  This work studies the typical behavior of random integer-valued Lipschitz
functions on expander graphs with sufficiently good expansion. We consider two
families of functions: M-Lipschitz functions (functions that change by at most
M along edges) and integer-homomorphisms (functions that change by exactly 1
along edges). We prove that such functions typically exhibit very small
fluctuations. For instance, we show that a uniformly chosen M-Lipschitz
function takes only M+1 values on most of the graph, with a double exponential
decay for the probability to take other values.
