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