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- W1808636941 abstract "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^{-1-2mu} 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." @default.
- W1808636941 created "2016-06-24" @default.
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- W1808636941 date "2013-08-14" @default.
- W1808636941 modified "2023-10-16" @default.
- W1808636941 title "Exact Statistics of the Gap and Time Interval between the First Two Maxima of Random Walks and Lévy Flights" @default.
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- W1808636941 doi "https://doi.org/10.1103/physrevlett.111.070601" @default.
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