First Gap Statistics of Long Random Walks with Bounded Jumps - Archive ouverte HAL
Article Dans Une Revue Journal of Physics A: Mathematical and Theoretical Année : 2017

First Gap Statistics of Long Random Walks with Bounded Jumps

Résumé

We study one-dimensional discrete as well as continuous time random walks, either with a fixed number of steps (for discrete time) $n$ or on a fixed time interval $T$ (for continuous time). In both cases, we focus on symmetric probability distribution functions (PDF) of jumps with a finite support $[-g_{max}, g_{max}]$. For continuous time random walks (CTRWs), the waiting time $\tau$ between two consecutive jumps is a random variable whose probability distribution (PDF) has a power law tail $\Psi(\tau) \propto \tau^{-1-\gamma}$, with $0<\gamma<1$. We obtain exact results for the joint statistics of the gap between the first two maximal positions of the random walk and the time elapsed between them. We show that for large $n$ (or large time $T$ for CTRW), this joint PDF reaches a stationary joint distribution which exhibits an interesting concentration effect in the sense that a gap close to its maximum possible value, $g\approx g_{max}$, is much more likely to be achieved by two successive jumps rather than by a long walk between the first two maxima. Our numerical simulations confirm this concentration effect.
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Dates et versions

hal-01503740 , version 1 (16-12-2023)

Identifiants

Citer

Philippe Mounaix, Gregory Schehr. First Gap Statistics of Long Random Walks with Bounded Jumps. Journal of Physics A: Mathematical and Theoretical, 2017, 50 (18), pp.185001. ⟨10.1088/1751-8121/aa65f2⟩. ⟨hal-01503740⟩
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