Lipschitz functions on weak expanders

Given a connected finite graph $G$, an integer-valued function $f$ on $V(G)$ is called $M$-Lipschitz if the value of $f$ changes by at most $M$ along the edges of $G$. In 2013, Peled, Samotij, and Yehudayoff showed that random $M$-Lipschitz functions on graphs with sufficiently good expansion typically exhibit small fluctuations, giving sharp bounds on the typical range of such functions, assuming $M$ is not too large. We prove that the same conclusion holds under a relaxed expansion condition and for larger $M$, (partially) answering questions of Peled et al. Our techniques involve a combination of Sapozhenko's graph container methods and entropy methods from information theory.