  We report further findings on the size distribution of the largest neutral
segments in a sequence of N randomly charged monomers [D. Ertas and Y. Kantor,
Phys. Rev. E53, 846 (1996); cond-mat/9507005]. Upon mapping to one--dimensional
random walks (RWs), this corresponds to finding the probability distribution
for the size L of the largest segment that returns to its starting position in
an N--step RW. We primarily focus on the large N, \ell = L/N << 1 limit, which
exhibits an essential singularity. We establish analytical upper and lower
bounds on the probability distribution, and numerically probe the distribution
down to \ell \approx 0.04 (corresponding to probabilities as low as 10^{-15})
using a recursive Monte Carlo algorithm. We also investigate the possibility of
singularities at \ell=1/k for integer k.
