We consider one-dimensional discrete-time random walks (RWs) of n steps, starting from x0=0\documentclass[12pt]{minimal}
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\begin{document}$$x_0=0$$\end{document}, with arbitrary symmetric and continuous jump distributions f(η)\documentclass[12pt]{minimal}
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\begin{document}$$f(\eta )$$\end{document}, including the important case of Lévy flights. We study the statistics of the gaps Δk,n\documentclass[12pt]{minimal}
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\begin{document}$$\varDelta _{k,n}$$\end{document} between the kth\documentclass[12pt]{minimal}
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\begin{document}$$k\text {th}$$\end{document} and (k+1)th\documentclass[12pt]{minimal}
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\begin{document}$$(k+1)\text {th}$$\end{document} maximum of the set of positions {x1,…,xn}\documentclass[12pt]{minimal}
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\begin{document}$$\{x_1,\ldots ,x_n\}$$\end{document}. We obtain an exact analytical expression for the probability distribution Pk,n(Δ)\documentclass[12pt]{minimal}
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\begin{document}$$P_{k,n}(\varDelta )$$\end{document} valid for any k and n, and jump distribution f(η)\documentclass[12pt]{minimal}
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\begin{document}$$f(\eta )$$\end{document}, which we then analyse in the large n limit. For jump distributions whose Fourier transform behaves, for small q, as f^(q)∼1-|q|μ\documentclass[12pt]{minimal}
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\begin{document}$${\hat{f}} (q) \sim 1 - |q|^\mu $$\end{document} with a Lévy index 0<μ≤2\documentclass[12pt]{minimal}
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\begin{document}$$0< \mu \le 2$$\end{document}, we find that the distribution becomes stationary in the limit of n→∞\documentclass[12pt]{minimal}
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\begin{document}$$n\rightarrow \infty $$\end{document}, i.e. limn→∞Pk,n(Δ)=Pk(Δ)\documentclass[12pt]{minimal}
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\begin{document}$$\lim _{n\rightarrow \infty } P_{k,n}(\varDelta )=P_k(\varDelta )$$\end{document}. We obtain an explicit expression for its first moment E[Δk]\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {E}[\varDelta _{k}]$$\end{document}, valid for any k and jump distribution f(η)\documentclass[12pt]{minimal}
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\begin{document}$$f(\eta )$$\end{document} with μ>1\documentclass[12pt]{minimal}
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\begin{document}$$\mu >1$$\end{document}, and show that it exhibits a universal algebraic decay E[Δk]∼k1/μ-1Γ1-1/μ/π\documentclass[12pt]{minimal}
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\begin{document}$$ \mathbb {E}[\varDelta _{k}]\sim k^{1/\mu -1} \varGamma \left( 1-1/\mu \right) /\pi $$\end{document} for large k. Furthermore, for μ>1\documentclass[12pt]{minimal}
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\begin{document}$$\mu >1$$\end{document}, we show that in the limit of k→∞\documentclass[12pt]{minimal}
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\begin{document}$$k\rightarrow \infty $$\end{document} the stationary distribution exhibits a universal scaling form Pk(Δ)∼k1-1/μPμ(k1-1/μΔ)\documentclass[12pt]{minimal}
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\begin{document}$$P_k(\varDelta ) \sim k^{1-1/\mu } \mathcal {P}_\mu (k^{1-1/\mu }\varDelta )$$\end{document} which depends only on the Lévy index μ\documentclass[12pt]{minimal}
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\begin{document}$$\mu $$\end{document}, but not on the details of the jump distribution. We compute explicitly the limiting scaling function Pμ(x)\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {P}_\mu (x)$$\end{document} in terms of Mittag–Leffler functions. For 1<μ<2\documentclass[12pt]{minimal}
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\begin{document}$$1< \mu <2$$\end{document}, we show that, while this scaling function captures the distribution of the typical gaps on the scale k1/μ-1\documentclass[12pt]{minimal}
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\begin{document}$$k^{1/\mu -1}$$\end{document}, the atypical large gaps are not described by this scaling function since they occur at a larger scale of order k1/μ\documentclass[12pt]{minimal}
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\begin{document}$$k^{1/\mu }$$\end{document}. This atypical part of the distribution is reminiscent of a “condensation bump” that one often encounters in several mass transport models.
