# Boosting macroscopic diffusion with local resetting

Henry Alston  Department of Mathematics, Imperial College London, South
Kensington, London SW7 2AZ, United Kingdom  Thibault Bertrand
<EMAIL_ADDRESS>Department of Mathematics, Imperial College London,
South Kensington, London SW7 2AZ, United Kingdom

###### Abstract

Stochastic interactions generically enhance self-diffusivity in living and
biological systems, e.g. optimizing navigation strategies and controlling
material properties of cellular tissues and bacterial aggregates. Despite
this, the physical mechanisms underlying this nonequilibrium behavior are
poorly understood. Here, we introduce a model of interactions between an agent
and its environment in the form of a local stochastic resetting mechanism, in
which the agent’s position is set to the nearest of a predetermined array of
sites with a fixed rate. We derive analytic results for the self-diffusion
coefficient, showing explicitly that this mechanism enhances diffusivity.
Strikingly, we show analytically that this enhancement is optimized by regular
arrays of resetting sites. Altogether, our results ultimately provide the
conditions for the optimization of the macroscopic transport properties of
diffusive systems with local random binding interactions.

Dynamic interactions drive biological systems away from thermodynamic
equilibrium Alston  _et al._ (2022a). Pili-mediated forces between bacteria
such as Neisseria Meningitidis Kuan  _et al._ (2021); Pönisch and Zaburdaev
(2022), where interactions are inherently stochastic due to the dynamic
binding-unbinding of type-IV pili Bonazzi  _et al._ (2018), lead to the
formation of nonequilibrium structures Alston  _et al._ (2022b). Biological
agents with ligand-receptor contacts Bressloff and Newby (2013); Marbach  _et
al._ (2022), such as cells Ley  _et al._ (2007), protein cargos Allen  _et
al._ (2000) and viruses Müller  _et al._ (2019), employ similar cycles of
attachment-detachment with their local environment to drive transport and
self-assembly at much smaller lengthscales.

Diffusive motion interposed by random binding interactions to a confining
surface can be studied in the wider context of so-called stochastic resetting,
a paradigmatic model in nonequilibrium statistical physics Evans and Majumdar
(2011a); Evans  _et al._ (2020). Resetting, where the particle position is
moved instantaneously (“reset”) to one of a predetermined set of sites,
generically leads to a nonequilibrium steady-state Evans and Majumdar (2011b);
Eule and Metzger (2016); Méndez and Campos (2016); Bello  _et al._ (2023) and
finds applications in a broad range of simple yet biological relevant
processes Boyer and Solis-Salas (2014); Genthon  _et al._ (2022); da Silva and
Fragoso (2021); Roldán  _et al._ (2016); Reuveni  _et al._ (2014); Basu  _et
al._ (2019); Karthika and Nagar (2020). One striking example is the problem of
optimizing target search strategies Condamin  _et al._ (2007); Bénichou  _et
al._ (2005); Majumdar  _et al._ (2021); De Bruyne  _et al._ (2022) where
resetting events can lower the mean first-passage time to reach a target
compared to simple diffusion across a range of lengthscales Redner (2001);
Janson and Peres (2012); Reuveni (2016); Pal and Reuveni (2017); Evans  _et
al._ (2013); Bray  _et al._ (2013); Faisant  _et al._ (2021); Besga  _et al._
(2020); Bénichou  _et al._ (2009, 2011), from proteins binding to DNA to
foraging animals.

Stochastic interactions generically enhance self-diffusion in living matter.
Pili-mediated interactions increase diffusivity at the surface of bacterial
colonies Bonazzi  _et al._ (2018), imparting the aggregates with liquid-like
properties Bonazzi  _et al._ (2018); Kuan  _et al._ (2021); similarly,
dynamically fluctuating cell-cell adhesion strengths lead to the fluidization
of embryonic tissues Mongera  _et al._ (2018). Further, type-IV pili-mediated
stochastic interactions can also drive persistent motion on surfaces through
twitching motility Merz  _et al._ (2000); Marathe  _et al._ (2014); Simsek
_et al._ (2019). Binding-unbinding interactions between cargo and crosslinks
in biopolymer networks, such as the extracellular matrix, can significantly
increase the cargo’s diffusivity by overcoming local trapping effects Goodrich
_et al._ (2018). Similar enhancement has been observed when a particle is
confined in a channel with polymer chains grafted to each side: activity-
driven attachment-detachment cycles can generate an increased effective
diffusion of the target particle Lalitha Sridhar  _et al._ (2021). Finally,
enzyme-substrate chemical reactions have been argued to enhance the effective
diffusivity of enzymes in solution Rezaei-Ghaleh  _et al._ (2022); Golestanian
(2015); Agudo-Canalejo  _et al._ (2018a, b); Jee  _et al._ (2020).

