Complete visitation statistics of one-dimensional random walks

We develop a framework to determine the complete statistical behavior of a fundamental quantity in the theory of random walks, namely, the probability that $n_1,n_2,n_3,...$ distinct sites are visited at times $t_1,t_2,t_3,...$. From this multiple-time distribution, we show that the visitation statistics of one-dimensional random walks are temporally correlated, and we quantify the non-Markovian nature of the process. We exploit these ideas to derive unexpected results for the two-time trapping problem and to determine the visitation statistics of two important stochastic processes, the run-and-tumble particle and the biased random walk.

Figure 1

(a) Space-time trajectory of a one-dimensional discrete random walk and (b) its corresponding span $N\left(t\right)$. At times $t_1,t_2,t_3$, and $t_4$ (dashed lines), we are interested in the joint statistics of $N\left(t_1\right),N\left(t_2\right),N\left(t_3\right)$, and $N\left(t_4\right)$.

Figure 2

The conditional two-time distribution (simulations, blue dots; theory, red curve) and its convergence to the single-time distribution (dashed). Shown is $\mathbb{P}(N\left(t_2\right)=n_2|N\left(t_1\right)=n_1)$ versus $n_2$ with fixed $n_1=10$ and $t_1=200$ with (a) $t_2=400$ and (b) $t_2=3200$.

Figure 3

Non-Markovian property of the span $\left\{N\right(t\left)\right\}$. The distribution $\mathbb{P}(N\left(t_3\right)=n_3|N\left(t_1\right)=n_1;N\left(t_2\right)=n_2)$ of the quantity $N(t_3=200)$ conditioned on $N(t_1=100)=5$ and (a) $N(t_2=110)=15$ and (b) $N(t_2=110)=6$ (blue curves). The distribution $\mathbb{P}(N\left(t_3\right)=n_3|N\left(t_2\right)=n_2)$ of $N\left(t_3\right)$ conditioned only on $N\left(t_2\right)$ is represented by the red dashed curves.

Figure 4

Long-time limit of the two-time survival probability, $S\left(t_2\right|t_1)$ [see Eq. (1)], of the 1D random walk at time $t_2$ knowing that it survived until time $t_1$. Here $z=t_2/t_1$ is a constant with $t_1,t_2\to \infty$. The black line represents the asymptotic theoretical value, Eq. (C5) of Appendix pp3. Symbols correspond to numerical simulations, for different values of the fraction $c$ of immobile and randomly distributed traps.

Figure 5

Comparison of the Laplace transform of the two point expectation $H(s_1,s_2)$, obtained in Ref. [24] and the same quantity $L(s_1,s_2)$ obtained using our formalism Eq. (A15).

Figure 6

Cumulative distribution of the span at times $t=5,$ 10, and 15 for parameters $v=1$ and $D=1$. The curves are $\frac{1}{2}[y_l(t,n)+y_r(t,n)]$, whereas the circles are from Ref. [30].

Figure 7

Two-time distribution for the persistent random walk for (a) parameters $T=v=1$ and $t_1,t_2=10,30$ as well as for (b) a biased random walk of parameters $v=1$ and diffusion $D=1$ at times $t_1,t_2=20,30$. Numerical simulations are represented by the dots.