论文标题
随机步行和Lévy航班的通用订单统计数据
Universal order statistics for random walks & Lévy flights
论文作者
论文摘要
我们认为一维离散时间随机步行（RWS）为$ n $ steps，从$ x_0 = 0 $开始，任意对称和连续的跳跃分布$ f（η）$，包括LévyFlights的重要情况。我们研究$ k^\ text {th} $和$（k+1）^\ text {th} $最大位置$ \ \ {x_1，\ ldots，x_n \} $之间的$ k^\ text {th} $和$（k+1）$之间的差距$δ_{k，n} $的统计数据。我们获得了概率分布$ p_ {k，n}（δ）$有效的概率分布的精确分析表达式，并为任何$ k $和$ n $有效，然后跳转分配$ f（η）$，然后我们以$ n $ limit进行分析。对于其傅立叶变换行为的跳跃分布，对于小$ q $，AS $ \ hat f（q）\ sim 1 - | q |^μ$带有lévy索引$ 0 <μ\ leq 2 $，我们发现，我们发现，该分布在$ n \ to \ infty $ n \ infty $ n \ fo.e.e.e.e. p_ {k，n}（δ）= p_k（δ）$。我们获得了第一刻的明确表达，$ \ mathbb {e} [δ_{k}] $，对任何$ k $有效，并以$μ> 1 $ $> 1 $有效，并表明它表现出一个通用代数衰减$ \ mathbb {e} e} [e} [e} [Δ_{k} f（k}] γ\左（1-1/μ\右）/π$用于大$ k $。此外，以$μ> 1 $的价格，我们表明，在$ k \ to \ infty $的限制中，固定分布表现出$ p_k（δ）\ sim k^{1-1/μ} \ Mathcal {p}_μ（k^{1-1-1/μ）$的$ lun的$ nounucy nounus的$ p_k（Δ）\ sim k^{1-1/μ} \ Mathcal {p}_μ（k^{1-1/μ）我们在Mittag-Leffler函数方面明确计算限制缩放函数$ \ MATHCAL {P}_μ（X）$。对于$ 1 <μ<2 $，我们表明，尽管此缩放函数捕获了尺度上典型间隙的分布$ k^{1/μ-1} $，但由于它们以较大的顺序出现$ k^{1/μ} $。
We consider one-dimensional discrete-time random walks (RWs) of $n$ steps, starting from $x_0=0$, with arbitrary symmetric and continuous jump distributions $f(η)$, including the important case of Lévy flights. We study the statistics of the gaps $Δ_{k,n}$ 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 $P_{k,n}(Δ)$ valid for any $k$ and $n$, and jump distribution $f(η)$, which we then analyse in the large $n$ limit. For jump distributions whose Fourier transform behaves, for small $q$, as $\hat f (q) \sim 1 - |q|^μ$ with a Lévy index $0< μ\leq 2$, we find that, the distribution becomes stationary in the limit of $n\to \infty$, i.e. $\lim_{n\to \infty} P_{k,n}(Δ)=P_k(Δ)$. We obtain an explicit expression for its first moment $\mathbb{E}[Δ_{k}]$, valid for any $k$ and jump distribution $f(η)$ with $μ>1$, and show that it exhibits a universal algebraic decay $ \mathbb{E}[Δ_{k}]\sim k^{1/μ-1} Γ\left(1-1/μ\right)/π$ for large $k$. Furthermore, for $μ>1$, we show that in the limit of $k\to\infty$ the stationary distribution exhibits a universal scaling form $P_k(Δ) \sim k^{1-1/μ} \mathcal{P}_μ(k^{1-1/μ}Δ)$ which depends only on the Lévy index $μ$, but not on the details of the jump distribution. We compute explicitly the limiting scaling function $\mathcal{P}_μ(x)$ in terms of Mittag-Leffler functions. For $1< μ<2$, we show that, while this scaling function captures the distribution of the typical gaps on the scale $k^{1/μ-1}$, the atypical large gaps are not described by this scaling function since they occur at a larger scale of order $k^{1/μ}$.