Though driven by diverse underlying processes, minimal physical models of
stochastic interactions can establish generic constraints for realizing high-
diffusivity behavior. In this Letter, we introduce a local (nearest-neighbor)
resetting mechanism, where the position of a diffusive particle is
stochastically reset to the nearest of a predetermined set of resetting sites.
We propose this mechanism as a minimal implementation of local stochastic
interactions such as attachment-cycles to fixed binding sites or intermittent
attractive forces with static targets or neighboring agents. We confirm that
resetting reduces the mean first-passage time to reach adjacent sites, then
analytically exhibit an enhancement of the macroscopic self-diffusion
coefficient of the tagged particle, offering a perspective that is generally
overlooked for stochastic resetting models. Remarkably, we show explicitly
that this enhancement is maximised by a regular array of resetting sites.

Nearest-neighbor resetting mechanism. — Consider $N$ resetting sites
$\\{x_{1},\dots x_{N}\\}$ on the 1D domain $[0,L)$ with periodic boundary
conditions, where without loss of generality, we assume that $x_{1}=0$. We
will denote $\rho=N/L$ the density of resetting targets. We then consider the
dynamics of a Brownian particle, whose motion is entirely characterized by the
diffusion coefficient $D$, subjected to resetting events occuring with
Poissonian rate $k_{r}$. At these events, the particle is moved to the nearest
resetting site as shown in Fig. 1.

In Note (1), we further motivate the instantaneous nature of the resetting
mechanism through a formal separation of the diffusion- and interaction-
timescales. While in what follows, we study analytically the case of
instantaneous resetting events, we also probe the parameter space beyond this
timescale separation (e.g. when resetting events have finite duration) and
confirm that our results stand for a range of physically realizable parameters
(see 111See Supplemental Material at [] for further analytical and
computational details, which includes Refs. [] for an extended discussion).

Figure 1: Diffusion with nearest-neighbor resetting — (a) Minimal model of
diffusive agent subjected to stochastic binding and unbinding to its confining
environment at static sites (e.g. modelling ligand-receptor contacts). (b)
Example trajectory of a diffusive particle subjected to local resetting
events. At each event, which are separated by exponentially distributed
waiting times, the particle is moved instantaneously, reset, to its nearest
resetting site.  Figure 2: Dynamics of nearest-neighbor resetting on the unit
interval — (a) We consider three adjacent resetting sites placed at
$\\{0,y,1\\}$, respectively. Initializing the particle at site $y$, we measure
the first-passage time to reach either of the two adjacent sites, via (i) pure
diffusion or (ii) a resetting event. (b) We compare the mean first-passage
time for the resetting process against that of pure diffusion. (c) Comparing
the two mean first passage times for different locations of the $y$ site. For
small $\alpha=\sqrt{k_{r}\ell^{2}/4D}$, the resetting mechanism always wins,
but this disappears at large $\alpha$ with the crossover value depending on
$y$ as shown in (b). (d) The splitting probability to hit the $x=0$ boundary
before the $x=1$ boundary. In (c) and (d), our results are compared to
numerical simulations of the microscopic process (see details in Note (1)).

Effective diffusion coefficient. — To assess the role of the resetting
mechanism on the particle’s dynamics, we study its long-time effective
diffusion coefficent. To do so, we formally map our model onto a continuous-
time random walk problem. Without loss of generality, we can consider that at
time $t$ our particle is located at resetting site $x_{n}$. Working on the
interval $[x_{n-1},x_{n+1}]$, we need to study the statistics of the jumps
$x_{n}\rightarrow x_{n+1}$ and $x_{n}\rightarrow x_{n-1}$, i.e. an escape from
site $n$ to one of its nearest-neighbor resetting sites. This can happen
either by diffusion and resetting or by diffusion alone.

