  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.
