  We introduce and study a simple Markovian model of random separable
permutations. Our first main result is the almost sure convergence of these
permutations towards a random limiting object in the sense of permutons, which
we call the recursive separable permuton. We then prove several results on this
new limiting object: a characterization of its distribution via a fixed-point
equation, a combinatorial formula for its expected pattern densities, an
explicit integral formula for its intensity measure, and lastly, we prove that
its distribution is absolutely singular with respect to that of the Brownian
separable permuton, which is the large size limit of uniform random separable
permutations.
