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

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 η_i's of the RW, f(η), is symmetric and its Fourier transform has the small k behavior 1-\hat{f}(k)\sim| k|^μwith 0 < μ\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 - μ} for g \gg 1 and 0<μ< 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^{-γ(μ)} for l \gg 1 with γ(1<μ\leq 2) = 1 + 1/μ, and γ(0<μ<1) = 2. For l, g \gg 1 with fixed l g^{-μ}, p(g,l) takes the scaling form p(g,l) \sim g^{-1-2μ} \tilde p_μ(l g^{-μ}) where \tilde p_μ(y) is a (μ-dependent) scaling function. We also present numerical simulations which verify our analytic results.