Randomly Charged Polymers, Random Walks, and Their Extremal Properties

TL;DR

This paper investigates the size distribution of largest charged segments in polymers by mapping to one-dimensional random walks, revealing complex distribution features and singularities through analytical and computational methods.

Contribution

It introduces a detailed analysis of the largest Q-segments in charged polymers using random walk models, uncovering new distribution properties and singularities.

Findings

Largest neutral segment size distribution has a square-root singularity at l=1

Distribution exhibits an essential singularity at l=0

Discontinuous derivative at l=1/2

Abstract

Motivated by an investigation of ground state properties of randomly charged polymers, we discuss the size distribution of the largest Q-segments (segments with total charge Q) in such N-mers. Upon mapping the charge sequence to one--dimensional random walks (RWs), this corresponds to finding the probability for the largest segment with total displacement Q in an N-step RW 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 neutral segment has a distribution with a square-root singularity at l=L/N=1, an essential singularity at l=0, and a discontinuous derivative at l=1/2. The behavior near l=1 is related to a another interesting RW problem which we call the "staircase problem". We also discuss the generalized problem for d-dimensional RWs.