Statistics of persistent events in the binomial random walk: Will the drunken sailor hit the sober man?

TL;DR

This paper analyzes the survival probability of a 1D lattice random walk with a moving obstacle, revealing complex long-term behaviors and discontinuities linked to number theory and algebraic structures.

Contribution

It introduces a novel algebraic approach to study persistent events in random walks with moving obstacles, uncovering discontinuities at rational velocities.

Findings

Survival probability exhibits non-trivial limits depending on obstacle velocity.

Discontinuities occur at all rational obstacle velocities.

Mathematical analysis involves number theory and algebraic curves.

Abstract

The statistics of persistent events, recently introduced in the context of phase ordering dynamics, is investigated in the case of the 1D lattice random walk in discrete time. We determine the survival probability of the random walker in the presence of an obstacle moving ballistically with velocity v, i.e., the probability that the walker remains up to time n on the left of the obstacle. Three regimes are to be considered for the long-time behavior of this probability, according to the sign of the difference between v and the drift velocity V of the random walker. In one of these regimes (v>V), the survival probability has a non-trivial limit at long times, which is discontinuous at all rational values of v. An algebraic approach allows us to compute these discontinuities, as well as several related quantities. The mathematical structure underlying the solvability of this model combines elementary number theory, algebraic functions, and algebraic curves defined over the rationals.