Survival probability of stochastic processes beyond persistence exponents

TL;DR

This paper derives explicit formulas for the prefactor in the algebraic decay of survival probability for stochastic processes, linking it to mean first-passage times, and confirms the results with simulations, including for non-Markovian processes.

Contribution

It provides the first explicit expressions for the prefactor in survival probability decay, extending understanding to non-Markovian processes like Fractional Brownian Motion.

Findings

Analytical expressions for the prefactor $S_0$ are derived.

Results agree well with numerical simulations.

The approach applies to strongly correlated processes like Fractional Brownian Motion.

Abstract

For many stochastic processes, the probability $S(t)$ of not-having reached a target in unbounded space up to time $t$ follows a slow algebraic decay at long times, $S(t)\sim S_0/t^\theta$. This is typically the case of symmetric compact (i.e. recurrent) random walks. While the persistence exponent $\theta$ has been studied at length, the prefactor $S_0$, which is quantitatively essential, remains poorly characterized, especially for non-Markovian processes. Here we derive explicit expressions for $S_0$ for a compact random walk in unbounded space by establishing an analytic relation with the mean first-passage time of the same random walk in a large confining volume. Our analytical results for $S_0$ are in good agreement with numerical simulations, even for strongly correlated processes such as Fractional Brownian Motion, and thus provide a refined understanding of the statistics of longest first-passage events in unbounded space.

Figures

Equations

$$p(0,t|{\mathbf{r}}_0)={\int }_0^{t}d\tau F(\tau ;{\mathbf{r}}_0)p(0,t-\tau |0),$$

$$\tilde{p}(0,s|{\mathbf{r}}_0)=\tilde{F}(s;{\mathbf{r}}_0)\tilde{p}(0,s|0),$$

$$p(0,t|\mathbf{r})\underset{t\to \infty }{\sim }K/t^{d_f/d_w},$$

$$\frac{K\Gamma (1-\frac{d_f}{d_w})}{s^{1-d_f/d_w}}-\tilde{p}(0,s|\mathbf{r})\underset{s\to 0}{\sim }{\int }_0^{\infty }dt\left[\frac{K}{t^{d_f/d_w}}-p(0,t|\mathbf{r})\right],$$

$$1-\tilde{F}(s;{\mathbf{r}}_0)\underset{s\to 0}{\sim }{\int }_0^{\infty }dt\left[p(0,t|0)-p(0,t|{\mathbf{r}}_0)\right]\frac{s^{1-d_f/d_w}}{K\Gamma \left(1-\frac{d_f}{d_w}\right)}.$$

$$S_0=\frac{\sin (\pi d_f/d_w)}{K\pi }{\int }_0^{\infty }dt\left[p(0,t|0)-p(0,t|{\mathbf{r}}_0)\right].$$

$$\frac{⟨\mathbf{T}⟩}{V}\underset{V\to \infty }{\sim }{\int }_0^{\infty }dt\left[p(0,t|0)-p(0,t|{\mathbf{r}}_0)\right].$$

$$S_0=\frac{\sin (\pi d_f/d_w)}{\pi K}\overline{T},\text{with}\overline{T}=\underset{V\to \infty }{\lim }⟨\mathbf{T}⟩/V.$$

$$S_0\simeq \frac{\sin (\pi d_f/d_w)r_0^{d_w-d_f}}{\pi K}.$$

$$\overline{T}={\int }_0^{\infty }dt[p_{\pi }(0,t)-p(0,t)].$$

$$p_{\pi }(0,t)={\int }_0^{\infty }d\tau p(0,t+\tau |\mathrm{FPT}=\tau )F(\tau ;r_0),$$

$$p(0,t)={\int }_0^{t}d\tau F(\tau ;r_0)p(0,t|\mathrm{FPT}=\tau ).$$

$$\displaystyle\overline{T}=\int_{0}^{\infty}dt\,\Big{[}$$

$$\displaystyle-\int_{0}^{t}d\tau\,p(0,t\,|\,\text{FPT}=\tau)\,F(\tau;r_{0})\Big{]}.$$

$$\overline{T}=\underset{A\to \infty }{\lim }{\int }_0^{A}dt{\int }_{A-t}^{\infty }d\tau F(\tau ;r_0)p(0,t+\tau |\text{ FPT}=\tau ).$$

