i}$.

We start with the grand joint distribution $P(\\{x_{i}\\},\\{y_{i}\\},n|t)$
given in Eq. (59). As we have mentioned earlier, the term
$\prod_{i=1}^{n}g_{s}(x_{i},y_{i})$ inside the square bracket on the right
hand side of this equation can be interpreted as the joint probability
distribution of the jumps $x_{i}$ and $y_{i}$ for $i=1,2,...,n$ of a random
walk in two dimension of $n$ steps. Once again we emphasise that $g_{s}(x,y)$,
given explicitly in Eq. (58), can be interpreted as an effective joint
distribution of elementary jumps along $x$ and $y$ directions, similar to
$p(x,y)$ as considered earlier in sec. 4 except now it is parametrised by $s$.
As a result we see that for a given trajectory of duration $t$ containing $n$
jump steps there is a trajectory of $n$ jumps generated by the joint
distribution $\prod_{i=1}^{n}g_{s}(x_{i},y_{i})$. Hence, if the maximum
displacement in the $x$-direction occurs at step $k^{*}$ of a trajectory of
duration $t$ containing $n$ jump steps, then in the auxiliary random walk
problem generated by $g_{s}(x,y)$ the maximum displacement along $x$-direction
occurs at the same step $k^{*}$. Moreover the displacements $X_{i}$ and
$Y_{i}$ (starting from the origin) along $x$ and $y$-directions at $i^{\rm
th}$ step are exactly same for $i=1,2,...,n$. Hence, we have

$\langle Y(t_{m})^{2}\rangle(t)=\int_{\Gamma}\frac{ds}{2\pi
i}~{}e^{st}~{}\frac{1}{\gamma}~{}\sum_{n=1}^{\infty}\left(\frac{\gamma}{\gamma+s}\right)^{n}~{}\langle
Y_{k^{*}}^{2}\rangle_{s}(n),$ (71)

where $\langle Y_{k^{*}}^{2}\rangle_{s}(n)$ should be computed following the
procedure given in sec. 4.2 with only difference being the joint distribution
$p(x,y)$ is replaced by $g_{s}(x,y)$ which is given in Eq. (58). That is why
we now have a subscript $s$ in the notation of $\langle
Y_{k^{*}}^{2}\rangle_{s}(n)$. Executing the computation steps from Eqs. (46) -
(48) with $g_{s}(x,y)$ we get

$\displaystyle\langle
Y_{k^{*}}^{2}\rangle_{s}(n)=\frac{v_{0}^{2}}{(\gamma+s)^{2}}\frac{n}{2},$ (72)

where we have used $\langle
y^{2}\rangle_{g_{s}}=\int_{-\infty}^{\infty}dx\int_{-\infty}^{\infty}dy~{}y^{2}~{}g_{s}(x,y)=\frac{v_{0}^{2}}{(\gamma+s)^{2}}$.
Inserting the above expression from Eq. (72) in Eq. (71) and carrying out the
sum over $n$ we get

$\displaystyle\langle Y(t_{m})^{2}\rangle(t)$
$\displaystyle=\frac{v_{0}^{2}}{2}\int_{\Gamma}\frac{ds}{2\pi
i}~{}\frac{e^{st}}{s^{2}(\gamma+s)}.$ (73)

which upon performing inverse Laplace transformation with respect to $s$ gives
the final expression

$\displaystyle\langle Y(t_{m})^{2}\rangle(t)$
$\displaystyle=\frac{v_{0}^{2}}{2\gamma^{2}}\left(\gamma t-1+e^{-\gamma
t}\right).$ (74)

###  5.3 Mean area for fixed-$t$ ensemble

The expressions of $\langle M^{2}(t)\rangle$ and $\langle
Y(t_{m})^{2}\rangle(t)$ in Eqs. (70) and (74) respectively guide us to write
the mean area $\langle A(t)\rangle$ via Eq. (22). Inserting these forms
explicitly, it is straightforward to show that $\langle A(t)\rangle$ indeed
possesses the scaling form of Eq. (8) with the scaling function
$\mathcal{J}(w)$ given in Eq. (9). In Figure 2 (right panel), we have plotted
$\langle A(t)\rangle$ and compared against the numerical simulations. We
observe excellent agreement. Here also, the convex hull is constructed
numerically by using Andrew’s monotone chain algorithm and we have used Eq.
(52) to calculate the area.

