Asymptotic mutual-information conjecture for the sparse stochastic block model

Let $f$ denote the unique viscosity solution of the infinite-dimensional Hamilton–Jacobi equation governing the enriched free energy, and let $\mathbf{G}_N$ and $\sigma^*$ denote the observed graph and latent spins in the sparse stochastic block model. Mutual-information conjecture. The limiting free energy and mutual information are

$$\lim_{N\to\infty}\overline{F}_N=f(1,0),$$

and

$$\lim_{N\to\infty}\frac{1}{N}I(\mathbf{G}_N;\sigma^*)=\frac{1}{2}\operatorname{\mathbb{E}}\big(c+\Delta \sigma_1^*\sigma_2^*\big)\log\big(c+\Delta \sigma_1^*\sigma_2^*\big)-\frac{c}{2}-\frac{\Delta\overline{m}^2}{2}-f(1,0).$$

The free-energy upper bound is proved, but the matching lower bound—and hence the full asymptotic mutual-information formula—remains open. In the disassortative regime, the paper gives supporting evidence by matching a known variational formula.