  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.
