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Article

Topological transition in disordered planar matching: combinatorial arcs expansion

Nechaev S., Tamm M., Valba O. V., Lokhov A.

In this paper, we investigate analytical properties of planar matching on a line in the disordered Bernoulli model. This model is characterized by a topological phase transition, yielding the 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 the physics of the RNA secondary structures, we suggest generalized models that incorporate a one-to-one correspondence between the contact matrix and the nucleotide sequence, thus giving sense to the notion of effective non-integer alphabets