To get the Brownian limit of the expression of $\langle A(t)\rangle$ in Eq.
(8), we look at the asymptotic behaviours of the scaling function
$\mathcal{J}(w)$ which read

$\displaystyle\mathcal{J}(w)$
$\displaystyle\simeq\frac{w^{3}}{3\pi}+\frac{w^{4}}{4\pi}\left(\frac{\pi}{6}-1\right),~{}~{}~{}~{}~{}~{}~{}\text{as
}w\to 0,$ (75) $\displaystyle\simeq\pi
w,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{as
}w\to\infty.$ (76)

Inserting these forms in Eq. (8), we find that $\langle A(t)\rangle$ exhibits
crossover from $\sim t^{3}$ scaling for $t\ll\gamma^{-1}$ to $\sim t$ scaling
for $t\gg\gamma^{-1}$:

$\displaystyle\langle A(t)\rangle$ $\displaystyle\simeq\frac{\gamma
v_{0}^{2}t^{3}}{6\pi},~{}~{}~{}~{}~{}\text{for }t\ll\gamma^{-1}$ (77)
$\displaystyle\simeq\frac{\pi v_{0}^{2}}{2\gamma}t,~{}~{}~{}~{}~{}~{}\text{for
}t\gg\gamma^{-1}.$ (78)

For $t\gg\gamma^{-1}$, we recover the result for Brownian motion with
effective diffusion constant $D=v_{0}^{2}/2\gamma$. However, at small times,
the behaviour is remarkably different than that of the Brownian motion as
illustrated by the $\sim t^{3}$ growth in Eq. (77). This qubic growth can be
easily understood by noting that at small times the RTP experiences only few
tumbles. The minimum number of tumbling required to constract a convex hull is
two tumbling events till time $t$ (counting the starting point as a tumble).
Then, the convex hull is essentially a triangle with two sides of length
$v_{0}\tau$ and $v_{0}(t-\tau)$ and some angle $\zeta$ between them. The area
is given by $A(t)=|\frac{v_{0}^{2}\tau(t-\tau)\sin\zeta}{2}|$. To calculate
mean, we recall that $\tau$ is drawn from exponential distribution
$p(\tau)=\gamma e^{-\gamma\tau}$ and $\zeta$ is chosen uniformly from
$[0,2\pi]$. It is then easy to show that the resulting mean exactly matches
with the short time asymptotics in Eq. (77). Although at large times, $\langle
A(t)\rangle$ behaves identical to that of the Brownian motion, the short time
behaviour is rather different. Another way to demonstrate this difference is
to define

$\displaystyle\alpha(t)=\frac{\langle A(t)\rangle}{\gamma t\sigma^{2}}.$ (79)

For $t\to\infty$, $\alpha(t)$ saturates to the value $\frac{\pi}{2}$. In
Figure 3 (right panel), we have plotted $\alpha(t)$ for two different values
of $\gamma$ and also compared them against the numerical simulations. We see
agreement of the numerical data to the analytic expressions for both cases.
Also, we obtain that $\alpha(t)$ approaches the value $\frac{\pi}{2}$ in both
cases.

Figure 4: Left: Simulation results for the distribution of the area $A_{n}$
for fixed-$n$ ensemble for different values of $n$. We have rescaled the
distribution with mean area $\langle A_{n}\rangle$ and compared it with that
of the Brownian motion. We have chosen $v_{0}=1$ and $\gamma=1$ for all values
of $n$. Right: The same analysis is conducted for the fixed-$t$ ensemble with
same choice of parameters.

