Atypical Decay Rates for Atypical Heights in Random Recursive Trees

TL;DR

This paper analyzes the large deviation probabilities for the height of random recursive trees, uncovering unique decay behaviors in the tails that differ from typical exponential decay.

Contribution

It establishes novel polynomial and stretched-exponential decay rates for the height distribution, highlighting an atypical prefactor in the lower tail.

Findings

Polynomial upper-tail decay for tree height

Stretched-exponential lower-tail decay with atypical prefactor

Prefactor grows slower than any iterated logarithm of n

Abstract

We establish the large deviation probabilities for the height of random recursive trees, revealing polynomial upper-tail decay and stretched-exponential lower-tail decay. Remarkably, the lower tail features an atypical prefactor that grows to infinity more slowly than any $n$-fold iterated logarithm.