Persistence of Random Walk Records

Abstract: We study records generated by Brownian particles in one dimension. Specifically, we investigate an ordinary random walk and define the record as the maximal position of the walk. We compare the record of an individual random walk with the mean record, obtained as an average over infinitely many realizations. We term the walk "superior" if the record is always above average, and conversely, the walk is said to be "inferior" if the record is always below average. We find that the fraction of superior walks, S, decays algebraically with time, S ~ t^-beta, in the limit t --> infty, and that the persistence exponent is nontrivial, beta=0.382258.... The fraction of inferior walks, I, also decays as a power law, I ~ t^-alpha, but the persistence exponent is smaller, alpha=0.241608.... Both exponents are roots of transcendental equations involving the parabolic cylinder function. To obtain these theoretical results, we analyze the joint density of superior walks with given record and position, while for inferior walks it suffices to study the density as function of position.