##  6 Numerical study of the probability distribution

In the previous sections, we explicitly derived the exact analytic expressions
of the mean area of the convex hull in fixed-$n$ and fixed-$t$ ensembles and
compared them against the numerical simulations. These expressions are given
respectively in Eqs. (5) and (8). We now investigate the probability
distribution of the area of the convex hull for a single RTP. Deriving
analytic forms of the distribution seems a difficult problem. In view of this,
we perform a rigorous numerical study for the distribution in the two
statistical ensembles. Here, we only look at the distribution corresponding to
the typical fluctuations in area. By this, we mean the parts of distribution
that lie within few standard deviations around the mean. To compare the
distribution for different values of $n$ or $t$, it turns out useful to
rescale it with the mean area. In Figure 4, we have illustrated the simulation
data for the rescaled distribution for different values of $n$ and $t$. For
both ensembles, we find that the distribution converges to that of the
Brownian motion in the asymptotic regime, i.e. $n\gg 1$ for fixed-$n$ ensemble
and $t\gg\gamma^{-1}$ for fixed-$t$ ensemble. However, for other (small and
intermediate) values of $n$ and $t$, we expectedly see clear departure from
the Brownian motion as elucidated by blue symbols in both panels of Figure 4.
To construct the distributions in Figure 4, we have adopted the simple
sampling techniques where we take a realisation of RTP depending on the
ensemble that we are interested in. Given this trajectory, we construct the
convex hull using Andrew’s monotone chain algorithm along with Akl’s heuristic
and use Eq. (52) to calculate the area. This procedure is then repeated for
$10^{5}$ realisations to finally construct the histogram.

Similarly, we have also studied the variance of the area in Figure 5 for two
ensembles. As done for the mean area in Eqs. (53) and (79), we define the
following two quantities:

$\displaystyle\beta_{n}$
$\displaystyle=\frac{\text{Var}(A_{n})}{n^{2}\sigma^{4}},~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{for
fixed-}n$ (80) $\displaystyle\beta(t)$
$\displaystyle=\frac{\text{Var}(A(t))}{(\gamma
t)^{2}\sigma^{4}}.~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{for fixed-}t$ (81)

As seen before, this rescaling of the variance helps in better visualisation
of the data since all of them converge to the same value in the asymptotic
regime for both ensembles and for different values of the parameters. For both
ensembles, we see in Figure 5 that $\beta_{n}$ and $\beta(t)$ tend towards the
same value for different values of $\gamma$.

Figure 5: Simulation data for the variance of area for fixed-$n$ (left) and
fixed-$t$ (right) ensembles. For fixed-$n$ ensemble (left), we have plotted
$\beta_{n}=\text{Var}(A_{n})/n^{2}\sigma^{4}$ vs. $n$ and for fixed-$t$
ensemble (right), we have plotted $\beta(t)=\text{Var}(A(t))/(\gamma
t)^{2}\sigma^{4}$ vs. $t$. Parameters used in these plots are $v_{0}=1$,
$\gamma=1$ (green) and $\gamma=0.5$ (red) for both panels.

##  7 Conclusion

We have investigated the area of the convex hull of a run-and-tumble particle
in two dimensions. We have considered this problem in two different ensembles:
(i) fixed-$n$ ensemble and (ii) fixed-$t$ ensemble. We have obtained explicit
expressions of the mean area $\langle A_{n}\rangle$ and $\langle A(t)\rangle$
in these two ensembles and verified them numerically. To study mean area
analytically, we have used a mapping of the run-and-tumble motion to a random
walk model in two dimensions similar to what was used previously in [35].
After exploiting the connection between the extreme value statistics and the
computation of the mean area through Cauchy formulae [Eqs. (21) and (22)], we
use this mapping to employ the Sparre Anderson theorem which finally lead us
arrive at the explicit expressions of the mean area in Eqs. (5) to (9). We
observed that at large times the mean area grows linearly whereas at small
times it grows as $\sim t^{3}$ with $t$. We have obtained a scaling function
that describes the crossover from the qubic growth to linear growth around the
natural time scale $\gamma^{-1}$ provided by the tumbling rate.

