We investigate the flat phase of quenched disordered polymerized membranes by means of a two-loop, weak-coupling computation performed near their upper critical dimension $D_{uc}=4$, generalizing the one-loop computation of Morse et al. [D. C. Morse et al., Phys. Rev. A 45, R2151 (1992); D. C. Morse and T. C. Lubensky, Phys. Rev. A 46, 1751 (1992)]. Our work confirms the existence of the finite-temperature, finite-disorder wrinkling transition, which has been recently identified by Coquand et al. [O. Coquand et al., Phys. Rev. E 97, 030102(R) (2018)] using a nonperturbative renormalization group approach. We also point out ambiguities in the two-loop computation that prevent the exact identification of the properties of the novel fixed point associated with the wrinkling transition, which very likely requires a three-loop order approach.

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• Critical phenomena
• Wrinkling
• Membrane structures
• Renormalization