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We establish rigorous bounds on dissipative transport in the disorderedKlein-Gordon chain with quartic on-site potential, governed by a parameter$\lambda$. At $\lambda = 0$, the chain is harmonic and transport is fullysuppressed by Anderson localization. For the anharmonic system at $\lambda >0$, our results show that dissipative effects set in on time scales that growfaster than any polynomial in $1/\lambda$, as $\lambda\to 0$. From a technicalperspective, the main novelty of our work is that we do not restrict ourselvesto the atomic limit, i.e. we develop perturbation theory around the harmonicsystem with a fixed hopping strength between near oscillators. This allows usto compare our mathematical results with previous numerical work and contributeto the resolution of an ongoing debate in the physics community, as we explainin a companion paper arXiv:2308.10572.

Long persistence of localization in a disordered anharmonic chain beyond the atomic limit