Physics – Condensed Matter – Statistical Mechanics
Scientific paper
1999-05-18
J. Stat. Phys. 96 (1999), 963-1019.
Physics
Condensed Matter
Statistical Mechanics
54 pages, 16 figures. To appear in J. Stat. Phys
Scientific paper
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.
Bauer Marianne
Godreche Claude
Luck Mck. J.
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