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 polymers with N monomers (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 another interesting RW problem, which we call the ``staircase problem.'' We also discuss the generalized problem for d-dimensional RWs. {} 1996 The American Physical Society.