Tree/Endofunction Bijections and Concentration Inequalities

TL;DR

This paper introduces a bijection-based method to establish precise concentration inequalities in random trees, enabling probabilistic proofs of properties like unimodality of independent set sequences.

Contribution

It presents a novel bijection approach for deriving concentration inequalities in uniform random trees, bridging combinatorial and probabilistic techniques.

Findings

Proves concentration inequalities for vertices connected to independent sets in random trees.

Demonstrates partial unimodality of the independent set sequence in random trees.

Provides a probabilistic framework for inequalities traditionally proven combinatorially.

Abstract

We demonstrate a method for proving precise concentration inequalities in uniformly random trees on $n$ vertices, where $n\geq1$ is a fixed positive integer. The method uses a bijection between mappings $f\colon\{1,\ldots,n\}\to\{1,\ldots,n\}$ and doubly rooted trees on $n$ vertices. The main application is a concentration inequality for the number of vertices connected to an independent set in a uniformly random tree, which is then used to prove partial unimodality of its independent set sequence. So, we give probabilistic arguments for inequalities that often use combinatorial arguments.