  We investigate the probability for the largest segment in 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 a square-root
singularity at $\ell\equiv L/N=1$, an essential singularity at $\ell=0$, and a
discontinuous derivative at $\ell=1/2$.
