EXTREMAL SEGMENTS IN RANDOM SEQUENCES

We investigate the probability for the largest segment with total displacement Q in an N-step random walk to have length L. Using analytical, exact enumeration, and Monte Carlo methods, we reveal the complex structure of the probability distribution in the large-hr limit. In particular, the size of the longest loop has a distribution with a square-root singularity at l = LIN = 1, an essential singularity at l = 0, and a discontinuous derivative at l = 1/2.