$$p(0,(u+v)A|\text{FPT}=vA)\underset{A\to \infty }{\sim }\frac{KG(u/v)}{(Au{)}^{1/d_w}},$$

$$\overline{T}=\underset{A\to \infty }{\lim }{A}^{1-\theta -1/d_w}K{\int }_0^{1}du{\int }_{1-u}^{\infty }dv\frac{\theta S_0}{v^{\theta +1}}\frac{G(u/v)}{u^{1/d_w}}.$$

$$S_0=\frac{\overline{T}}{K(1-1/d_w)}{\left({\int }_0^{1}du{\int }_{1-u}^{\infty }dv\frac{G(u/v)}{u^{1/d_w}{v}^{2-1/d_w}}\right)}^{-1}.$$

$$S_0\simeq \frac{\sin (\pi /d_w)}{K\pi }\overline{T},$$

$$S_0={\beta }_H\sin (\pi H)\sqrt{\frac{2}{\pi }}{\left(\frac{r_0}{{\kappa }^{1/2}}\right)}^{\frac{1}{H}-1}$$

Full text

Introduction

In order to determine the time it takes for a random walker to find a target, or the probability that a stochastic signal has not reached a threshold up to time $t$, it is required to analyse the First-Passage Time (FPT) statistics. This has attracted considerable attention from physicists and mathematicians in the last decades Redner:2001a ; Condamin2007 ; pal2017first ; grebenkov2016universal ; benichou2010optimal ; vaccario2015first notably because of the relevance of FPT related quantities in contexts as varied as diffusion controlled reactions, finance, search processes, or biophysics metzler2014first ; Berg1985 .

A single-target first-passage problem is entirely characterized by the so-called “survival probability” $S(t)$ (the probability that the target has not been reached up to time $t$), or equivalently by the FPT distribution $F(t)=-\partial_{t}S(t)$. For a symmetric random walk in a confined domain, the mean FPT is in general finite and has been studied at length. This led recently to explicit results for broad classes of stochastic processes Condamin2005 ; Condamin2007 ; Benichou2008 ; Schuss2007 ; Guerin2012a . The opposite case of unconfined random walks is drastically different. In this case, either the walker has a finite probability of never finding the target (non-compact random walks), or it reaches it with probability one (compact random walk) and the survival probability decays algebraically with time, $S(t)\sim S_{0}/t^{\theta}$, with $\theta$ the persistence exponent that does not depend on the initial distance to the target. In this case the mean FPT is often infinite so that the relevant observable to quantify FPT statistics is the long time algebraic decay of the probability $S(t)$ that the target has not been reached up to $t$. This, additional to the fact that $\theta$ can be non-trivial for non-Markovian random walks, has triggered a considerable amount of work to characterize the persistence exponent $\theta$ in a wide number of models of non-equilibrium statistical mechanics. Indeed, $S(t)$ is an essential observable to quantify the kinetics of transport controlled reactions and the dynamics of coarsening in phase transitions in general Majumdar1999 ; ReviewBray .

However, if one aims to evaluate the time $t$ to wait for observing a first-passage event with a given likelihood, or to determine the dependence of the survival probability on the initial distance to the target, one needs to know the prefactor $S_{0}$, which turns out to be much less characterized than the persistence exponent $\theta$. Even for Markovian random walks this problem is not trivial Aurzada2015 , as exemplified by recent studies for one dimensional Levy flights Majumdar2017 , while only scaling relations for $S_{0}$ (with the initial distance to the target) are known meroz2011distribution in fractal domains. However, if the dynamics of the random walker results from interactions with other degrees of freedom, the process becomes non-Markovian and the determination of $S_{0}$ becomes much more involved VanKampen1992 . In this case, the only explicit results are derived from perturbation expansion around Markovian processes delorme2015maximum ; delorme2016perturbative , or have been obtained for particular processes such as “run and tumble” motion (driven by telegraphic noise Masoliver1986 ) or the random acceleration process Burkhardt1993 . For long range correlated processes, such as fractional Brownian Motion, the existence of $S_{0}$ is not even established rigourously Aurzada2011 ; Aurzada2015 , and it has been found that straightforward adaptation of Markovian methods can lead to order-of-magnitude overestimations of $S_{0}$ and even to erroneous scalings Sanders2012 .

