Hostname: page-component-8448b6f56d-cfpbc Total loading time: 0 Render date: 2024-04-19T04:24:43.188Z Has data issue: false hasContentIssue false
  • English
  • Français

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)
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Benjamini, I., Häggström, O. and Mossel, E. (2000) On random graph homomorphisms into $\mathbb{Z}$. J. Combin. Theory Ser. B 78 86–114.Google Scholar
[2]Benjamini, I. and Schechtman, G. (2000) Upper bounds on the height difference of the Gaussian random field and the range of random graph homomorphisms into $\mathbb{Z}$. Random Struct. Alg. 17 20–25.Google Scholar
[3]Benjamini, I., Yadin, A. and Yehudayoff, A. (2007) Random graph-homomorphisms and logarithmic degree. Electron. J. Probab. 12 926–950.Google Scholar
[4]Engbers, J. and Galvin, D. (2012) H-coloring tori. J. Combin. Theory Ser. B 102 1110–1133.Google Scholar
[5]Engbers, J. and Galvin, D. (2012) H-colouring bipartite graphs. J. Combin. Theory Ser. B 102 726–742.Google Scholar
[6]Friedman, J. (2008) A Proof of Alon's Second Eigenvalue Conjecture and Related Problems, Vol. 195 of Mem. Amer. Math. Soc., AMS.Google Scholar
[7]Galvin, D. (2003) On homomorphisms from the Hamming cube to $\mathbb{Z}$. Israel J. Math. 138 189–213.Google Scholar
[8]Hoory, S., Linial, N. and Wigderson, A. (2006) Expander graphs and their applications. Bull. Amer. Math. Soc. (NS) 43 439–561.Google Scholar
[9]Kahn, J. (2001) An entropy approach to the hard-core model on bipartite graphs. Combin. Probab. Comput. 10 219–237.CrossRefGoogle Scholar
[10]Kahn, J. (2001) Range of cube-indexed random walk. Israel J. Math. 124 189–201.CrossRefGoogle Scholar
[11]Peled, R. (2010) High-dimensional Lipschitz functions are typically flat. arXiv:1005.4636Google Scholar
[12]Peled, R., Samotij, W. and Yehudayoff, A. Grounded Lipschitz functions on trees are typically flat. Submitted.Google Scholar
[13]Peled, R., Samotij, W. and Yehudayoff, A.H-coloring expander graphs. In preparation.Google Scholar