Randomly Charged Polymers, Random Walks, and Their Extremal Properties

Details Randomly Charged Polymers, Random Walks, and Their Extremal Properties Randomly Charged Polymers, Random Walks, and Their Extremal Properties

1995-07-04

arxiv.org/abs/cond-mat/9507005v1

Phys. Rev. E53, 846-860 (1996)

Physics

Condensed Matter

33 pages, 19 Postscript figures, RevTeX

Scientific paper

10.1103/PhysRevE.53.846

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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