Grounded Lipschitz functions on trees are typically flat

Ron Peled; Wojciech Samotij; Amir Yehudayoff

doi:https://doi.org/10.1214/ECP.v18-2796

Grounded Lipschitz functions on trees are typically flat

Ron Peled, Wojciech Samotij, Amir Yehudayoff

Tel Aviv University, Technion - Israel Institute of Technology

Research output: Contribution to journal › Article › peer-review

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 v₀ of the tree. We prove that the probability that the value of a uniformly chosen random function at v₀ is more than M + t is doubly-exponentially small in t. We also show a similar bound for continuous (real-valued) grounded Lipschitz functions.

Original language	English
Journal	Electronic Communications in Probability
Volume	18
DOIs	https://doi.org/10.1214/ECP.v18-2796
State	Published - 2013

Keywords

Random Lipschitz functions
Rooted trees

All Science Journal Classification (ASJC) codes

Statistics and Probability
Statistics, Probability and Uncertainty

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Cite this

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title = "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 v0 of the tree. We prove that the probability that the value of a uniformly chosen random function at v0 is more than M + t is doubly-exponentially small in t. We also show a similar bound for continuous (real-valued) grounded Lipschitz functions.",

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doi = "https://doi.org/10.1214/ECP.v18-2796",

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AU - Peled, Ron

AU - Samotij, Wojciech

AU - Yehudayoff, Amir

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N2 - 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 v0 of the tree. We prove that the probability that the value of a uniformly chosen random function at v0 is more than M + t is doubly-exponentially small in t. We also show a similar bound for continuous (real-valued) grounded Lipschitz functions.

AB - 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 v0 of the tree. We prove that the probability that the value of a uniformly chosen random function at v0 is more than M + t is doubly-exponentially small in t. We also show a similar bound for continuous (real-valued) grounded Lipschitz functions.

KW - Random Lipschitz functions

KW - Rooted trees

UR - http://www.scopus.com/inward/record.url?scp=84880012379&partnerID=8YFLogxK

U2 - https://doi.org/10.1214/ECP.v18-2796

DO - https://doi.org/10.1214/ECP.v18-2796

M3 - مقالة

SN - 1083-589X

VL - 18

JO - Electronic Communications in Probability

JF - Electronic Communications in Probability

ER -

Grounded Lipschitz functions on trees are typically flat

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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