  In this paper, we investigate analytically the properties of the disordered
Bernoulli model of planar matching. This model is characterized by a
topological phase transition, yielding complete planar matching solutions only
above a critical density threshold. We develop a combinatorial procedure of
arcs expansion that explicitly takes into account the contribution of short
arcs, and allows to obtain an accurate analytical estimation of the critical
value by reducing the global constrained problem to a set of local ones. As an
application to a toy representation of the RNA secondary structures, we suggest
generalized models that incorporate a one-to-one correspondence between the
contact matrix and the RNA-type sequence, thus giving sense to the notion of
effective non-integer alphabets.
