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摘要翻译：
本文研究了马尔可夫一维随机步行者(RW)在n个时间步长后的两个最右位置之间的间隙G_n和将这两个极值位置的出现分开的持续时间L_n的统计量。RW的跳数Eta_i的分布是对称的，其Fourier变换具有小k行为1-\hat{f}(k)\sim k^\mu,0<\mu\leq2。我们计算了G_n和L_n的联合概率密度函数(pdf)P_n(g，l)，证明了当N_to-infty时，它接近一个极限pdf p(g，l)。对于g\gg1和0<\mu<2,g\gg1和0<\mu<2,对应的边缘pdf为p_{\rm gap}(g)\sim g^{-1-\mu}。对于l\gg1,当\gamma(1<\mu\leq2)=1+1/\mu,\gamma(0<\mu<1)=2时，L_n的极限边际分布p_{\rm时间}(l)有一个代数尾p_{\rm时间}(l)\sim L_{-\gamma(\mu)}。对于l,g\gG1,具有固定的l g^{-\mu},p(g，l)采用p(g，l)\sim g^{-1-2\mu}\tilde P_\mu（l g^{-\mu}）的标度形式，其中tilde P_\mu(y)是(\mu相关的）标度函数。我们也给出了数值模拟，验证了我们的分析结果。
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英文标题：
《Exact Statistics of the Gap and Time Interval Between the First Two
Maxima of Random Walks》
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作者：
Satya N. Majumdar, Philippe Mounaix, Gregory Schehr
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最新提交年份：
2013
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分类信息：
一级分类：Physics 物理学
二级分类：Statistical Mechanics 统计力学
分类描述：Phase transitions, thermodynamics, field theory, non-equilibrium phenomena, renormalization group and scaling, integrable models, turbulence
相变，热力学，场论，非平衡现象，重整化群和标度，可积模型，湍流
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一级分类：Physics 物理学
二级分类：Disordered Systems and Neural Networks 无序系统与神经网络
分类描述：Glasses and spin glasses; properties of random, aperiodic and quasiperiodic systems; transport in disordered media; localization; phenomena mediated by defects and disorder; neural networks
眼镜和旋转眼镜；随机、非周期和准周期系统的性质；无序介质中的传输；本地化；由缺陷和无序介导的现象；神经网络
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一级分类：Mathematics 数学
二级分类：Probability 概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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一级分类：Quantitative Finance 数量金融学
二级分类：Statistical Finance 统计金融
分类描述：Statistical, econometric and econophysics analyses with applications to financial markets and economic data
统计、计量经济学和经济物理学分析及其在金融市场和经济数据中的应用
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英文摘要：
We investigate the statistics of the gap, G_n, between the two rightmost positions of a Markovian one-dimensional random walker (RW) after n time steps and of the duration, L_n, which separates the occurrence of these two extremal positions. The distribution of the jumps \eta_i's of the RW, f(\eta), is symmetric and its Fourier transform has the small k behavior 1-\hat{f}(k)\sim| k|^\mu with 0 < \mu \leq 2. We compute the joint probability density function (pdf) P_n(g,l) of G_n and L_n and show that, when n \to \infty, it approaches a limiting pdf p(g,l). The corresponding marginal pdf of the gap, p_{\rm gap}(g), is found to behave like p_{\rm gap}(g) \sim g^{-1 - \mu} for g \gg 1 and 0<\mu < 2. We show that the limiting marginal distribution of L_n, p_{\rm time}(l), has an algebraic tail p_{\rm time}(l) \sim l^{-\gamma(\mu)} for l \gg 1 with \gamma(1<\mu \leq 2) = 1 + 1/\mu, and \gamma(0<\mu<1) = 2. For l, g \gg 1 with fixed l g^{-\mu}, p(g,l) takes the scaling form p(g,l) \sim g^{-1-2\mu} \tilde p_\mu(l g^{-\mu}) where \tilde p_\mu(y) is a (\mu-dependent) scaling function. We also present numerical simulations which verify our analytic results.
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PDF链接：
https://arxiv.org/pdf/1303.4607
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