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.
- Random Lipschitz functions
- Rooted trees