Range of random $\mathbb Z$-homomorphisms on weak expanders

TL;DR

This paper demonstrates that random $bZ$-homomorphisms on weakly expanding bipartite graphs are highly flat, with their range typically bounded by a small logarithmic double-logarithmic factor, extending previous results on expanders.

Contribution

It extends the flatness phenomenon of $bZ$-homomorphisms from expanders to weak expanders, providing tight bounds and applications to the Hamming cube.

Findings

Range of random $bZ$-homomorphisms is at most $O(\log \log n)$ with high probability.

This flatness bound is tight up to a constant factor.

On the middle layers of the Hamming cube, the homomorphism takes at most 5 values with high probability.

Abstract

We prove that random $\mathbb{Z}$-homomorphisms on weakly expanding bipartite graphs exhibit a strong "flatness" phenomenon. Extending prior work of Peled, Samotij, and Yehudayoff for expanders, we first show that on any bipartite $(n, d, \lambda)$-graph with $\lambda \leq (1-\delta)d$, a uniformly chosen $\mathbb{Z}$-homomorphism has a range at most $O(\log \log n)$ with high probability, which is tight up to a constant factor. This provides an affirmative answer to their question in the spectral setting. As a concrete application, we prove that a random $\mathbb{Z}$-homomorphism on the middle layers of the Hamming cube takes at most $5$ values with high probability. This shows that the $O(1)$-flatness for the full Hamming cube, proved by Kahn and Galvin, persists even when the rigid structural properties are relaxed.