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Randomly Charged Polymers, Random Walks, and Their Extremal Properties
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.Comment: 33 pages, 19 Postscript figures, RevTe
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