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Randomly Charged Polymers, Random Walks, and Their Extremal Properties

Deniz Ertas·Yacov Kantor
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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.

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Deniz Ertas, Yacov Kantor. 1995-07-04. Randomly Charged Polymers, Random Walks, and Their Extremal Properties. https://doi.org/10.1103/physreve.53.846

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