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