Obtaining analytic results for higher order moments and distribution seems a
challenging task. We have numerically studied the variance of the mean area as
time ( number of tumbles in the fixed ensemble and $t$ in the fixed time
ensemble) and found that it grows quadratically with time at long times. We
have also studied the distribution of the area numerically. While in the
asymptotic regime, i.e. $n\gg 1$ for fixed-$n$ ensemble and $t\gg\gamma^{-1}$
for fixed-$t$ ensemble, the distribution converges to that of the Brownian
motion when area is scaled with its mean, we find clear difference at small or
intermediate regime.

As mentioned before, computing the higher order moments and the full
distribution of the area is a challenging problem and still remains an open
problem even for Brownian particle. It is worth emphasizing that we have
looked at the simple version of run and tumble model where tumbles are
instantaneous. However, it has been experimentally found that active particles
in reality spends small but non-zero time while tumbling [73, 19]. Extending
our results for these realistic systems remains a promising future direction.
Finally, in this work, we have focused on one run and tumble model active
particles. It would be interesting to explore how our results get generalised
for other models of active particles like active Brownian particle and active
Ornstein-Uhlenbeck particle [74, 3].

##  8 Acknowledgement

AK and PS acknowledge support of the Department of Atomic Energy, Government
of India, under project no.12-R&D-TFR-5.10-1100. AK acknowledges support from
DST, Government of India grant under project No. ECR/2017/000634. We thank B.
De Bruyne and F. Mori for useful discussions.

##  Appendix A Derivation of $\bar{\mathcal{P}}\left(\xi,k^{*}|n\right)$ in
Eq. (43)

In this appendix we derive the expression of
$\bar{\mathcal{P}}\left(\xi,k^{*}|n\right)$ given in Eq. (43). To begin with,
we start with the joint distribution $\mathscr{P}(M,Y,k^{*}|n)$ in Eq. (39).
Performing Fourier transform with respect to $Y$, one can write
$\bar{\mathcal{P}}\left(\xi,k^{*}|n\right)$ defined in Eq. (42) as

$\displaystyle\bar{\mathcal{P}}\left(\xi,k^{*}|n\right)$
$\displaystyle=\int_{0}^{\infty}dM~{}I_{\rm left}(M,\xi,k^{*})~{}I_{\rm
right}(M,k^{*},n),$ (82)

where we have defined

$\displaystyle I_{\rm
right}(M,k^{*},n)=\int\prod_{i=k^{*}+1}^{N}dy_{i}\,dx_{i}\,p(x_{i},y_{i})\,\Theta\left(M-\sum_{j=1}^{i}x_{j}\right),$
(83)

and

$\displaystyle I_{\rm
left}(M,\xi,k^{*})=\int\prod_{i=1}^{k^{*}}dx_{i}\,{\tilde{p}}(x_{i},\xi)\,\prod_{i=1}^{k^{*}-1}\Theta\left(M-\sum_{j=1}^{i}x_{j}\right)\delta\left(M-\sum_{j=1}^{k^{*}}x_{j}\right)\,,$
(84)

with the definition

${\tilde{p}}(x,\xi)=\int_{-\infty}^{\infty}dye^{i\xi y}p(x,y)\,.$ (85)

Let us first consider the integral $I_{\rm right}(M,k^{*},n)$ defined in Eq.
(83). Since,

