First-Passage Properties of Bursty Random Walks

We investigate the first-passage properties of bursty random walks on a finite one-dimensional interval of length L, in which unit-length steps to the left occur with probability close to one, while steps of length b to the right – “bursts” – occur with small probability. This stochastic process provides a crude description of the early stages of virus spread in an organism after exposure. The interesting regime arises when b is of the order of but less than 1, where the conditional exit time to reach L, corresponding to an infected state, has a non-monotonic dependence on initial position. Both the exit probability and the infection time exhibit complex dependences on the initial condition due to the interplay between the burst length and interval length.

💡 Research Summary

The paper introduces a stochastic model called the “bursty random walk” to study first‑passage properties on a finite one‑dimensional interval of length L. At each discrete time step a particle moves one unit to the left with a high probability p≈1, while with a small probability q=1−p it makes a rightward jump of length b (the “burst”). The left boundary (x=0) is reflecting (or absorbing) and the right boundary (x=L) is absorbing, representing an infected state. By formulating the master equation for this asymmetric Markov chain, the authors derive exact expressions for the exit probability E(x) and the mean exit time T(x). They solve the resulting difference equations using generating functions, eigenvalue analysis of the transition matrix, and, where appropriate, continuous approximations with boundary‑layer corrections.

A central finding is that when the burst length b is comparable to but smaller than L (b/L≈0.5–0.9), the mean exit time T(x) becomes non‑monotonic in the initial position x. Specifically, T(x) attains a minimum near the midpoint of the interval, while it grows sharply toward both ends. This behavior contrasts with the monotonic decrease seen in ordinary biased random walks. The authors explain the phenomenon as a competition between frequent small left steps, which tend to trap the particle near the left edge, and rare large rightward bursts, which can quickly propel the particle to the absorbing boundary when it occurs near the centre. The exit probability E(x) remains monotonic but exhibits a steep rise when b approaches L, reflecting the high chance of reaching L after a single burst.

The analytical results are validated by extensive Monte‑Carlo simulations for a wide range of parameters (L up to 10⁴, various b and q). The simulations confirm the predicted non‑monotonic T(x) and the scaling of the minimum’s location with the ratio b/L. Moreover, the authors show that as q becomes very small the system reverts to the classic biased walk behavior, while larger q accentuates the non‑linear features.

In the discussion, the model is mapped onto the early stage of viral spread within an organism. The leftward steps represent the natural decay or clearance of viral particles, whereas the bursts correspond to occasional replication events that produce many new virions. The analysis suggests that if the initial infection site is near the centre of the tissue (mid‑interval), the infection can reach a systemic state (x=L) more rapidly than if it starts near a peripheral boundary, due to the higher likelihood that a burst will bridge the remaining distance. This insight provides a mechanistic explanation for spatial heterogeneity observed in early infection dynamics.

The paper concludes that bursty random walks display fundamentally different first‑passage characteristics from standard biased walks, with the burst length relative to the system size being a crucial control parameter. The authors propose extensions to higher dimensions, time‑dependent burst probabilities, and quantitative comparison with experimental viral load data as promising directions for future research.