In the context of order statistics of discrete time random walks (RW), we investigate the statistics of the gap, G n , and the number of time steps, L n , between the two highest positions of a Markovian one-dimensional random walker, starting from x 0 = 0, after n time steps (taking the x-axis vertical).The jumps η i = x i − x i−1 are independent and identically distributed random variables drawn from a symmetric probability distribution function (PDF), f (η), the Fourier transform of which has the small k behavior 1 −f (k) ∝ |k| µ , with 0 < µ ≤ 2. For µ = 2, the variance of the jump distribution is finite and the RW (properly scaled) converges to a Brownian motion. For 0 < µ < 2, the RW is a Lévy flight of index µ. We show that the joint PDF of G n and L n converges to a well defined stationary bi-variate distribution p(g, l) as the RW duration n goes to infinity. We present a thorough analytical study of the limiting joint distribution p(g, l), as well as of its associated marginals p gap (g) and p time (l), revealing a rich variety of behaviors depending on the tail of f (η) (from slow decreasing algebraic tail to fast decreasing super-exponential tail). We also address the problem for a random bridge where the RW starts and ends at the origin after n time steps.We show that in the large n limit, the PDF of G n and L n converges to the same stationary distribution p(g, l) as in the case of the free-end RW. Finally, we present a numerical check of our analytical predictions. Some of these results were announced in a recent letter [S. N. Majumdar, Ph. Mounaix, G. Schehr, Phys. Rev. Lett. 111, 070601 (2013)].
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.
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