In this article, we rely on a non-perturbative strategy to determine $S_{0}$, which is of crucial interest to quantify the statistics of long FPT events. Our main result is a relation between the prefactor $S_{0}$ in the long time survival probability in free space and the mean FPT for the same process in a large confining volume. Our formula thus shows how to make use of the wealth of explicit results obtained recently on first-passage properties in confinement Condamin2005 ; Condamin2007 ; Benichou2008 ; Schuss2007 ; Guerin2012a ; guerin2016mean to determine the decay of the free-space survival probability. This formula is shown to be robust and holds for Markovian or non-Markovian processes with stationary increments, that are scale invariant at long times with diverging moments of the position, in one or higher spatial dimensions, and also for processes displaying transient aging (i. e. processes with finite memory time, whose initial state is not stationary, see below). This theory is confirmed by simulations for a variety of stochastic processes, including highly correlated ones such as Fractional Brownian Motion.

Results

Markovian case. We consider a symmetric random walker of position $\mathbf{r}(t)$ moving on an infinite discrete lattice (potentially fractal) of dimension $d_{f}$ [see Fig. 1(a) for the continuous space counterpart] in continuous time $t$, in absence of external field. The initial position is $\mathbf{r}_{0}$. We assume that the increments are stationary (no aging), which means in particular that $\sigma(t,\tau)\equiv\langle|\mathbf{r}(t+\tau)-\mathbf{r}(t)|^{2}\rangle$ is independent of the elapsed time $t$. Note that in the case of fractal spaces, we use the standard “chemical” distance defined as the minimal number of steps to link two points on the lattice. We define the walk-dimension $d_{w}$ such that $\sigma(t,\tau)\propto\tau^{2/d_{w}}$ for $\tau\to\infty$. Note that (i) this scale invariance is assumed only at long times, and that (ii) it implies that all even moments of the position diverge with time. We assume $d_{w}>d_{f}$ so that the process is compact DEGENNES1982 ; benAvraham2000 (and eventually reaches any point with probability one). We also introduce the Hurst exponent $H=1/d_{w}$.

We first consider the case of Markovian (memoryless) random walks. One can then define a propagator $p(\mathbf{r},t|\mathbf{r}_{0})$, which represents the probability to observe the walker at site $\mathbf{r}$ at time $t$ given that it started at $\mathbf{r}_{0}$ at initial time. Note that $p$ is defined in absence of target. We now add an absorbing target at site $\mathbf{r}=0$ (different from $\mathbf{r}_{0}$). We start our analysis with the standard renewal equation VanKampen1992 ; Redner:2001a ; hughes1995random :

[TABLE]

which relates the propagator $p$ to the FPT distribution $F$ that depends on $\mathbf{r}_{0}$. This equation is obtained by partitioning over the FPT to the target, and can be rewritten in Laplace space as

[TABLE]

where $\tilde{F}(s)=\int_{0}^{\infty}dtF(t)e^{-st}$ stands for the Laplace transform of $F(t)$. Here, we only focus on the long-time behavior of $F(t)$, that can be obtained by expanding Eq. (2) for small $s$. Scale invariance at long times implies benAvraham2000 that for any site $\mathbf{r}$

[TABLE]

where the notation “$\sim$” represents asymptotic equivalence, and $K$ is a positive coefficient. Note that $K$ is known to be position independent and is well characterized (at least numerically) for a large class of stochastic processes, including diffusion in a wide class of fractals grabner1997functional ; kron2004asymptotics ; weber2010random ; meroz2011distribution . We find that the small-$s$ behavior of the propagator is

[TABLE]

where $\Gamma(\cdot)$ is the Gamma function. Eqs. (2) and (4) (written for $\mathbf{r}=\mathbf{0}$ and $\mathbf{r}=\mathbf{r}_{0}$) lead to

[TABLE]

Taking the inverse Laplace transform (and using $F(t)=-\dot{S}$) leads to $S(t)\sim S_{0}/t^{\theta}$ with $\theta=1-d_{f}/d_{w}$ (as found in meroz2011distribution ), and to

[TABLE]

This expression is exact and characterizes the decay of the survival probability of unconfined scale invariant Markovian random walks.