$X_{k^{*}+j}=x_{1}+x_{2}+\ldots+x_{k^{*}}+x_{k^{*}+1}+\ldots+x_{k^{*}+j}=M+x_{k^{*}+1}+\ldots+x_{k^{*}+j}$
(86)

upon using $M=\sum_{i=1}^{k^{*}}x_{i}$, we can re-write Eq. (83) as

$I_{\rm
right}(M,k^{*},n)=\int\left[\prod_{i=k^{*}+1}^{n}dx_{i}p_{1}(x_{i})\right]\prod_{i=k^{*}+1}^{n}\Theta\left(-\sum_{j=k^{*}+1}^{i}x_{j}\right)\,$
(87)

where $p_{1}(x)=\int_{-\infty}^{\infty}p(x,y)\,dy$ is a normalized (to unity)
probability density function for the increment in the $x$-direction. However,
the integral in Eq. (87) is simply the probability that a random walk in one
dimension (in the $x$-direction) starting at the origin, with independent and
identically distributed increment $x_{i}$’s drawn from $p_{1}(x_{i})$, stays
below the origin up to step $n-k^{*}$. This is precisely given by
$q_{n-k^{*}}$ via the Sparre Andersen theorem (independently of the jump
distribution $p_{i}(x)$, where $q_{n}={2n\choose n}\,2^{-2n}$. Hence, we have

$I_{\rm right}(M,k^{*},n)=q_{n-k^{*}}\,.$ (88)

Note that the integral $I_{\rm right}(M,k^{*},n)$ does not depend on $M$, but
only on $(n-k^{*})$.

We now turn to the left integral $I_{\rm left}(M,\xi,k^{*})$ in Eq. (84). Let
us first re-write ${\tilde{p}}(x,\xi)$ in Eq. (85) in a different way. Let us
first consider the integral

$\int_{-\infty}^{\infty}dx\,{\tilde{p}}(x,\xi)=\int_{-\infty}^{\infty}dx\,\int_{-\infty}^{\infty}dy\,p(x,y)\,e^{i\xi
y}=\int_{-\infty}^{\infty}dy\,p_{2}(y)\,e^{i\xi y}={\tilde{p}}_{2}(\xi)\,,$
(89)

where $p_{2}(y)=\int_{-\infty}^{\infty}dx\,p(x,y)$ is the marginal
distribution for the $y$-increment. Now, let us re-write

${\tilde{p}}(x,\xi)=\frac{{\tilde{p}}(x,\xi)}{\int_{-\infty}^{\infty}dx\,{\tilde{p}}(x,\xi)}\times{\tilde{p}}_{2}(\xi)=f(x,\xi)\,{\tilde{p}}_{2}(\xi)$
(90)

where we used the identity in Eq. (89) and

$f(x,\xi)=\frac{{\tilde{p}}(x,\xi)}{\int_{-\infty}^{\infty}dx\,{\tilde{p}}(x,\xi)}\,.$
(91)

Note that $f(x,\xi)$ is normalized to unity (when integrated over $x$) and can
be thought of as an effective jump distribution in the $x$ direction that is
just parametrized by $\xi$ assuming it is positive for all $x$. We use this
expression of ${\tilde{p}}(x,\xi)$ from Eq. (90) into the integral expression
for $I_{\rm left}(M,\xi,k^{*})$ in Eq. (84), to get

$I_{\rm
left}(M,\xi,k^{*})=\left[{\tilde{p}}_{2}(\xi)\right]^{k^{*}}\,\int\prod_{i=1}^{k^{*}}dx_{i}\,f(x_{i},\xi)\,\prod_{i=1}^{k^{*}-1}\Theta\left(M-\sum_{j=1}^{i}x_{j}\right)\delta\left(M-\sum_{j=1}^{k^{*}}x_{j}\right).$
(92)

Now, substituting this expression and the result in Eq. (88) on the right hand
side of Eq. (82) and carrying out the integral over $M$ gives

$\displaystyle\begin{split}{\tilde{P}}(\xi,k^{*}|n)=q_{n-k^{*}}\,&\left[{\tilde{p}}_{2}(\xi)\right]^{k^{*}}\,\int\left[\prod_{i=1}^{k^{*}}dx_{i}\,f(x_{i},\xi)\right]\,\prod_{i=1}^{k^{*}}\Theta\left(\sum_{j=1}^{i}x_{k^{*}+1-j}\right)\end{split}$
(93)

