Freezing transition of the random bond RNA model: statistical properties of the pairing weights

To characterize the pairing-specificity of RNA secondary structures as a function of temperature, we analyse the statistics of the pairing weights as follows : for each base $(i)$ of the sequence of length N, we consider the $(N-1)$ pairing weights $w_i(j)$ with the other bases $(j \neq i)$ of the sequence. We numerically compute the probability distributions $P_1(w)$ of the maximal weight, the probability distribution $\Pi(Y_2)$ of the parameter $Y_2(i)= \sum_j w_i^2(j)$, as well as the average values of the moments $Y_k(i)= \sum_j w_i^k(j)$. We find that there are two important temperatures $T_c<T_{gap}$. For $T>T_{gap}$, the distribution $P_1(w)$ vanishes at some value $w_0(T)<1$, and accordingly the moments $\bar{Y_k(i)}$ decay exponentially in $k$. For $T<T_{gap}$, the distributions $P_1(w)$ and $\Pi(Y_2)$ present the characteristic Derrida-Flyvbjerg singularities at $w,Y_2=1/n$ for $n=1,2..$. In particular, there exists a temperature-dependent exponent $\mu(T)$ that governs these singularities and the decay of the moments $ \bar{Y_k(i)} \sim 1/k^{\mu(T)}$. The exponent $\mu(T)$ grows from $\mu(T=0)=0$ up to $\mu(T_{gap}) \sim 2$. The study of spatial properties indicates that the critical temperature $T_c$ where the roughness exponent changes from the low temperature value $\zeta \sim 0.67$ to the high temperature value $\zeta \sim 0.5$ corresponds to the exponent $\mu(T_c)=1$. For $T<T_c$, there exists frozen pairs of all sizes, whereas for $T_c< T <T_{gap}$, there exists frozen pairs, but only up to some characteristic length diverging as $\xi(T) \sim 1/(T_c-T)^{\nu}$ with $\nu \simeq 2$. The similarities and differences with the weight statistics in L\'evy sums and in Derrida's Random Energy Model are discussed.