We now consider the target search problem for the same random walk, with the only difference that it takes place in a confining volume $V$ (that is equal to the number of sites $N$ in our discrete formulation) [see Fig. 1(b)]. For this problem, the mean FPT $\langle\mathbf{T}\rangle$ is in general finite and it is known that it scales linearly with the volume and reads Condamin2005 ; Condamin2007

[TABLE]

We recognize in the above expression the time integral of propagators appearing in Eq. (6), leading to

[TABLE]

Hence, for compact Markovian random walks, we have identified a proportionality relation between the prefactor $S_{0}$ that characterizes the long time survival probability in free space and the rescaled mean FPT to the target in unconfined space. The proportionality coefficient involves the walk-dimension $d_{w}$ and the coefficient $K$ which characterizes the long time decay of the propagator [see Eq. (3)]. Formula (8) is the key result of this paper. As we proceed to show, it is very robust and is not limited to Markovian walks.

As a first application, consider the case of scale invariant Markovian random walks (such as diffusion on fractals), for which it was shown benichou2008zero that $\overline{T}\simeq r_{0}^{d_{w}-d_{f}}$, where the mean waiting time on a given site is taken as unity, and $r_{0}$ is the initial source-target (chemical) distance. Inserting this formula into Eq. (8) thus leads to

[TABLE]

In this case, we thus recover the scaling result of Ref. meroz2011distribution but in addition obtain the value of the prefactor. We have checked this relation for the Sierpinski gasket: simulation results are shown on Fig. 2(a). The long time persistence is perfectly described by our formula without any fitting parameter for different source-target distances, confirming the validity of our approach (see SI for other examples).

As a second application, we can consider the 1-dimensional Lévy stable process of index $\alpha$, which is defined as the only Markovian process whose jump distribution is given by $p(\Delta x,t)=1/(2\pi)\int_{-\infty}^{\infty}e^{i\Delta x.k-t|k|^{\alpha}}dk$. This process, defined in continuous space, is the continuous time limit of the Lévy Flight with same index $\alpha$. Its walk dimension is $d_{w}=\alpha$ and it is compact for $\alpha>1$, so that the first passage to a point target is well defined (note that we consider here the first arrival at the target, and not the first crossing event al:2003 ; Tejedor2011 ). For such a process, the prefactor $S_{0}$ for an unconfined random walk starting at a distance $r_{0}$ from the target has been shown to be $S_{0}=\alpha\sin(\pi\alpha/2)\sin(\pi/\alpha)\Gamma(2-\alpha)r_{0}^{\alpha-1}/(\pi\Gamma(1/\alpha)(\alpha-1))$ Blumenthal1961 . By computing the rescaled MFPT in confinement with Eq. (7), one can check that the relation (8), which can be readily generalised to continuous space, is still verified for this process.

Extension to non-Markovian processes. We now relax the Markov property and generalize our theory to the case of non-Markovian processes, i. e. displaying memory. In the following, we argue that the relation (8) yields much more accurate results for $S_{0}$ than Markov approximations; it is exact for processes with finite memory time, and is very precise (even though not exact) for strongly correlated processes such as the Fractional Brownian Motion. As the mean FPT in confinement has recently been characterized for non-Markovian Gaussian processes guerin2016mean , this equation (8) provides a means to estimate $S(t)$ at long times, beyond persistence exponents, for a wide class of random walks with memory.

For simplicity, we consider 1-dimensional processes and we switch to continuous space description. The stochastic trajectories $x(t)$ are assumed to be continuous but non-smooth (the method in fact also applies to compact and not continuous processes, such as $1d$ Levy stable processes of index $\alpha>1$ as discussed above), mathematically meaning that $\langle\dot{x}^{2}\rangle=\infty$ and physically corresponding to very rough trajectories, similar to those of Brownian motion. We assume that the increments of the walk are stationary (meaning that there is no aging, even transient 111In particular the case of continuous time random walks (CTRWs) is not directly covered by our analysis; persistence exponents and prefactors for CTRWs can be obtained from the subordination principle). This hypothesis is known to have two consequences: (i) the persistence exponent for the unconfined problem is exactly given by $\theta=1-1/d_{w}$ levernier2018universal ; Molchan1999 ; ReviewBray ; Krug1997 ; Aurzada2011 ; Aurzada2015 ; (ii) the mean FPT for the confined problem varies linearly with the confinement volume $V$, so that $\overline{T}$ is finite and has been identified as guerin2016mean :

[TABLE]

Here, $p(0,t)$ is the probability density of $x=0$ at a time $t$ after the initial state (where $x(0)=r_{0}$), and $p_{\pi}(0,t)$ denotes the probability density of finding the walker on the target at a time $t$ after the first-passage:

[TABLE]

where $p(0,t+\tau|\mathrm{FPT}=\tau)$ is the probability density of $x=0$ at time $t+\tau$, given that the FPT is $\tau$.

