Extremal Properties of Random Systems

TL;DR

This paper uncovers universal extremal properties of the probability distribution for the largest intervals in various random systems, revealing singularities at specific points and their persistence across dimensions.

Contribution

It demonstrates that the distribution p(l) exhibits universal singularities in diverse random models, including higher-dimensional cases, highlighting fundamental extremal behaviors.

Findings

p(l) has singularities at l=1/k for k=2,3,...

p(l) exhibits an essential singularity at l=0

Universal properties observed across different systems and dimensions

Abstract

We find that the probability distribution for the largest intervals $p(l)$ exhibits universal properties for different systems including random walk and random cutting models. In particular, $p(l)$ has an infinite set of singularities at $l=1/k$ with $k=2,3,\ldots$ which become weaker and weaker as $k \to \infty$; additionally, $p(l)$ has an essential singularity at $l=0$. These properties are found also in many dimensional situation.