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[over ^](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 g>>1 and 0<μ<2, and p(time)(l)~l(-γ(μ)) for l>>1 with γ(1<μ≤2)=1+1/μ and γ(0<μ<1)=2. For l, g>>1 with fixed lg(-μ), p(g,l) takes the scaling form p(g,l)~g(-1-2μ)p[over ˜](μ)(lg(-μ)), where p[over ˜](μ)(y) is a (μ-dependent) scaling function. We also present numerical simulations which verify our analytic results.