Derrida–Griffiths–Higgs self-averaging and weak freezing transition conjecture

Let $\beta>0$ be the inverse temperature, let $\bar Z_n^{\beta,\omega}$ be the quenched partition function, and let the finite-volume free energy be the corresponding normalized logarithm of $\bar Z_n^{\beta,\omega}$. Define

$$p_{i,n}^{\beta,\omega}:={\mathrm{E}}_n^{\beta,\omega}(\Delta S_i),\qquad 1\leq i<n,$$

and the empirical cumulative distribution function

$$\mathcal F_n^{\beta,\omega}(p)=\frac{1}{n-1}\sum_{1\leq i<n}{\sf 1}\{p_{i,n}^{\beta,\omega}\leq p\},\qquad p\in[0,1].$$

Derrida–Griffiths–Higgs conjecture. The following assertions hold: (i) the specific heat, namely the derivative of the finite-volume free energy, converges $\mathbb P$-almost surely as $n\to\infty$ to a non-random limit; (ii) $\mathcal F_n^{\beta,\omega}(p)$ converges $\mathbb P$-almost surely as $n\to\infty$ to a non-random limit $\mathcal F_\beta(p)$; and (iii) there exists a critical inverse temperature $\beta_c\in(0,\infty)$ such that, for every $\beta<\beta_c$, there exists $p_{\rm \min}(\beta)\in(0,1)$ with $\mathcal F_\beta(p)=0$ for every $p<p_{\rm \min}(\beta)$, whereas for every $\beta>\beta_c$ there exists an exponent $\gamma(\beta)>0$ such that

$$\mathcal F_\beta(p)\sim \mathrm{(cst.)}\,p^{\gamma(\beta)}\qquad\text{as }p\to0.$$

This conjecture predicts self-averaging of the specific heat and the local-transition-probability distribution, together with a high-temperature regime bounded away from zero and a low-temperature power-law regime. The claimed low-temperature phase, including for binary or other charge distributions, remains open to the authors' knowledge.