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

The statistics of persistent events, recently introduced in the context of phase ordering dynamics, is investigated in the case of the one-dimensional 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 upsilon, i.e., the probability that the random walker remains up to lime 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 upsilon and the drift velocity (V) over bar of the random walker. In one of these regimes (upsilon > (V) over bar), the survival probability has a nontrivial limit at long times which is discontinuous at all rational values of upsilon. 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.