Wednesday, March 20, 2013

1303.4607 (Satya N. Majumdar et al.)

Exact Statistics of the Gap and Time Interval Between the First Two
Maxima of Random Walks

Satya N. Majumdar, Philippe Mounaix, Gregory Schehr
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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