Exact statistics of the gap and time interval between the first two maxima of random walks and Lévy flights

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-f(k)∼|k|μ, with 0<μ≤2. For μ=2, the RW converges, for large n, to Brownian motion, while for 0<μ<2 it corresponds to a Lévy flight of index μ. We compute the joint probability density function (PDF) P_n(g,l) of G_n and L_n and show that, when n→∞, it approaches a limiting PDF p(g,l). The corresponding marginal PDFs of the gap, p_gap(g), and of L _n, p_time(l), are found to behave like p _gap(g)∼g^-1^- μ for gâ‰1 and 0<μ<2, and p_time(l)∼l^{- γ}μ for lâ‰1 with γ(1<μ≤2)=1+1/μ and γ(0<μ<1)=2. For l, gâ‰1 with fixed lg ^-μ, p(g,l) takes the scaling form p(g,l)∼g^-1 ^-2μp_Ëœμ(lg^-μ), where p _Ëœμ(y) is a (μ-dependent) scaling function. We also present numerical simulations which verify our analytic results.