Extremal Segments in Random Sequences

TL;DR

This paper analyzes the probability distribution of the longest segment in a random walk with total displacement Q, revealing complex singularities and structural properties in the large N limit through analytical, enumeration, and Monte Carlo methods.

Contribution

It provides a detailed characterization of the probability distribution of the longest segment in random walks, highlighting singularities and structural features.

Findings

Distribution has a square-root singularity at bblbbl=1

Essential singularity at bblbbl=0

Discontinuous derivative at bblbbl=1/2

Abstract

We investigate the probability for the largest segment in with total displacement $Q$ in an $N$-step random walk to have length $L$. Using analytical, exact enumeration, and Monte Carlo methods, we reveal the complex structure of the probability distribution in the large $N$ limit. In particular, the size of the longest loop has a distribution with a square-root singularity at $\ell\equiv L/N=1$, an essential singularity at $\ell=0$, and a discontinuous derivative at $\ell=1/2$.