For a much-studied model of random copolymer at a selective interface we prove that the slope of the critical curve in the weak-disorder limit is strictly smaller than 1, which is the value given by the annealed inequality. The proof is based on a coarse-graining procedure, combined with upper bounds on the fractional moments of the partition function.
"Coarse graining, fractional moments and the critical slope of random copolymers." Electron. J. Probab. 14 531 - 547, 2009. https://doi.org/10.1214/EJP.v14-612