We study analytically the order statistics of a time series generated by the positions of a symmetric random walk of $n$ steps with step lengths of finite variance ${\sigma }^2$. We show that the statistics of the gap $d_{k,n}=M_{k,n}-M_{k+1,n}$ between the $k$th and the $(k+1)$th maximum of the time series becomes stationary, i.e., independent of $n$ as $n\to \infty$ and exhibits a rich, universal behavior. The mean stationary gap exhibits a universal algebraic decay for large $k$, $⟨d_{k,\infty }⟩/\sigma \sim 1/\sqrt{2\pi k}$, independent of the details of the jump distribution. Moreover, the probability density (pdf) of the stationary gap exhibits scaling, $\mathrm{Pr}(d_{k,\infty }=\delta )\simeq (\sqrt{k}/\sigma )P(\delta \sqrt{k}/\sigma )$, in the regime $\delta \sim ⟨d_{k,\infty }⟩$. The scaling function $P(x)$ is universal and has an unexpected power law tail, $P(x)\sim x^{-4}$ for large $x$. For $\delta \gg ⟨d_{k,\infty }⟩$ the scaling breaks down and the pdf gets cut off in a nonuniversal way. Consequently, the moments of the gap exhibit an unusual multiscaling behavior.

• Received 15 November 2011

DOI:https://doi.org/10.1103/PhysRevLett.108.040601