For each site $x_{n}$, we write the increase in the variance of the particle
position due to the jump to one of the neighboring sites as

$\displaystyle\langle(\Delta x)^{2}\rangle_{n}$
$\displaystyle=(x_{n+1}-x_{n})^{2}\mathbb{P}(n\rightarrow n+1)$
$\displaystyle\quad+(x_{n}-x_{n-1})^{2}\mathbb{P}(n\rightarrow n-1).$ (1)

where $\mathbb{P}(n\rightarrow n\pm 1)$ are the splitting probabilities
corresponding to the two possible events (reaching either end of the
interval).

Due to the Markovian nature of the process, we can write the mean-squared
displacement for this continuous-time random walk Hughes (1995) as

$\langle(\Delta x)^{2}\rangle(t)=\sum_{n=1}^{N}\Lambda_{n}(t)\langle(\Delta
x)^{2}\rangle_{n}$ (2)

where $\Lambda_{n}$ is the number of times that site $n$ has been escaped from
up to time $t$. The effective diffusion coefficient can then be derived as

$D_{\rm eff}=\lim_{t\to\infty}\frac{\langle(\Delta
x)^{2}\rangle}{2t}=\sum_{n=1}^{N}\pi_{n}\frac{\langle(\Delta
x)^{2}\rangle_{n}}{2\langle\tau\rangle_{n}}$ (3)

where $\pi_{n}$ is the fraction of the total trajectory time spent escaping
from each site:
$\pi_{n}=\lim_{t\to\infty}{\Lambda_{n}(t)\langle\tau\rangle_{n}}/{t}$ and
$\langle\tau\rangle_{n}$ is the (unconditional) mean first-passage time (MFPT)
to reach either of the nearest neighbors of site $n$. Eq. (3) can also be read
as a time average of the local diffusion coefficients $D_{\rm
eff}^{(n)}={\langle(\Delta x)^{2}\rangle_{n}}/2{\langle\tau\rangle_{n}}$. To
make further analytical progress, we thus need to consider the local dynamics,
namely the MFPT $\langle\tau\rangle_{n}$ and the splitting probabilities
$\mathbb{P}(n\rightarrow n\pm 1)$.

First-passage time to adjacent sites. — We will first study the first-passage
time distribution from site $x_{n}$ to adjacent resetting sites
$\\{x_{n-1},x_{n+1}\\}$. Without loss of generality, we can translate this
interval so that $x_{n}=0$, denoting $x_{n-1}=-a<0$ and $x_{n+1}=b>0$,
respectively. We define $Q(t)$ as the probability that a particle has not
reached either site before time $t$. We write this survival probability as a
sum over the different realizations of the resetting process:

$Q(t)=\sum_{m=0}^{\infty}\mathcal{P}(m,t)Q_{m}(t)$ (4)

where $\mathcal{P}(m,t)$ is the probability of observing exactly $m$ resetting
events before time $t$ given by the Poisson distribution

$\mathcal{P}(m,t)=\frac{(k_{r}t)^{m}e^{-k_{r}t}}{m!}$ (5)

and $Q_{m}(t)$ is the probability of a particle surviving to time $t$ given
that there have been $m$ resetting events, all of which have reset the
particle to $x_{n}$ rather than its adjacent sites.

For a particle to survive to time $t$, its position must satisfy
$x(t^{\prime})\in(-a,b)$ for $t^{\prime}<t$, plus
$x(\tau_{m^{\prime}})\in(-a/2,b/2)$ at all resetting times $\tau_{m^{\prime}}$
to ensure resetting to the site $x_{n}=0$. If we suppose that there were $m$
resetting events before time $t$ occuring at $\\{\tau_{1},\dots,\tau_{m}\\}$
(where $\\{\tau_{m^{\prime}}\\}$ is a strictly increasing sequence), the
conditional survival probability is

$\displaystyle Q_{m}(t;\\{\tau_{m^{\prime}}\\})=P_{s}(\tau_{1})$
$\displaystyle P_{s}(\tau_{2}-\tau_{1})\cdots$ $\displaystyle
P_{s}(\tau_{m}-\tau_{m-1})Q_{0}(t-\tau_{m})$ (6)

where we have defined $P_{s}(\tau)$, the probability that the particle
survives a resetting event at time $\tau$ given that it was initially at $x=0$
and $Q_{0}(\tau)$, the probability that a diffusive particle initially at
$x=0$ does not escape the interval $(-a,b)$ by a time $\tau$.

