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arXiv · 2604.03119

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

Dingding Dong·Jinyoung Park
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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.

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Dingding Dong, Jinyoung Park. 2026-04-03. Range of random $\mathbb Z$-homomorphisms on weak expanders. https://arxiv.org/abs/2604.03119

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