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

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.

54 pages, 16 figures. To appear in J. Stat. Phys

References in corpus (4)

• Exact Large Deviation Function in the Asymmetric Exclusion Process
• Large deviations and nontrivial exponents in coarsening systems
• Generalized persistence exponents: an exactly soluble model
• Stationary definition of persistence for finite temperature phase ordering

Cited by in corpus (17)

• Persistence and First-Passage Properties in Non-equilibrium Systems
• Driven particle in a random landscape: disorder correlator, avalanche distribution and extreme value statistics of records
• Record statistics for biased random walks, with an application to financial data
• Record statistics of a strongly correlated time series: random walks and Lévy flights
• Spectral properties of zero temperature dynamics in a model of a compacting granular column
• Time-Dependent Random Walks and the Theory of Complex Adaptive Systems
• Local Persistence in the Directed Percolation Universality Class
• Zero-Temperature Coarsening in the Two-Dimensional Long-Range Ising Model
• Sparre-Andersen theorem with spatiotemporal correlations
• Brownian motion in time-dependent logarithmic potential: Exact results for dynamics and first-passage properties
• Statistics of leaders and lead changes in growing networks
• Persistence exponents of non-Gaussian processes in statistical mechanics
• Analytical results for generalized persistence properties of smooth processes
• Statistics of the occupation time for a random walk in the presence of a moving boundary
• Sign-time distribution for a random walker with a drifting boundary
• Survival probabilities in time-dependent random walks
• Occupation time of a renewal process coupled to a discrete Markov chain