These probabilities can be evaluated using the solution $u(x,t)$ to the
diffusion equation in one spatial dimension, with absorbing boundary
conditions $u(-a,t)=u(b,t)=0$ and initial condition $u(x,t)=\delta(x).$ The
probability distributions then required for evaluating Eq. (6) can then be
defined through

$P_{s}(\tau)=\int_{-a/2}^{b/2}dx\,u(x,\tau)~{}~{}\text{and}~{}~{}Q_{0}(\tau)\int_{-a}^{b}dx\,u(x,\tau),$
(7)

where $u(x,t)$ can be obtained through separation of variables Note (1).

Armed with these, we can express $Q_{m}(t)$ as an average over all possible
configurations of the resetting times sequence. For this, we argue that the
probability of realizing a given set of resetting times
$\\{\tau_{m^{\prime}}\\}$ given that $m$ resetting events occur in a time $t$
is exactly given by $m!/t^{m}$; indeed, it is the probability of sampling $m$
uniformly-distributed times between $0$ and $t$ in any order. It follows that

$Q_{m}(t)=\frac{m!}{t^{m}}\int_{0}^{\tau_{2}}d\tau_{1}\dots\int_{0}^{t}d\tau_{m}\>Q_{m}(t;\\{\tau_{m^{\prime}}\\}).$
(8)

Defining here the Laplace transform of the survival probability as
$\tilde{Q}(s)=\int_{0}^{\infty}dt\>e^{-st}Q(t)$, we can rewrite Eq. (4) in
Laplace space as

$\displaystyle\tilde{Q}(s)$
$\displaystyle=\sum_{m=0}^{\infty}(k_{r}\tilde{P}_{s}(s+k_{r}))^{m}\tilde{Q}_{0}(s+k_{r})$
(9)
$\displaystyle=\frac{\tilde{Q}_{0}(s+k_{r})}{1-k_{r}\tilde{P}_{s}(s+k_{r})}$
(10)

from which one can derive the unconditional mean first-passage time to reach
one of the nearest-neighbor resetting sites as

$\langle\tau\rangle=\tilde{Q}(s=0)=\frac{\tilde{Q}_{0}(k_{r})}{1-k_{r}\tilde{P}_{s}(k_{r})}.$
(11)

As shown in Note (1), we can find a general expression for the first-passage
time density in Laplace space using Eqs. (7). While not in general tractable,
this expression takes a particularly simple form in the case where $\ell\equiv
a=b$. In this case, the first-passage time density $\tilde{\tau}(s)\equiv
1-s\tilde{Q}(s)$ reads

$\displaystyle\tilde{\tau}(s)=\frac{s+k_{r}\cosh\bigg{(}\sqrt{\frac{(s+k_{r})l^{2}}{4D}}\bigg{)}}{k_{r}\cosh\bigg{(}\sqrt{\frac{(s+k_{r})l^{2}}{4D}}\bigg{)}+s\cosh\bigg{(}2\sqrt{\frac{(s+k_{r})l^{2}}{4D}}\bigg{)}},$
(12)

leading to the following succinct expression for the mean first-passage time

$\langle\tau\rangle=\frac{2}{k_{r}}\sinh(\alpha)\tanh(\alpha).$ (13)

where we have defined the dimensionless parameter
$\alpha=\sqrt{k_{r}\ell^{2}/4D}$. This can be understood as the ratio of the
two competing timescales in the problem: the time between two resetting
events, $1/k_{r}$, and the typical time to diffuse outside of the resetting
neighborhood of the initial site, which scales as
$(\ell/2)^{2}/D=1/(4\rho^{2}D)$.

Through a suitable re-scaling of the interval $(-a,b)$, we consider for
simplicity and without loss of generality the dynamics on the interval $(0,1)$
with a site at intermediate point $y=a/(a+b)$ [see Fig. 2(a)] and identify in
Fig. 2(b) that the resetting mechanism can lower the first passage time for a
wide range of parameter values but can be a hindrance when
$k_{r}\gg\rho^{2}D$, i.e. when the timescale between resetting events is very
short compared to the timescale for the particle to diffuse over length
$1/\rho$. We also note that the reduction in mean first-passage time compared
to pure diffusion is most notable when the initial site is near the
boundaries, as shown in Fig. 2(c).