However, we immediately identify the $k^{*}$-fold integral in Eq. (93) as the
probability that a one dimensional random walker, starting at the origin and
with jump distribution drawn from $f(x,\xi)$ (which is normalised to unity),
stays above the origin up to $k^{*}$ steps. By Sparre Andersen theorem, this
is universal and is simply $q_{k^{*}}$ and is independent of $f(x,\xi)$, and
in particular then does not depend on $\xi$. Hence we finally have

${\tilde{P}}(\xi,k^{*}|n)=\int_{\infty}^{\infty}P(Y,k^{*}|n)\,e^{i\xi
Y}\,dY=q_{k^{*}}\,q_{n-k^{*}}\,{\left[{\tilde{p}}_{2}(\xi)\right]}^{k^{*}}\,,$
(94)

which upon Fourier inversion, yields the result in Eq. (38). This result is
true for arbitrary joint distribution $p(x,y)$ as long as it is symmetric and
continuous in $x$.

##  Appendix B Proof $\mathcal{S}_{n}\simeq\pi n$ as $n\to\infty$

In this appendix, we derive the asymptotic form of $\mathcal{S}_{n}$ for large
$n$ which was used to obtain the large $n$ behaviour of $\langle A_{n}\rangle$
in Eq. (51). To this end, we consider the expression of $\mathcal{S}_{n}$ in
Eq. (35) and change the variable $m=zn$ to yield

$\displaystyle\mathcal{S}_{n}$
$\displaystyle=\frac{\sqrt{\pi}}{\sigma}\sum_{z=1/n}^{(n-1)/n}~{}~{}\frac{\Gamma\left(\frac{n(1-z)+1}{2}\right)}{\Gamma\left(\frac{n(1-z)+2}{2}\right)}~{}\langle
M_{nz}\rangle.$ (95)

Note that $z\in\\{\frac{1}{n},\frac{2}{n},...,\frac{n-1}{n}\\}$. For large
$n$, we change the summation in Eq. (95) to integration as
$\sum_{z=1/n}^{(n-1)/n}\to n\int_{0}^{1}dz$ and rewrite it as

$\displaystyle\mathcal{S}_{n}$
$\displaystyle\simeq\frac{\sqrt{\pi}}{\sigma}n\int_{0}^{1}dz\frac{\Gamma\left(\frac{n(1-z)+1}{2}\right)}{\Gamma\left(\frac{n(1-z)+2}{2}\right)}~{}\langle
M_{nz}\rangle,~{}~{}~{}~{}\text{as }n\to\infty.$ (96)

We next use the result of [65] to write $\langle M_{nz}\rangle$ for large $n$
as $\langle M_{nz}\rangle\simeq\sigma\sqrt{\frac{2zn}{\pi}}$. In addition, we
approximate
$\frac{\Gamma\left(\frac{n(1-z)+1}{2}\right)}{\Gamma\left(\frac{n(1-z)+2}{2}\right)}\simeq\sqrt{\frac{2}{n(1-z)}}$
as $n\to\infty$. Inserting these forms in Eq. (96) and performing the
integration over $z$, we get

$\displaystyle\mathcal{S}_{n}\simeq\pi n,~{}~{}~{}~{}~{}~{}~{}\text{as
}n\to\infty.$ (97)

##  Appendix C Derivation of $\langle M_{s}^{2}(n)\rangle$ in Eq. (69)

Here, we show that the expression of $\langle M_{s}^{2}(n)\rangle$ in Eq. (69)
can be derived using the Pollaczek-Spitzer formula in Eq. (26). We first
recall that $Q_{s}(M,n)$ in Eq.(67) represents the cumulative distribution
that the maximum is less than $M$ up to $n$ steps for a random walker with
independent and identically distributed increments $\\{x_{i}\\}$ chosen from
the symmetric and continous distribution $g_{s}(x_{i})$ in Eq. (62). For this,
the Pollaczek-Spitzer formula gives [67, 46]

