Decelle–Krzakala–Moore–Zdeborová parameter-estimation threshold conjecture

Let $\mathcal{G}(n,a/n,b/n)$ be the sparse planted partition model with two equal-sized clusters and parameters $a>b>0$. A consistent estimator of $a$ and $b$ is an estimator whose estimates converge to the true parameters as $n\to\infty$. Decelle–Krzakala–Moore–Zdeborová's parameter-estimation conjecture. If

$$(a-b)^2>2(a+b),$$

then there is a consistent estimator for $a$ and $b$ under $\mathcal{G}(n,a/n,b/n)$; conversely, if

$$(a-b)^2<2(a+b),$$

then there is no consistent estimator. This predicts that parameter estimation has the same phase transition as community recovery in the sparse two-block model. The source presents the claim as a conjectural extension of statistical work on parameter estimation and gives no proof of either direction.