Random words in free groups, non-crossing matchings and RNA secondary structures

Consider a random word $$X^n=(X_1,\ldots ,X_n)$$ in an alphabet consisting of 4 letters, with the letters viewed either as A, U, G and C (i.e., nucleotides RNA sequence) or $$\alpha $$ , $$\bar{\alpha }$$ $$\beta $$\bar{\beta generators free group $$\langle \alpha ,\beta \rangle their inverses). We show that expected fraction $$\rho (n)$$ unpaired bases optimal secondary structure (with only Watson-Crick bonds no pseudo-knots) converges to constant $$\lambda _2$$ $$0<\lambda _2<1$$ $$n\rightarrow \infty . Thus, positive proportion string do not form hydrogen bonds. know exact value but we derive upper lower bounds for it. In terms groups, is ratio length shortest representing X generating set conjugates inverses respect standard inverses. Thus typical (infinite) grows linearly generators. fact, similar result holds all non-abelian finitely generated groups _1,\dots ,\alpha _k\rangle $$k\ge 2$$