Random walks in nonuniform environments with local dynamic interactions

We consider a class of lattice random walk models in which the random walker is initially confined to a finite connected set of allowed sites but has the opportunity to enlarge this set by colliding with its boundaries, each such collision having a given probability of breaking through. The model is motivated by an analogy to cell motility in tissue, where motile cells have the ability to remodel extracellular matrix, but is presented here as a generic model for stochastic erosion. For the one-dimensional case, we report some exact analytic results, some mean-field type analytic approximate results and simulations. We compute exactly the mean and variance of the time taken to enlarge the interval from a single site to a given size. The problem of determining the statistics of the interval length and the walker's position at a given time is more difficult and we report several interesting observations from simulations. Our simulations include the case in which the initial interval length is random and the case in which the initial state of the lattice is a random mixture of allowed and forbidden sites, with the walker placed at random on an allowed site. To illustrate the extension of these ideas to higher-dimensional systems, we consider the erosion of the simple cubic lattice commencing from a single site and report simulations of measures of cluster size and shape and the mean-square displacement of the walker.

Figure 1

The site occupied by the walker is shown in black and the star denotes a blocked site. A walker at position $i$ attempts to move right onto the blocked site $i+1$. It is successful with the probability $p_{\mathrm{s}}$, in which case the walker makes the step, and $i+1$ becomes an allowed site. With probability $1-p_{\mathrm{s}}$ the walker fails to clear this site $i+1$, and the move is aborted, leaving the walker at site $i$.

Figure 2

The erosion model for a walker with all sites other than the starting site initially blocked for four values of the snipping probability $p_{\mathrm{s}}$. Results shown are averaged over ${10}^4$ simulations of ${10}^3$ step walks. The case $p_{\mathrm{s}}=1$ corresponds to the ordinary random walk. (a) Mean length of the interval. For $p_{\mathrm{s}}=1$ our simulations are indistinguishable at this level of resolution from a straight line of slope $8/\pi$ (the asymptotic form for the square of the mean span of an ordinary random walk). (b) Mean-square displacement. For $p_{\mathrm{s}}=1$ we recover the elementary exact result $\langle R_n^2\rangle =n$.

Figure 3

The erosion model for a walker with all sites other than the starting site initially blocked. For $p_{\mathrm{s}}=0.01,0.02,...,1$ we show estimates of the coefficients $C\left(p_{\mathrm{s}}\right)$ and $2D\left(p_{\mathrm{s}}\right)$. The estimates are obtained by assuming that Eqs. (10) and (11) are equalities at the largest value of $n$ used. The results for ${10}^3$ step walks, shown as blue (gray) curves, and for ${10}^4$ step walks, shown as black curves, are virtually indistinguishable. In each case ${10}^4$ individual walks were averaged. The gray horizontal lines indicate the exact value for $p_{\mathrm{s}}=1$. The approximation $C\left(p_{\mathrm{s}}\right)\approx 8p_{\mathrm{s}}/\pi$ is shown as a red (gray) line.

Figure 4

The growth over time of the cluster size $\langle L_n\rangle$ for erosion from a random interval (dashed lines) together with the empirical interpolation formula (36) (solid lines) for ${10}^4$ steps and $2\times {10}^4$ realizations and three sets of parameter values.

Figure 5

The transition to effective diffusion for erosion from a random interval. Mean-square displacement $\langle R_n^2\rangle$ versus $n$ for ${10}^4$ steps and averaged over $2\times {10}^4$ realizations, with $p=0.9$ and $p_{\mathrm{s}}=0.1$. We see that $\langle R_n^2\rangle \sim 2D\left(p_{\mathrm{s}}\right)n$ as $n\to \infty$, where the slope of the line, $2D\left(p_{\mathrm{s}}\right)$, is given by the empirically determined formula (12).

Figure 6

Diffusivity for eroding a random environment with initial allowed site density $p$ and snipping probability $p_{\mathrm{s}}$ (Sec. 4) for $0.05\le p\le 0.95$ and $0.05\le p_{\mathrm{s}}\le 0.95$ (with increments of $0.05$ in each case). For each choice of $p$ and $p_{\mathrm{s}}$, we generated $2\times {10}^6$ random walks of ${10}^4$ steps and the diffusivity was inferred by a linear regression fit (not constrained to pass through the origin) assuming that $\langle R_n^2\rangle =2\mathcal{D}(p,p_{\mathrm{s}})n+\mathrm{constant}$ for $n>2500$. (a) Each curve shows $2\mathcal{D}(p,p_{\mathrm{s}})$ for a fixed value of $p_{\mathrm{s}}$ between from $0.05$ (lowest curve) to $0.95$ (highest curve). (b) Each curve shows $2\mathcal{D}(p,p_{\mathrm{s}})$ for a fixed value of $p$ from $0.05$ (lowest curve) to $0.95$ (highest curve). The curves plotted are linear interpolations. Except for the lowest three curves in (a), a higher-order interpolation scheme would produce no visible difference in the figure.

Figure 7

Simulation results for erosion of a one-dimensional random environment. (a) The growth over time of the mean cluster size $\langle L_n\rangle$ for ${10}^4$ steps and $2\times {10}^6$ realizations for the same three sets of parameter values as in Fig. 4. (b) Ratio of cluster sizes in Fig. 7a above those in Fig. 4. The results are always greater than unity. The ratio increases as $p$ increases.

Figure 8

The growth of the mean cluster size $\chi (n,p_{\mathrm{s}})$ for erosion of the simple cubic lattice ${\mathbb{Z}}^3$ starting from a single site.

Figure 9

The second moment of the mass distribution about the starting position, ${\rho }_0{\left(n\right)}^2$, for the cluster of allowed sites in erosion of the simple cubic lattice ${\mathbb{Z}}^3$ starting from a single site.

Figure 10

The second moment of the mass distribution about the center of mass, ${\rho }_{\mathrm{CM}}{\left(n\right)}^2$, for the cluster of allowed sites in erosion of the simple cubic lattice ${\mathbb{Z}}^3$ starting from a single site.

Figure 11

The mean-square displacement $\langle |{\mathbf{R}}_n{|}^2\rangle$ for erosion of the simple cubic lattice ${\mathbb{Z}}^3$ starting from a single site.

Figure 12

Plot of root-mean-square displacement estimated by simulations over ${10}^3$ steps and $5\times {10}^4$ realizations for three values of $p$. The solid lines are simulation results and dashed lines are the square root of Eq. (A7). This shows that ${\langle R_n^2\rangle }^{1/2}\to {\langle R_{\infty }^2\rangle }^{1/2}$, given by Eq. (A7), as $n\to \infty$.

Figure 13

Convergence of the mean-square displacement ${\langle R_n^2\rangle }^{1/2}$ to its equilibrium value, for a noneroding walker ($p_{\mathrm{s}}=0$) in a single interval of random length controlled by the allowed site density $p$. Solid curves show simulation data, while dashed curves show the empirical interpolation (A9) with $\beta =2$, which we found to be the most satisfactory choice. (a) Long-time results, $0\le n\le 3000$. (b) Early time results, $0\le n\le 200$.