For large combinatorial structures, two main notions of convergence can be defined: scaling limits and local limits. In particular for graphs, both notions are well-studied and well-understood. For permutations only a notion of scaling limits, called permutons, has been recently introduced. The convergence for permutons has also been characterized by frequencies of pattern occurrences.

We set up a new notion of local convergence for permutations and we prove a characterization in terms of proportions of *consecutive* pattern occurrences. We are also able to characterize random limiting objects introducing a “shift-invariant” property (corresponding to the notion of unimodularity for random graphs). We finally show two applications in the framework of random pattern-avoiding permutations, computing the local limits of uniform ρ-avoiding permutations for |ρ|=3. For this last result we use bijections between ρ-avoiding permutations and ordered rooted trees, a local limit result for Galton-Waltson trees, the Second moment method and singularity analysis.

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