The starting point to relate $\overline{T}$ to $S_{0}$ consists in writing the generalization of Eq. (1) to non-smooth non-Markovian processes:

[TABLE]

To proceed further, we insert into Eq. (10) the expressions (11) and (12) of $p_{\pi}(0,t)$ and $p(0,t)$:

[TABLE]

To avoid diverging integrals in the change of variables $t=u+\tau$, we replace $\int_{0}^{\infty}dt(...)$ by $\underset{A\to\infty}{\lim}\;\;\int_{0}^{A}dt(...)$, so that

[TABLE]

Setting $t=uA$ and $\tau=vA$, we note that when $A\to\infty$, only the large time behavior are involved in these integrals, where one can use the asymptotics $F(Av;r_{0})\sim S_{0}\theta/(Av)^{1+\theta}$ and

[TABLE]

which is a form imposed by dimensional analysis. As previously, $K$ is the constant which characterizes the long time behavior of the one point probability distribution function [i.e. $p(x,t)\underset{t\to\infty}{\sim}K/t^{1/d_{w}}$], $G$ is a scaling function, with $G(\infty)=1$, that does not depend on the geometrical parameters of the problem. Inserting these asymptotic behaviors into Eq. (14), we get:

[TABLE]

The fact that the above integral exists and is finite leads to the (known) relation $\theta=1-1/d_{w}$. This finally leads to the exact relation:

[TABLE]

We stress that the dependency of $S_{0}$ on the source-to-target distance, even when not-trivial, is entirely contained in the term $\overline{T}$. Indeed, the scaling function $G$ depends only on the large scale properties of the random walk and not on the geometrical parameters.

While the exact determination of $G$ is a challenging task, the following decoupling approximation turns out to be very accurate. In this approximation, the return probability to the target at a time $t$ after the first passage time is independent of the actual value of the FPT, which leads to $p(0,t+\tau|\text{FPT}=\tau)\simeq p_{\pi}(t)$ for self-consistence reason. Within this decoupling approximation, $G\simeq 1$ and we obtain

[TABLE]

which generalizes Eq. (8) to non-Markovian processes. We now comment on the validity of this key relation.

First, we stress that Eq. (18) is exact for processes with finite memory time (i.e. for which the correlation function of increments decays exponentially at long times). This comes from the very definition of the function $G$, which involves only large time scales in Eq. (15), over which this finite memory time becomes irrelevant. This case is illustrated here by considering a Gaussian process whose Mean Square Displacement function $\psi(t)=\langle[x(t+\tau)-x(\tau)]^{2}\rangle$ is given by $\psi(t)=Dt+B(1-e^{-\lambda t})$. This “bidiffusive” process involves two diffusive behaviors at long and short time scales, and presents only one relaxation time $\lambda^{-1}$. This is typically relevant to tracers moving in viscoelastic Maxwell fluids grimm2011brownian , nematics turiv2013 or solutions of non-adsorbing polymers ochab2011scale . We also consider the effect of multiple relaxation times with the case that $x(t)$ is the position of the first monomer of a flexible polymer chain with $N$ monomers, in the most simple (Rouse, bead-spring) polymer model. We use recently obtained estimates of $\overline{T}$ in Ref. guerin2016mean to obtain estimates of $S_{0}$ through Eq. (18) and compare with numerical simulations on Fig. 2(b),(c). We also compare with a pseudo-Markovian approximation (using Eq. (6) with effective “propagators” PM ). Our prediction for $S_{0}$ is in good agreement with numerical simulations, and shows that even if the memory time is finite, memory effects are strong.

Second, it is showed in SI that Eq. (18) is also exact at first order in $\varepsilon=H-1/2$ for the fractional Brownian Motion (FBM), which is an emblematic example of processes with infinite memory time. The FBM is used in fields as varied as hydrology mandelbrot1968noah , finance cutland1995stock and biophysics ernst2012fractional ; burnecki2012universal . This Gaussian process is characterized by $\left\langle[x(t+\tau)-x(t)]^{2}\right\rangle=\kappa\tau^{2H}$, with $0<H<1$.