Splitting probabilities to adjacent sites. — To evaluate the splitting
probabilities, we note that a particle starting in $x=0$ at time $t=0$ can
reach an adjacent sites by either diffusion or a resetting event. The
probability that the particle diffuses through one of the boundaries before a
resetting event, denoted $\mathbb{P}(0\leadsto s)$ with $s\in\\{-a,b\\}$, is
evaluated as the integrated-current through the boundary up until the initial
resetting time which is exponentially distributed leading to

$\displaystyle\mathbb{P}(0\leadsto-a)$
$\displaystyle=D\int_{0}^{\infty}dt\,k_{r}e^{-k_{r}t}\int_{0}^{t}d\tau\,\partial_{x}u\big{|}_{x=-a},$
(14a) $\displaystyle\mathbb{P}(0\leadsto b)$
$\displaystyle=-D\int_{0}^{\infty}dt\,k_{r}e^{-k_{r}t}\int_{0}^{t}d\tau\,\partial_{x}u\big{|}_{x=b}.$
(14b)

Next, the probability that the tagged particle is reset to one of the adjacent
sites, denoted $\mathbb{P}(0\mapsto s)$, is exactly

$\displaystyle\mathbb{P}(0\mapsto-a)$
$\displaystyle=\int_{0}^{\infty}dt\,k_{r}e^{-k_{r}t}\int_{-a}^{-a/2}dx\,u(x,t),$
(15a) $\displaystyle\mathbb{P}(0\mapsto b)$
$\displaystyle=\int_{0}^{\infty}dt\,k_{r}e^{-k_{r}t}\int_{b/2}^{b}dx\,u(x,t).$
(15b)

Finally, we can evaluate the splitting probabilities to reach each adjacent
site $s\in\\{-a,b\\}$ by either diffusion or resetting as

$\mathbb{P}(0\rightarrow s)=\frac{\mathbb{P}(0\leadsto s)+\mathbb{P}(0\mapsto
s)}{\sum_{r\in\\{-a,b\\}}\mathbb{P}(0\leadsto r)+\mathbb{P}(0\mapsto r)}.$
(16)

Given $u(x,t)$ the solution to the above one-dimensional diffusion problem, we
can evaluate these splitting probabilities Note (1) as shown in Fig. 2(d). At
small values of $\alpha$, the splitting probabilities are linear, as in the
case of pure diffusion. This disappears at large $\alpha$, where we observe
non-linear scaling and a strong preference to reach the nearest site.

Macroscopic diffusion coefficient for a regular array of resetting sites. —
Armed with these results, we evaluate the local diffusion coefficients
$D^{(n)}_{\rm eff}$ for all possible configurations of three adjacent sites.
As before, we consider the dynamics on the unit interval and use the fact that
the results can be extended to an arbitrary interval $\\{-a,0,b\\}$ after a
suitable re-scaling of space. As shown in Fig. 3(a), $D^{(n)}_{\rm eff}$ can
be evaluated numerically from the analytic expressions above for any values of
$\alpha$ and $y\in(0,1)$ to show the existence of a global maximum at
$\alpha\approx 1.606$ and $y=0.5$.

Figure 3: Effective diffusion coefficient on unit interval — (a) Local
effective diffusion coefficient constructed from the splitting probabilities
and unconditional mean first-passage time for the unit interval with resetting
sites at $x=\\{0,y,1\\}$, showing a boost in local diffusivity over a wide
range of parameters and a global maximum (yellow triangle). (b) Macroscopic
diffusivity as defined in Eq. 3 is maximized for equidistant adjacent
resetting sites (i.e for $y=0.5$) and $\alpha^{*}\approx 1.606$ where it takes
the value $D_{\rm eff}\approx 1.169D$.

We now turn to the macroscopic diffusion coefficient for a regular array of
resetting sites. Obtaining closed form analytic results for arbitrary arrays
of resetting sites requires the evaluation of the stationary probabilities
$\bm{\pi}=[\pi_{n}]$, which in general is not tractable analytically.
Nevertheless, for the case of a regular array of $N$ resetting sites (with
spacing $\ell=1/\rho$) we have $\pi_{n}\equiv 1/N$ and hence the large-scale
effective diffusion coefficient can be expressed as

$D_{\rm eff}/D=\alpha^{2}\operatorname{csch}(\alpha)\coth(\alpha),$ (17)

which we plot in Fig. 3(b).

