On the Gap and Time Interval between the First Two Maxima of Long Random Walks

Satya N. Majumdar 1 Philippe Mounaix 2 Gregory Schehr 1

Journal of Statistical Mechanics, 2014, pp.P09013

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 $\eta_i = x_i - x_{i-1}$ are independent and identically distributed random variables drawn from a symmetric probability distribution function (PDF), $f(\eta)$, the Fourier transform of which has the small $k$ behavior $1 - \hat f(k) \propto |k|^\mu$, with $0 < \mu \leq 2$. For $\mu=2$, the variance of the jump distribution is finite and the RW (properly scaled) converges to a Brownian motion. For $0<\mu<2$, the RW is a L\'evy flight of index $\mu$. 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 $p_{\rm gap}(g)$ and $p_{\rm time}(l)$, revealing a rich variety of behaviors depending on the tail of $f(\eta)$ (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 {\it 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. {\bf 111}, 070601 (2013)].

  • 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
  • 2. CPHT - Centre de Physique Théorique [Palaiseau]