Abstract. We formulate and justify rigorously a numerically ac-cessible criterion for the computation of the analyticity breakdown of quasi-periodic solutions in Symplectic maps and 1-D Statistical Mechanics models. Depending on the physical interpretation of the model, the analyticity breakdown may correspond to the onset of mobility of dislocations, or of spin waves (in the 1-D models) and to the onset of global transport in symplectic twist maps. The criterion we propose here works whenever there is an a posteriori KAM theorem that asserts the existence of a KAM torus provided that we can find a function that satisfies very ap-proximately the invariance equation and satisfies some mild non-degeneracy conditions. We formulate two precise theorems that implement these ideas: one which applies to statistical mechanics models (possibly with long range interactions) an another one which applies to symplectic mappings. The proof of both theorems uses an abstract implicit function theorem that unifies several such theorems in the literature. Content
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