Typical-starting-vertex cutoff on the giant component of a supercritical Erdős–Rényi graph

Establish whether repeated averaging started from a typical vertex on the giant component of a supercritical Erdős–Rényi graph with fixed expected degree exhibits $L^1$ cutoff.

Background

The paper identifies the giant component of a supercritical Erdős–Rényi graph as a natural setting beyond bounded-degree expanders. Its local limit is a Poisson Galton–Watson tree conditioned to survive, and the associated random walk has an entropic cutoff time.

However, the giant component has unbounded degrees, dangling trees, and long degree-two paths, so it lacks the uniform spectral-gap structure used in the paper. The authors therefore leave unresolved whether repeated averaging, when initialized at a typical rather than worst-case vertex, still has an L1L^1 cutoff.

References

In light of these, we expect that repeated averaging, when starts from a typical vertex (rather than the worst initial case), should still have $L1$ cutoff, and we leave it to future exploration.

— Repeated averaging on expanders I: random $d$-regular graph  (2609.25729 - Yao et al., 22 Sep 2026) in Section 1, subsection 'Robustness of the method and subsequent works'