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

We set up a new notion of local convergence for permutations and we prove a characterization in terms of proportions of \emph{consecutive} pattern occurrences. We also characterize random limiting objects for this new topology introducing a notion of "shift-invariant" property (corresponding to the notion of unimodularity for random graphs). We then study two models in the framework of random pattern-avoiding permutations. We compute the local limits of uniform $\rho$-avoiding permutations, for $|\rho|=3,$ when the size of the permutations tends to infinity. The core part of the argument is the description of the asymptotics of the number of consecutive occurrences of any given pattern. For this result we use bijections between $\rho$-avoiding permutations and rooted ordered trees, local limit results for Galton--Watson trees, the Second moment method and singularity analysis.