$\displaystyle\sum_{n=0}^{\infty}z^{n}\langle e^{-\lambda
M_{s}(n)}\rangle=\sum_{n=0}^{\infty}z^{n}\int_{0}^{\infty}dMe^{-\lambda
M}Q_{s}^{\prime}(M,n)=\frac{\phi_{s}(z,\lambda)}{\sqrt{1-z}},$ (98)

where $0\leq z\leq 1$ and $\lambda\geq 0$ and the function
$\phi_{s}(z,\lambda)$ is defined as

$\displaystyle\phi_{s}(z,\lambda)=\text{exp}\left(-\frac{\lambda}{\pi}\int_{0}^{\infty}d\xi~{}\frac{\ln(1-z\hat{p}_{s}(\xi))}{\lambda^{2}+\xi^{2}}\right),~{}\text{
with}$ (99) $\displaystyle\hat{p}_{s}(\xi)=\int_{-\infty}^{\infty}dx~{}e^{i\xi
x}g_{s}(x)~{}=\frac{1}{\sqrt{1+\xi^{2}\sigma_{s}^{2}}}.$ (100)

Here $\sigma_{s}=v_{0}/(\gamma+s)$. As seen for fixed-$n$ ensemble in Eqs.
(29) and (30), one can extend this formula to determine the generating
function for the moments [69]. For this case, one gets

$\displaystyle h^{(1)}_{s}(z)=\sum_{n=0}^{\infty}z^{n}\langle
M_{s}(n)\rangle=\frac{1}{\pi(1-z)}\int_{0}^{\infty}\frac{d\xi}{\xi^{2}}~{}\ln\left(\frac{1-z\hat{p}_{s}(\xi)}{1-z}\right),$
(101) $\displaystyle h^{(2)}_{s}(z)=\sum_{n=0}^{\infty}z^{n}\langle
M_{s}^{2}(n)\rangle=(1-z)\left[h^{(1)}_{s}(z)\right]^{2}+\frac{\sigma_{s}^{2}~{}z}{2(1-z)^{2}}.$
(102)

Taking derivative of $h^{(2)}_{s}(z)$ $n$-times, we get

$\displaystyle\langle M_{s}^{2}(n)\rangle=\sum_{m=1}^{n-1}\langle
M_{s}(n)\rangle\left[\langle M_{s}(n-m)\rangle-\langle
M_{s}(n-m-1)\rangle\right]+\frac{n\sigma_{s}^{2}}{2}.$ (103)

We next use the results of [65] to write $\langle M_{s}(n)\rangle$ as

$\displaystyle\langle
M_{s}(n)\rangle=\frac{\sigma_{s}}{2\sqrt{\pi}}\sum_{j=1}^{n}\frac{\Gamma\left(\frac{j+1}{2}\right)}{\Gamma\left(\frac{j+2}{2}\right)}.$
(104)

and using this, we get

$\displaystyle\langle M_{s}(n-m)\rangle-\langle
M_{s}(n-m-1)\rangle=\frac{\sigma_{s}}{2\sqrt{\pi}}~{}\frac{\Gamma\left(\frac{n-m+1}{2}\right)}{\Gamma\left(\frac{n-m+2}{2}\right)}.$
(105)

Finally, we insert Eqs. (104) and (105) in the expression of $\langle
M_{s}^{2}(n)\rangle$ in Eq. (103) and perform the sum over $m$ explicitly to
yield

$\displaystyle\langle
M_{s}^{2}(n)\rangle=\frac{\sigma_{s}^{2}}{2}\left(\frac{\mathcal{S}_{n}}{\pi}+n\right),$
(106)

where $\mathcal{S}_{n}$ is given in Eq. (6). Identifying
$\sigma_{s}=v_{0}/(\gamma+s)$, we recover the result in Eq. (69).

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