Lipschitz Functions on Expanders are Typically Flat

Details Lipschitz Functions on Expanders are Typically Flat Lipschitz Functions on Expanders are Typically Flat

2012-03-18

arxiv.org/abs/1203.3930v1

Mathematics

Probability

26 pages

Scientific paper

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.

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