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

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.