Exact Statistics of the Gap and Time Interval between the First Two Maxima of Random Walks and Levy 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 eta(i)'s of the RW, fd(eta), is symmetric and its Fourier transform has the small k behavior 1 - (f) over tilde (k) similar to vertical bar k vertical bar(mu), with 0 < mu <= 2. For mu = 2, the RW converges, for large n, to Brownian motion, while for 0 < mu < 2 it corresponds to a Levy flight of index mu. We compute the joint probability density function (PDF) P-n(g,l); of Gn and Ln and show that, when n -> infinity, 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) similar to g(-1-mu) for g >> 1 and 0 < mu < 2, and p(time)(l) similar to l(-gamma(mu)) for l >> 1 with gamma(1< mu <= 2) = 1 + 1/mu and (0< mu < 1) = 2. For l, g >> 1 with fixed lg(-mu), p(g, l) takes the scaling form p(g, l) similar to g(-1-2 mu) (p) over bar (mu)(y) is a (mu-dependent) scaling function. We also present numerical simulations which verify our analytic results.