Solvable random-walk model with memory and its relations with Markovian models of anomalous diffusion

Motivated by studies on the recurrent properties of animal and human mobility, we introduce a path-dependent random-walk model with long-range memory for which not only the mean-square displacement (MSD) but also the propagator can be obtained exactly in the asymptotic limit. The model consists of a random walker on a lattice, which, at a constant rate, stochastically relocates at a site occupied at some earlier time. This time in the past is chosen randomly according to a memory kernel, whose temporal decay can be varied via an exponent parameter. In the weakly non-Markovian regime, memory reduces the diffusion coefficient from the bare value. When the mean backward jump in time diverges, the diffusion coefficient vanishes and a transition to an anomalous subdiffusive regime occurs. Paradoxically, at the transition, the process is an anticorrelated Lévy flight. Although in the subdiffusive regime the model exhibits some features of the continuous time random walk with infinite mean waiting time, it belongs to another universality class. If memory is very long-ranged, a second transition takes place to a regime characterized by a logarithmic growth of the MSD with time. In this case the process is asymptotically Gaussian and effectively described as a scaled Brownian motion with a diffusion coefficient decaying as $1/t$.

Figure 1

Top: At time $t$, the model walker performs a nearest-neighbor random-walk step with probability $1-q$ or relocates, with probability $q$, to the site it occupied at some earlier time $t^{\prime }$. Bottom: Probability distribution of $t^{\prime }$ for the uniform preferential visit case [$\beta =0$ in Eq. (4)] or in a case with memory decay ($\beta >0$).

Figure 2

Trajectories obtained from numerical simulations of the model in the normally diffusive regime (top), at the subdiffusive threshold (middle) and in the first subdiffusive regime (bottom). The thick (black) vertical lines indicate the relocation steps in the memory mode. The rate of memory use is $q=0.01$ in these cases.

Figure 3

Comparison between the exact MSD obtained from solving Eq. (19) numerically (solid red lines), and the asymptotic analytical results [dotted blue lines, from Eq. (8)], in the normal diffusive case $\beta >2$. Top: $M_2\left(t\right)$ vs $t$. The black dots correspond to Monte Carlo simulations. Bottom: The same curves, represented as $d{M}_2\left(t\right)/dt$ vs $t$. Higher-order corrections, which are not calculated in the theory, become more important at finite $t$ when $\beta$ approaches the subdiffusive transition point $\beta =2$. In all cases, $q=0.5$.

Figure 4

Top panels: $M_2\left(t\right)$ as a function of $t^{\beta -1}$ for two values of $\beta$ in the range $(1,2)$ and with $q=0.5$. The solid red lines are obtained from exact resolution of Eq. (19) and the dotted blue lines are given by Eq. (9). The black dots correspond to Monte Carlo simulations of the model. Bottom panels: The same curves, represented as $d{M}_2\left(t\right)/d\left(t^{\beta -1}\right)$ vs $t^{\beta -1}$.

Figure 5

Top: $M_2\left(t\right)$ as a function of $lnt$ for three values of $\beta <1$ and with $q=0.5$. The solid red lines are obtained from exact resolution of Eq. (19) and the dotted blue lines are given by Eq. (10). The black dots correspond to Monte Carlo simulations. Bottom: The same curves, represented as $d{M}_2\left(t\right)/d(lnt)$ vs $lnt$.

Figure 6

Moment ratios $a_p/a_{p-1}$ (for $p=2$, 3, and 4) as a function of $\beta$. The solid red lines correspond to the analytical results. The two Gaussian regimes ($a_p/a_{p-1}=cst=2p-1$) are separated by non-Gaussian scalings in the interval $1<\beta <2$, given by Eq. (15). The dots correspond to the quantity $Q_p\left(t\right)$ computed from simulations ($q=0.1$, $t={10}^6$, $5\times {10}^4$ runs). The green crosses correspond to $Q_p\left(t\right)$ computed from exact numerical resolution of Eq. (1), with $q=0.1$ and $t={10}^4$. The ratios $a_p/a_{p-1}$ corresponding to the CTRW model (dotted blue lines) are shown for comparison for the same values of $p$. In that case, $\beta$ is the waiting-time exponent.

Figure 7

Qualitative comparison between the current model (left) and a CTRW (right); see text.

Figure 8

Probability distribution function of the relocation length $\ell$ in simulations for several values of $\beta$ ($q=0.01$ in all cases). The solid lines have slopes $-(2\beta -1)$, i.e., $-4$, $-3$, and $-2$, from bottom to top. The variance of the distribution thus becomes infinite at the subdiffusion threshold $\beta =2$.