Strikingly, this implies that the macroscopic diffusion coefficient can indeed
exceed the particle’s bare diffusion coefficient when $\alpha<2.676$ and is
maximised for $\alpha^{*}\approx 1.606$ for which $D_{\rm
eff}(\alpha^{*})\approx 1.169D$. This implies that this enhancement of
diffusive properties is a generic feature of this model, going beyond the
example of a regular array of resetting sites, provided the resetting occurs
sufficiently rarely (see Note (1) for an extended discussion). We conclude
that our local resetting mechanism can induce a striking enhancement of the
transport properties of a simple diffusive particle over macroscopic scales
which we capture in an analytically tractable model of stochastic
interactions.

Regular arrays of resetting sites optimize diffusion enhancement. — Finally,
we study the conditions for optimization of the effective macroscopic
diffusivity of the particle. Assuming that the resetting rate is an external
parameter which we can set to maximize the self-diffusivity, we can write the
optimal diffusion coefficient for an array of resetting sites
$\\{x_{n}\\}_{n\in[1,N]}$ as

$\max_{k_{r}}\big{(}D_{\rm
eff}/D)=\max_{k_{r}}\left(\sum_{n=1}^{N}\pi_{n}\frac{\langle(\Delta
x)^{2}\rangle_{n}}{2D\langle\tau\rangle_{n}}\right).$ (18)

The right-hand-side can be bounded from above trivially by the optimal value
for the local effective diffusion coefficient as

$\max_{k_{r}}\bigg{(}\sum_{n=1}^{N}\pi_{n}\frac{\langle(\Delta
x)^{2}\rangle_{n}}{2D\langle\tau\rangle_{n}}\bigg{)}\leq\max(D^{(n)}_{\rm{eff}}/D).$
(19)

However, the results of Fig. 3(a) imply that the maximum realizable value of
$D^{(n)}_{\rm eff}/D$ is exactly the maximum achieved for a regular array of
resetting sites leading to:

$\max_{k_{r}}(D_{\rm
eff}/D)\leq(\alpha^{*})^{2}\operatorname{csch}(\alpha^{*})\coth(\alpha^{*})\approx
1.169.$ (20)

In other words, we have showed analytically that the maximal value for the
effective macroscopic diffusivity of our process is obtained for regular
arrays of resetting sites, giving in turn an upper-bound for the macroscopic
effective diffusion coefficient over any array of resetting sites.

We confirm this result two ways. First, we measure the diffusion coefficient
for randomly generated arrays of resetting sites $\\{x_{n}\\}_{n\in[1,N]}$; to
do so, we numerically compute the stationary probability vector
$\bm{\pi}=[\pi_{n}]$ which allows us to calculate the maximum enhancement in
diffusivity achievable through Eq. (18). We confirm that a regular array
outperforms random arrays and show that increasing the variance in the local
density of resetting sites decreases the diffusion enhancement. Second, we
show that an optimization algorithm aiming at maximizing the effective
diffusion coefficient of the particle over the arrangement of resetting sites
starting from random configurations always converges to evenly spaced
resetting sites Note (1).

Discussion & outlook. — We have introduced a local resetting mechanism as a
minimal implementation of fluctuating interactions between a tagged particle
and its local environment. Strikingly, we show that this local mechanism can
generically enhance the macroscopic self-diffusion coefficient of the tagged
particle for generic arrangements of resetting sites. We further show that
this is maximized for regular arrays of evenly-spaced resetting sites. This
implies that physical implementations of attachment-detachment cycles between
a tagged particle and e.g. binding sites on a surface or static neighboring
agents are optimized for enhancing self-diffusion when the sites are placed in
a ordered manner, with minimal variance in the distances between neighboring
sites. Our formal treatment of interactions with individual binding sites goes
beyond previous coarse-grained approaches Marbach  _et al._ (2022) and
captures explicitly the impact of stochastic interactions on the effective
diffusion coefficient. It also sufficiently captures the enhancement in
diffusion observed in a range of biological settings Bonazzi  _et al._ (2018);
Mongera  _et al._ (2018); Goodrich  _et al._ (2018); Lalitha Sridhar  _et al._
(2021); Merz  _et al._ (2000); Marathe  _et al._ (2014); Simsek  _et al._
(2019).

###### Acknowledgements.

HA was supported by a Roth PhD scholarship funded by the Department of
Mathematics at Imperial College London.

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