Electronic Communications in Probability

Grounded Lipschitz functions on trees are typically flat

Ron Peled, Wojciech Samotij, and Amir Yehudayoff

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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

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

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

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Zentralblatt MATH identifier

Primary: 05C60: Isomorphism problems (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.)
Secondary: 60C05: Combinatorial probability 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]

Random Lipschitz functions rooted trees

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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

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