New insights into the distribution of the topmost gap in random walks and L\'evy flights

TL;DR

This paper investigates the distribution of the gap between the top two positions in random walks and Lévy flights, providing new insights and more direct derivations of known results in this area.

Contribution

It introduces novel results on the statistics of the top gap in random walks and Lévy flights, building on the distribution of the first positive position.

Findings

Derived new results on the top gap distribution

Provided more direct derivations of existing results

Enhanced understanding of maximum position statistics in random walks

Abstract

Building upon the knowledge of the distribution of the first positive position reached by a random walker starting from the origin, one can derive new results on the statistics of the gap between the largest and second-largest positions of the walk, and recover known ones in a more direct manner.