Third, in the strongly non-Markovian regime, where $\varepsilon$ cannot be considered as small, it turns out that Eq. (18) provides a very accurate approximation [Fig. 2(e)] of $S_{0}$, which takes the explicit form

[TABLE]

where $\beta_{H}$ is a function of $H$ analyzed in Ref. guerin2016mean . It can indeed be seen on Fig. 2(e) that Eq. (18) correctly predicts the long time behavior of $S(t)$ when $H=0.34$. For this value non-Markovian effects are strong, as can be seen by comparing with the prediction of the pseudo-Markovian approximation, which is wrong by more than one order of magnitude [Fig. 2(e), dashed line]. The value of $S_{0}$ is slightly underestimated in the decoupling approximation, but can be made more precise by evaluating the scaling function $G$ (see SI).

Furthermore, our approach also holds in dimension higher than one, even for strongly correlated non-Markovian processes. Indeed, the $d-$dimensional version of Eq. (18) [i.e. Eq. (8)] correctly predicts (but slightly underestimates) the value of $S_{0}$ for an example of two dimensional FBM [Fig. 2(f)]. In this example, the target radius $a$ is not zero even if the $a\to 0$ limit is well defined for compact processes; the dependence of $S_{0}$ on the target radius is predicted to be the same as that of $\overline{T}$, which is available in the non-Markovian theory of Ref. guerin2016mean . Finally, in the case of processes with finite memory, we find that Eq. (18) also holds for non-stationary initial conditions. This is illustrated by considering the case of a flexible phantom polymer for which all monomers are placed initially at $r_{0}$ (instead of having the shape of a random equilibrium coil for stationary initial conditions). This non-stationary initial condition induces transiently aging dynamics, and $S_{0}$ is changed with respect to the case of stationary initial conditions, but is still predicted correctly by Eq. (18) [see Fig. 2(d)].

Finally, let us mention the case of the 1-dimensional run and tumble process, where a particle switches between phases of constant velocities $\pm v$ with rate $\alpha$. This process is smooth and is a priori not covered by our analysis. However, our relation (18) between $S_{0}$ and $T/V$ is still exact, as is made clear by comparing the results for the mean FPT in confinement Masoliver1986 and in semi-infinite space Malakar2018 . This agreement holds even for non-stationary initial conditions, where the probability $p$ that the initial velocity is positive differs from $1/2$: in this case, one can obtain $S_{0}=(r_{0}+pv/\alpha)\sqrt{2\alpha/(\pi v^{2})}=\overline{T}/(K\pi)$, with $K=\sqrt{\alpha/(2\pi v^{2})}$, and we can check that our relation still holds Masoliver1986 ; Malakar2018 . Furthermore, it also holds in the case of partially reflecting targets, as can be deduced from the results of Angelani2015 . This suggests that our analysis can be extended to smooth non-Markovian processes with partial absorption as well.

Discussion

The determination of the survival probability $S(t)$, and in particular its dependence on the initial distance to the target, requires the knowledge of its prefactor $S_{0}$, which has remained an elusive quantity up to now. In this article, we have bridged this gap by identifying a general relation between the long time persistence and the mean FPT in confinement. The latter can be calculated with various recently introduced methods, for a large class of Markovian Condamin2007 ; Benichou2008 ; Schuss2007 and non-Markovian random walks guerin2016mean . Our theory holds for compact, unbiased walks with stationary increments that are scale invariant at long times (without confinement), with moments of the position that diverge with time. Our main result is Eq. (8), which is exact for both Markovian processes (such as diffusion in fractals) and for non Markovian processes with finite memory time (for which memory effects are nevertheless quantitatively non-negligible). For long-ranged correlated processes such as FBM our formula provides a good approximation of $S_{0}$ in one or higher dimensions, and is found to be exact at first order in a perturbation expansion around Brownian motion. Together, our results thus improve our understanding of the impact of memory on the statistics of long first passage events.

Acknowledgments. This work was supported by ERC grant FPTOpt-277998. Computer time for this study was provided by the computing facilities MCIA (Mesocentre de Calcul Intensif Aquitain) of the Université de Bordeaux and of the Université de Pau et des Pays de l’Adour.

Data and code availability. The numerical data presented in Figure 2, as well as the code that generated these data, are available from the corresponding author on reasonable request.

Competing interests statement. The Authors declare no competing interests.