Loops in One Dimensional Random Walks

TL;DR

This paper studies the distribution and properties of loops in one-dimensional random walks, revealing universal behaviors and analytical characteristics that are independent of specific walk details.

Contribution

It provides a combined numerical and analytical analysis of loops in various one-dimensional random walks, including those with continuous step lengths, highlighting universal properties.

Findings

Probability density of the longest loop becomes universal for long walks.

Identifies crossover behaviors and convergence to limiting distributions.

Derives analytical properties of the universal probability density.

Abstract

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.