## Electronic Communications in Probability

### Grounded Lipschitz functions on trees are typically flat

#### Abstract

A grounded $M$-Lipschitz function on a rooted $d$-ary tree is an integer valued map on the vertices that changes by at most $M$ along edges and attains the value zero on the leaves. We study the behavior of such functions, specifically, their typical value at the root $v_0$ of the tree. We prove that the probability that the value of a uniformly chosen random function at $v_0$ is more than $M+t$ is doubly-exponentially small in $t$. We also show a similar bound for continuous (real-valued) grounded Lipschitz functions.

#### Article information

Source
Electron. Commun. Probab., Volume 18 (2013), paper no. 55, 9 pp.

Dates
Accepted: 6 July 2013
First available in Project Euclid: 7 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465315594

Digital Object Identifier
doi:10.1214/ECP.v18-2796

Mathematical Reviews number (MathSciNet)
MR3078018

Zentralblatt MATH identifier
1298.05306

Rights

#### Citation

Peled, Ron; Samotij, Wojciech; Yehudayoff, Amir. Grounded Lipschitz functions on trees are typically flat. Electron. Commun. Probab. 18 (2013), paper no. 55, 9 pp. doi:10.1214/ECP.v18-2796. https://projecteuclid.org/euclid.ecp/1465315594

#### References

• Benjamini, Itai; HÄ‚Â¤ggstrÄ‚Å›m, Olle; Mossel, Elchanan. On random graph homomorphisms into ${\bf Z}$. J. Combin. Theory Ser. B 78 (2000), no. 1, 86–114.
• Benjamini, Itai; Yadin, Ariel; Yehudayoff, Amir. Random graph-homomorphisms and logarithmic degree. Electron. J. Probab. 12 (2007), no. 32, 926–950.
• Brascamp, Herm Jan; Lieb, Elliot H.; Lebowitz, Joel L. The statistical mechanics of anharmonic lattices. Proceedings of the 40th Session of the International Statistical Institute (Warsaw, 1975), Vol. 1. Invited papers. Bull. Inst. Internat. Statist. 46 (1975), no. 1, 393–404 (1976).
• Galvin, David. On homomorphisms from the Hamming cube to ${\bf Z}$. Israel J. Math. 138 (2003), 189–213.
• Kahn, J. Range of cube-indexed random walk. Israel J. Math. 124 (2001), 189–201.
• R. Peled, phHigh-dimensional Lipschitz functions are typically flat, ARXIV1005.4636v1.
• R. Peled, W. Samotij, and A. Yehudayoff, phH-coloring expander graphs, in preparation.
• bysame, phLipschitz functions on expanders are typically flat, to appear in Combin. Probab. Comput.
• Velenik, Yvan. Localization and delocalization of random interfaces. Probab. Surv. 3 (2006), 112–169.