Refined scaling conjecture for unconditionally stable LDPC ensembles

Consider transmission over the binary erasure channel with erasure probability $\epsilon$ using random elements from an ensemble $\operatorname{LDPC}(n,\lambda,\rho)$ having a single critical point and being unconditionally stable. Let $\epsilon^*=\epsilon^*(\lambda,\rho)$ be the threshold, let $\nu^*$ be the fractional size of the residual graph at the threshold, and let $\gamma\in(0,1)$. Write $\operatorname{P}_{\rm b}(n,\lambda,\rho,\epsilon)$ for the expected bit-erasure probability and $\operatorname{P}_{\rm B,\gamma}(n,\lambda,\rho,\epsilon)$ for the expected block-erasure probability due to errors of size at least $\gamma\nu^*$. Set

$$z:=\sqrt{n}\left(\epsilon^*-\beta n^{-2/3}-\epsilon\right).$$

Refined scaling conjecture. As $n$ tends to infinity,

$$\operatorname{P}_{\rm B,\gamma}(n,\lambda,\rho,\epsilon)=Q\left(\frac{z}{\alpha}\right)\left(1+O(n^{-1/3})\right),$$

and

$$\operatorname{P}_{\rm b}(n,\lambda,\rho,\epsilon)=\nu^*Q\left(\frac{z}{\alpha}\right)\left(1+O(n^{-1/3})\right),$$

where $\alpha=\alpha(\lambda,\rho)$ and $\beta=\beta(\lambda,\rho)$ are constants depending on the ensemble. The preceding scaling lemma gives the leading Gaussian behavior, but the finite-length shift and its $O(n^{-1/3})$ refinement were not rigorously established in the source; the authors describe the remaining difficulty as technical rather than conceptual.