Phase Transition in a Random Fragmentation Problem with Applications to Computer Science

TL;DR

This paper investigates a fragmentation process with a size cutoff, revealing a phase transition in the fluctuation behavior of splitting events, and applies the findings to analyze search algorithms in computer science.

Contribution

It introduces a phase transition in the fluctuation behavior of a fragmentation process and applies this to analyze search algorithms in computer science.

Findings

Fluctuations undergo a phase transition at a critical branching number m_c.

Below m_c, fluctuations are Gaussian; above m_c, they become large and non-Gaussian.

The results provide insights into the behavior of certain search algorithms.

Abstract

We study a fragmentation problem where an initial object of size x is broken into m random pieces provided x>x_0 where x_0 is an atomic cut-off. Subsequently the fragmentation process continues for each of those daughter pieces whose sizes are bigger than x_0. The process stops when all the fragments have sizes smaller than x_0. We show that the fluctuation of the total number of splitting events, characterized by the variance, generically undergoes a nontrivial phase transition as one tunes the branching number m through a critical value m=m_c. For m<m_c, the fluctuations are Gaussian where as for m>m_c they are anomalously large and non-Gaussian. We apply this general result to analyze two different search algorithms in computer science.