  We study a quenched charged-polymer model, introduced by Garel and Orland in
1988, that reproduces the folding/unfolding transition of biopolymers. We prove
that, below the critical inverse temperature, the polymer is delocalized in the
sense that: (1) The rescaled trajectory of the polymer converges to the
Brownian path; and (2) The partition function remains bounded. At the critical
inverse temperature, we show that the maximum time spent at points jumps
discontinuously from 0 to a positive fraction of the number of monomers, in the
limit as the number of monomers tends to infinity. Finally, when the critical
inverse temperature is large, we prove that the polymer collapses in the sense
that a large fraction of its monomers live on four adjacent positions, and its
diameter grows only logarithmically with the number of the monomers. Our
methods also provide some insight into the annealed phase transition and at the
transition due to a pulling force; both phase transitions are shown to be
discontinuous.
