  We study the asymptotic properties of fracture strength distributions of
disordered elastic media by a combination of renormalization group, extreme
value theory, and numerical simulation. We investigate the validity of the
`weakest-link hypothesis' in the presence of realistic long-ranged interactions
in the random fuse model. Numerical simulations indicate that the fracture
strength is well described by the Duxbury-Leath-Beale (DLB) distribution which
is shown to flow asymptotically to the Gumbel distribution. We explore the
relation between the extreme value distributions and the DLB type asymptotic
distributions, and show that the universal extreme value forms may not be
appropriate to describe the non-universal low-strength tail.
