Strong Disorder Renewal Approach to DNA denaturation and wetting : typical and large deviation properties of the free energy

For the DNA denaturation transition in the presence of random contact energies, or equivalently the disordered wetting transition, we introduce a Strong Disorder Renewal Approach to construct the optimal contacts in each disordered sample of size $L$. The transition is found to be of infinite order, with a correlation length diverging with the essential singularity $\ln \xi(T) \propto |T-T_c |^{-1}$. In the critical region, we analyze the statistics over samples of the free-energy density $f_L$ and of the contact density, which is the order parameter of the transition. At the critical point, both decay as a power-law of the length $L$ but remain distributed, in agreement with the general phenomenon of lack of self-averaging at random critical points. We also obtain that for any real $q>0$, the moment $\overline{Z_L^q} $ of order $q$ of the partition function at the critical point is dominated by some exponentially rare samples displaying a finite free-energy density, i.e. by the large deviation sector of the probability distribution of the free-energy density.