On the gap and time interval between the first two maxima of long continuous time random walks

Abstract

We consider a one-dimensional continuous time random walk (CTRW) on a fixed time interval T where at each time step the walker waits a random time τ , before performing a jump drawn from a symmetric continuous probability distribution function (PDF) f (η), of Lévy index 0 < µ ≤ 2.Our study includes the case where the waiting time PDF Ψ(τ ) has a power law tail, Ψ(τ ) ∝ τ −1−γ , with 0 < γ < 1, such that the average time between two consecutive jumps is infinite. The random motion is sub-diffusive if γ < µ/2 (and super-diffusive if γ > µ/2). We investigate the joint PDF of the gap g between the first two highest positions of the CTRW and the time t separating these two maxima. We show that this PDF reaches a stationary limiting joint distribution p(g, t) in the limit of long CTRW, T → ∞. Our exact analytical results show a very rich behavior of this joint PDF in the (γ, µ) plane, which we study in great detail. Our main results are verified by numerical simulations. This work provides a non trivial extension to CTRWs of the recent study in the discrete time setting by Majumdar et al. (J. Stat. Mech. P09013, 2014).

Extreme value statistics of correlated random variables: A pedagogical review

“…beyond Brownian motion and its variants) involving (i) either a single degree of freedom, like the so called continuous time random walks (CTRW) [52,103], the random acceleration process [34,50] and more general 1/f α noise [27,104,105] as well as fractional Brownian motion [106][107][108][109][110] (ii) or several degrees of freedom like N independent [111,112] as well as non-intersecting Brownian motions [41-44, 53, 58, 59, 61, 113-115] and branching Brownian motions [63,[116][117][118][119][120][121][122][123][124]. Extreme value questions for these processes have recently found many applications ranging from the number of common and distinct visited sites by N random walkers [112], distribution of the cover time by N Brownian motions on an interval of length L [144] and the distribution of the cover time of a single transient walker in higher dimensions [126], combinatorics [41,42,44], fluctuating interfaces in the Kardar-Parisi-Zhang universality class [42,53,58,61,62], disordered systems [116,117], genetics [118], propagation of epidemics [63] or in random planar geometry…”

“…( 26)]. Of particular interest in this context are the gaps between two successive maxima and for which a lot of results have been obtained during the last years [104,[138][139][140][141][142][143][144][145].…”