TY - JOUR

T1 - Grounded Lipschitz functions on trees are typically flat

AU - Peled, Ron

AU - Samotij, Wojciech

AU - Yehudayoff, Amir

PY - 2013

Y1 - 2013

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 - 10.1214/ECP.v18-2796

DO - 10.1214/ECP.v18-2796

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AN - SCOPUS:84880012379

VL - 18

JO - Electronic Communications in Probability

JF - Electronic Communications in Probability

SN - 1083-589X

ER -