# Gap statistics close to the quantile of a random walk

## TL;DR

This paper derives the exact distribution of gaps between ordered maxima in a large random walk with Laplace jumps, revealing universal fluctuation behavior and non-universal large deviations.

## Contribution

It provides an explicit universal distribution for typical gaps and analyzes their moments, extending understanding of order statistics in correlated stochastic processes.

## Key findings

- Universal alpha-dependent gap distribution with inverse cubic tail
- Typical fluctuations scale as O(n^{-1/2})
- Large deviations are non-universal

## Abstract

We consider a random walk of $n$ steps starting at $x_0=0$ with a double exponential (Laplace) jump distribution. We compute exactly the distribution $p_{k,n}(\Delta)$ of the gap $d_{k,n}$ between the $k^{\rm th}$ and $(k+1)^{\rm th}$ maxima in the limit of large $n$ and large $k$, with $\alpha=k/n$ fixed. We show that the typical fluctuations of the gaps, which are of order $O( n^{-1/2})$, are described by a universal $\alpha$-dependent distribution, which we compute explicitly. Interestingly, this distribution has an inverse cubic tail, which implies a non-trivial $n$-dependence of the moments of the gaps. We also argue, based on numerical simulations, that this distribution is universal, i.e. it holds for more general jump distributions (not only the Laplace distribution), which are continuous, symmetric with a well defined second moment. Finally, we also compute the large deviation form of the gap distribution $p_{\alpha n,n}(\Delta)$ for $\Delta=O(1)$, which turns out to be non-universal.