Persisting randomness in randomly growing discrete structures: graphs and search trees

TL;DR

This paper investigates how randomness persists in the evolution of random graphs and search trees, using potential theory to detect long-term dependencies and establish strong limit theorems.

Contribution

It introduces a novel application of discrete potential theory to analyze and quantify persistence of randomness in Markov chains generated by sequential algorithms.

Findings

Persistence of randomness can be rigorously detected and measured.

Strong limit theorems can be derived for these structures.

The approach extends understanding beyond distributional convergence.

Abstract

The successive discrete structures generated by a sequential algorithm from random input constitute a Markov chain that may exhibit long term dependence on its first few input values. Using examples from random graph theory and search algorithms we show how such persistence of randomness can be detected and quantified with techniques from discrete potential theory. We also show that this approach can be used to obtain strong limit theorems in cases where previously only distributional convergence was known.