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Lipschitz Functions on Expanders are Typically Flat

Published online by Cambridge University Press:  11 June 2013

RON PELED ,
WOJCIECH SAMOTIJ  and
AMIR YEHUDAYOFF
RON PELED
Affiliation:
School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel (e-mail: peledron@post.tau.ac.il)
WOJCIECH SAMOTIJ
Affiliation:
School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel and Trinity College, Cambridge CB2 1TQ, UK (e-mail: ws299@cam.ac.uk)
AMIR YEHUDAYOFF
Affiliation:
Department of Mathematics, Technion–IIT, Haifa, Israel (e-mail: amir.yehudayoff@gmail.com)
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Abstract

This work studies the typical behaviour of random integer-valued Lipschitz functions on expander graphs with sufficiently good expansion. We consider two families of functions: M-Lipschitz functions (functions which change by at most M along edges) and integer-homomorphisms (functions which 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 of taking other values.

Keywords

Primary 60C0505C15Secondary 05C6082B41
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 22 , Issue 4 , July 2013 , pp. 566 - 591
Copyright
Copyright © Cambridge University Press 2013 

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