Universal survival probability for a correlated random walk and applications to records

TL;DR

This paper derives a universal survival probability for correlated one-dimensional random walks, demonstrating its independence from jump distribution and applying it to record statistics and the run-and-tumble particle model.

Contribution

It analytically computes the survival probability for correlated walks, revealing universality and connecting to record statistics and bacterial motion models.

Findings

Survival probability is independent of jump distribution for any finite number of steps.

Distribution of record numbers matches that of an effective uncorrelated walk with adjusted steps.

Model converges to run-and-tumble particle behavior in a specific limit.

Abstract

We consider a model of space-continuous one-dimensional random walk with simple correlation between the steps: the probability that two consecutive steps have same sign is $q$ with $0\leq q\leq 1$. The parameter $q$ allows thus to control the persistence of the random walk. We compute analytically the survival probability of a walk of $n$ steps, showing that it is independent of the jump distribution for any finite $n$. This universality is a consequence of the Sparre-Andersen theorem for random walks with uncorrelated and symmetric steps. We then apply this result to derive the distribution of the step at which the random walk reaches its maximum and the record statistics of the walk, which show the same universality. In particular, we show that the distribution of the number of records for a walk of $n\gg 1$ steps is the same as for a random walk with $n_{\rm eff}(q)=n/(2(1-q))$ uncorrelated and symmetrically distributed steps. We also show that in the regime where $n\to \infty$ and $q\to 1$ with $y=n(1-q)$, this model converges to the run-and-tumble particle, a persistent random walk often used to model the motion of bacteria. Our theoretical results are confirmed by numerical simulations.