Loops in one-dimensional random walks
School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel
Published online: 13 August 2012
Distribution of loops in a one-dimensional random walk (RW), or, equivalently, neutral segments in a sequence of positive and negative charges is important for understanding the low energy states of randomly charged polymers. We investigate numerically and analytically loops in several types of RWs, including RWs with continuous step-length distribution. We show that for long walks the probability density of the longest loop becomes independent of the details of the walks and definition of the loops. We investigate crossovers and convergence of probability densities to the limiting behavior, and obtain some of the analytical properties of the universal probability density.
PACS: 02.50.-r – Probability theory, stochastic processes, and statistics / 05.40.-a – Fluctuation phenomena, random processes, noise, and Brownian motion / 36.20.-r – Macromolecules and polymer molecules
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 1999