  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.
