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-N limit. In particular, the size of the longest loop has a distribution with square-root singularity at l identical to L/N=1, an essential singularity at l=0, and a discontinuous derivative at l=1/2.