Hidden symmetries and limit laws in the extreme order statistics of the Laplace random walk

TL;DR

This paper investigates the limit laws of extreme order statistics in symmetric Laplace walks, revealing hidden symmetries through branching and Bessel process representations, and connecting to Brownian motion properties.

Contribution

It introduces two novel descriptions of the limiting extreme order statistics, uncovering hidden symmetries and linking them to Brownian motion and Bessel processes.

Findings

Two different descriptions of the limiting extreme order statistics point process.

Identification of hidden symmetries in branching processes and Brownian motion.

Connection of the Bessel process of dimension 4 to Brownian motion path decompositions.

Abstract

This paper is concerned with the limit laws of the extreme order statistics derived from a symmetric Laplace walk. We provide two different descriptions of the point process of the limiting extreme order statistics: a branching representation and a squared Bessel representation. These complementary descriptions expose various hidden symmetries in branching processes and Brownian motion which lie behind some striking formulas found by Schehr and Majumdar (Phys. Rev. Lett., 108:040601). In particular, the Bessel process of dimension $4 = 2+2$ appears in the descriptions as a path decomposition of Brownian motion at a local minimum and the Ray-Knight description of Brownian local times near the minimum.