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We set up a new notion of local convergence for permutations and we prove acharacterization in terms of proportions of \emph{consecutive} patternoccurrences. We also characterize random limiting objects for this new topologyintroducing a notion of "shift-invariant" property (corresponding to the notionof unimodularity for random graphs). We then study two models in the frameworkof random pattern-avoiding permutations. We compute the local limits of uniform$\rho$-avoiding permutations, for $|\rho|=3,$ when the size of the permutationstends to infinity. The core part of the argument is the description of theasymptotics of the number of consecutive occurrences of any given pattern. Forthis result we use bijections between $\rho$-avoiding permutations and rootedordered trees, local limit results for Galton--Watson trees, the Second momentmethod and singularity analysis.

Local convergence for permutations and local limits for uniform $\rho$-avoiding permutations with $|\rho|=3$