Moving-subcritical macroscopic-layer asymptotics

Determine, in the moving subcritical region satisfying $|r_M|/\sqrt{\log M}\to\infty$, the relative probability and fine conditional law of macroscopic-core failures for the ideal Plain IBLT peeling model.

Background

The paper proves a moderate-deviation result only while ∣rM∣|r_M| is bounded by a constant multiple of log⁡M\sqrt{\log M}. It explicitly identifies the more distant moving-subcritical region, where macroscopic failures become much rarer and repeated-edge failures dominate the overall failure-conditioned law, as beyond the scope of the current proposition. The unresolved task is to obtain both the relative probability of the macroscopic layer and its detailed conditional measurement law there.

References

The following questions are left for future work: the error rate at finite size; in the moving subcritical region with $|r_M|/\sqrt{\log M}\to\infty$, the relative probability and the fine conditional law of the macroscopic layer itself, a region beyond the scope of Proposition~\ref{prop:mda}, where the failure-conditioned law is dominated by the repeated-edge layer; the complete classification of the sparse end $d=O(\sqrt M)$; $k$ growing with the size; and non-ideal checksums and other mapping families.

— Decoding Failure Biases IBLT Count Measurements: A Gaussian-Selection Law at the Peeling Threshold  (2610.21418 - Wu, 15 Oct 2026) in Section Discussion and conclusion; Appendix A, Section \ref{app:mixture}